Input: Who is the program chair of this conference?

Context:
HOFFMAN: I'm delighted to introduce the chair of the last session, Mara Liasson from the National Public Radio. Mara is Congressional correspondent for NPR, and covers activities in Congress in D.C. Right now, this week, she has been covering the tax bill, which people currently are going at hot and heavy. She took time off from her busy schedule to come here to help us sort out some of these key issues for today, and more importantly, for what happens in the next decade and beyond. I'll turn it over to Mara to get the panel going.
LIASSON: Thank you very much. I am probably the only person here who has absolutely no background in technology. Anyway, I am the only one who does not understand what the panelists are going to be talking about (laughter), and although they have already told me that they do not appreciate people who think that that's a great quality and look down on people who are technical, and I certainly do not, I will reserve the right to insist that they all talk in terms that people like me can understand, since there is more of me out there than you, although not in this room today. (laughter) What we are going to do is introduce each panelist, and each one will make a short three- to five-minute presentation. Then my instructions say that we are going to have a McLaughlin Group discussion, which I guess means lots of yelling and screaming and talking at once. (laughter) After that's over, about 4:10, we'll open up the panel for questions from the audience.
To my left is Susan Blackmore, who is a renowned psychologist and also a lecturer in consciousness studies. Susan Blackmore is the program chair of this conference, has also served as a prominent researcher in the field of memetics, and she is currently a frequent contributor to various scientific publications.
Simon Davies, to my right, also wears blue suits, but you can tell him from Mitch, because he wears a white hat. (laughter) He is from Sydney, Australia, and is the Director General of Privacy International, which is an international network of privacy advocates. He is also an author, a journalist, and radio commentator.
To his right is Roland Homet. He is an information policy writer and thinker who recently opened his own public policy writing firm here in Washington -- it's called Executive Ink, not Inc., as it is written in your programs, so you can scratch that out.
Esther Dyson, at the end of the panel, is among the most respected commentators on developing technology trends in the personal computer business. She publishes two newsletters, Release 1.0 and Rel-EAST. She has also been one of the driving forces promoting East-West relations through computer networks. She is a board member of the Electronic Frontier Foundation as well.
I'll ask Peter to start.
P. DENNING: Thank you. Starting around 1850, people of many countries looked to their governments to regulate commerce, erase inequities, and build societies of better human beings. For over a hundred years, many people, from peasants to intellectuals, had faith that strong governments would bring them a better life. This faith was part of the clearing in which Communist governments flourished; although the United States took an anti-Communist stand, the same faith fostered a strong government that promised salvation by great national programs including Social Security, welfare, food stamps, the War on Poverty, and the Great Society. This faith is now shattered. People no longer trust that powerful government can deliver a better life.
The dramatic collapse of Communism in Eastern Europe and the Soviet Union illustrates this, as does the growing disillusionment of the American people for federal, state, and local governments. The poor track record of government is not the only reason for the shift. Information technology has accelerated the process. Communications that took weeks in the last century now take fractions of a second. Business success depends on what happens around the globe, not only on local conditions. Radio, TV, fax, and now E-mail are common worldwide, so much so that not even a powerful government can control what information its citizens have. Because the space of opportunity for people to engage in transactions with each other has been so enormously enlarged during the past decade, faith in marketplace democracies is on the rise worldwide; correspondingly faith in central management mechanisms is on the decline. This shift has brought with it a shift of the power of institutions. Government institutions tend to try to hold onto their power by regulatory coercion to enforce the old ways. This can produce big tensions and even promote breakage.
Nowhere can this be seen more clearly than in the cryptographic area which we have just been talking about in the previous hour. This technology, cryptography, produces mechanisms for digital signatures, authentication, electronic money, certificates, and private communication -- all offering a way for standard business practices now based on paper to be shifted into the electronic media. The success of worldwide enterprises depends on this shift being completed rapidly and effectively. As more people realize this, the momentum for incorporating cryptographic technology into the information infrastructure is accelerating.
In this country, the National Security Agency has long been given the authority to regulate cryptography. This authority was granted in another time when the success of the country depended upon the ability of its government to gather intelligence and communicate in secret. These premises made sense in a world where most of the power resided in governments, but the world is changing. Much economic power is now accumulating in large apolitical transnational corporations. These corporations place their own concerns and strategies ahead of those of governments of the countries in which they do business. Like governments, they are interested in gathering intelligence about competitors and in conducting business in private. Unlike governments, they want open access to the technologies of authentication, electronic money, digital signatures, and certificates that will allow them to conduct business transactions across the network. So it is no longer true that national power and national security are increased when government has the sole right to gather intelligence and encipher communications. Now the strength of a country depends not only on its government, but also on its corporations. The old premises have fallen away in the new reality, but the old policy remains. It's time to rethink the policy, before tensions between a threatened government and corporations produce significant social tension and perhaps breakage.
Well, digital media -- computer-based communications -- are the printing press of the 21st century, and as the printing press transformed society, created the modern individual, gave rise to the basis of the democratic state and to the notion of individual rights, I suspect that we will see a similar, radical transformation of the very constitution of global society in the next century, facilitated by this enabling technology. I would be the last person to try to sketch out the details, or tell you what the issues are going to be, but I want to share with you some feelings about what is really going to matter, as we go about this -- and I'll start with something about myself.
You see a guy wearing a suit; most of you know I have a lot of money -- I'm a successful businessman. God knows what images propagate around the media and settle in people's minds, but I've always seen myself, and felt myself to the core of my being, as an outsider, every bit as much as a self-proclaimed outsider, as Tom Jennings -- who spoke so eloquently about this at the Pioneer awards* yesterday -- was. *The Electronic Freedom Foundation presented its first awards at a related, adjacent reception which was not formally a part of the conference.
I think we are all outsiders; we are all different, all unique. We're not the same. We share an underlying common humanity, but we should not be asked to subjugate ourselves to some form of mass society that causes us each to become indistinguishable from one another. I believe that computer- based communications technology is an enabling technology to liberate individuals and to free us from the oppressive influence of large institutions, whether those are public or private. And I am talking about an economic restructuring that results in a much more decentralized society, and social restructuring in an affirmation of the simple right to be left alone. I think Cyberspace is good for individuals, and I think that's important. I also think that the flip side of the coin, the creation of community, which we so sorely lack in this country today, can be facilitated through these technologies.
I have experienced that for myself, as many of you have on your various computer networks on conferencing systems like the WELL. It is enormously liberating to overcome the artificial boundaries of space and time. We are prisoners of geography in the physical world, and our communities are largely a product of who we can see face to face each day, even though our real comrades and colleagues may be scattered all over the world and our interests -- whether they are hobbies or political interests or religious interests, whatever they might be -- can be facilitated if we are able to get in touch with, to form bonds with, to exchange views and ideas with other kindred spirits. And I believe this technology is an enabling technology for the formation of community. My hope is that we will have the wisdom to create policies which enable individuals to flourish free from the chains of mass society, and which enable voluntary communities of people, individuals, groups who come together to be with each other and to work together. I hope both of those become possible.
DAVIES: I feel very warmed by the various visions of the future that have come out of this conference, but I am a cynic, and cynicism is good, because it adds fiber. (laughter) How nice the world would be if everyone was like Mitch, but they're not, because the future is in the hands of ruthless, greedy little men.
I want to paint the vision of the future that I have, and I hope it's not too depressing because there is a future, a good future... possibly. I agree, as many of you do, that the future is going to be like some giant informational Yggdrasil* *Reference from Old Norse mythology -- the Yggdrasil was a giant ash tree whose roots held together the universe.. We'll all be part of interconnectivity, the likes of which we can scarcely imagine right now. I imagine it will be like an organism where we're independent and interdependent, and so it's like a two-edged sword. That's all very nice, and we can see that we form part of that new community. But, I see a world with 15 billion beings scrambling for life, where four-fifths of the world lives on half a liter of water a day, where people grow up to see their children dying, where new political frontiers are destroying freedoms and the democracy that we have developed over the last two centuries. I see a world where there is very little hope for nearly everybody on the planet, except for the elite -- that's us -- except for those of us who are plugged into the informational Yggdrasil.
What I see is that 14 of those 15 billion people are a lot of pissed-off people who have their eyes set on what they see, not as a wonderful informational community, but as the beast. And they see that that is where the resources are, and that's where the opportunities are, and that's where the political power is. I can't see a future for us in a world where ultimately the great demon becomes information. It might be good for us, but for the disaffected four-fifths of the world, information is going to be something which, frankly, we can do without, because in a world with almost no resources left, surely information is selfishness.
HOMET: Thank you. I'm grateful to the organizers for including me in these proceedings -- they are reminiscent for me of some information policy conferences that I organized 15 to 20 years ago for the Aspen Institute. The particulars have certainly changed, but the dynamics remain much the same. For me, these are well-represented by Peter Denning's image of a changeable clearing in the woods. At any given time, as I see it, the clearing is an acceptable standoff between the forces of modernization and of traditional culture, between freedom and discipline, between structure and spontaneity. Now we voice these as opposites, but in fact, they need each other. It is the creative tension between technological innovation and established order that allows society to hold together and progress to take place. Take away freedom and order will be overthrown -- witness the Soviet Union. Take away tradition, and modernization will be crushed -- witness Iran. The clearing must be respected and it must move. Just as Benjamin Cardozo of the U.S. Supreme Court said 65 years ago, the genius of the American system is its penchant for ordered liberty. When both halves of the equation work against each other and together in Hegelian terms, the clearing that they produce is, at any given time, a prevailing hypothesis, which is challenged by a new antithesis. Together they can produce a fresh synthesis. And all that is very familiar. What is new and trying is the sweep and pace of innovation today, plus -- and this is what we sometimes forget -- the political volatility of the value systems that this can induce. If you doubt that, consider the Buchanan campaign and what's been going on with the Endowment for the Arts and public broadcasting. These are signs of people running scared, and they can cause damage.
So the answer for the 21st century is to proceed under power, but with restraint, to practice what Mitch Kapor in another connection called toleration for opposing forces and perspectives. We need each other to keep the enterprise together and on course. For computer practitioners represented in this room, this means restraint from provoking unnecessary and damaging social backlash. A good example might be New York telcos offering free per-call and per-line blocking with this caller identification service. For regulators and law enforcers, restraint means asking, "Do you know enough to freeze emerging conduct in a particular form or pattern?" I was very taken by the role reversal exercise organized by Michael Gibbons on Wednesday night. It led me to wonder what might have happened to the government's wiretapping and encryption proposals had they been subjected to a comparable advanced exercise before introduction.
Sixteen years ago in Aspen, Colorado, I convened a gathering of federal policymakers and invited them to consider a suggested matrix of policy values and processes in the information society. The first two of those values -- it will not surprise you to know -- were freedom of discourse and individual privacy. But there were more: freedom of economic choice is one; the general welfare another; popular sovereignty, worth pausing on, I described as avoiding concentrations of economic and political power in any sector of industry or government that impinge unduly on the freedoms or welfare of the citizenry. And then there is progress, social progress, the fostering, I said, of market incentives and opportunities for technological and service innovations and for widened consumer choice among technologies and services. Now obviously if you give just a moment's thought to it, you will recognize, as I think we have in this conference, that these values can collide with each other at key points, and therefore accommodations must be made. For that we need processes of accommodation. I also suggested some of those. After you identify the relevant values and goals, you then should ask yourself about the necessity and the appropriateness of having government make any decision on the matter. And this has to do with such things like the adequacy of decision-making standards, the availability of adequate information, and the adequacy of personnel resources to deal with it. Then you get into dividing up the possible roles of the various elements of government -- the regulatory agencies, the Executive Branch, the Judiciary, and the Congress. It doesn't stop there, because you need to ask about international implications, which we have done some of here. And federal/state implications -- very often allowing the state to make a stab at social ordering in the first instance is, as Justice Brandeis often said, the best way, through the social laboratory technique, to try out what is the right answer, without endangering the whole society. And as we have heard today, we need also to think about the availability of non-coercive instruments of accommodation, like a federal data protection board.
DYSON: I want to just say one thing about this business of crypto technology -- it is a very simple sentence, and everyone seems to slip slightly by it; that is, if you outlaw guns, only outlaws will have guns. Crypto technology is fundamentally a defensive weapon. It may protect murderers and thieves, but it is not a weapon that murders, kills, does anything bad; and so it is a very different kettle of fish from any other kind of weapon. The whole point is that information is powerful, and that the free flow of information, privacy-protected, empowers the powerless and is dangerous to the powerful -- and that's why we need our privacy protected.
Now let me just talk a wee bit about the future. A couple of days ago, a reporter called me and asked what the EFF stood for. I kind of floundered around and said, "Well, we want privacy, we want good hackers to be protected and bad crackers to be punished. We want people to understand the difference, and we want all these good things, but we really don't want to grab power." The guy kept on not quite getting it. The real answers were pro choice. We don't want someone else to make all these decisions for anybody. We don't even want the majority to rule. In every way that is possible, we want the minorities to control their own conditions in their own lives. There are very few things that are the province of government, but way too many things nowadays are being given to the government carelessly, fearfully, whatever. In my terms -- and I happen to be a right-wing person in terms of the economy and private freedoms -- I want more markets and fewer governments. Markets give choices to individuals. They let people trade what they don't want for what they do want. Again, to the extent possible, they want people to make individual choices.
What worries me is large concentrations of power, making choices for people. Big business, big government, even big media. The media until now have mostly been our protectors, because they go out and produce information, they use anonymous sources where necessary, and they make that information free. What protected global networking is going to do is give more and more of that power to individuals, and help reduce the power of big institutions of any kind. We are going to have small businesses flourishing, because it is easier for them to collect resources. You don't need to have a giant monolithic corporation to be efficient any more, and so a lot of marketplace economies of scale will even disappear, as we have better networking, better coordination. We have markets like the American Information Exchange, and if you don't know what that is, come and see me, or Hugh Daniel, or a couple of other people.
On the social side, I think 20 years ago... when you mentioned 15 years ago, I thought, Yes, that must have been about 1940. Then I realized... Anyway, some time ago there was all this talk about the global village. We're going to have mass broadcasting, we're going to have mass E-mail, we're going to have this global village. We don't. What we have is a lot of global villages, but as Mitch said, they're no longer geographical, physical villages. They're small, geographical villages of people with like interests. The big question becomes, How do we avert tribalism? It might not be nation against nation any more, but it certainly will be rich against poor, and franchised versus disenfranchised.
LIASSON: Thank you all very much. Now we can all try to stir up the pot a little bit. Somewhere between Mitch's paradise and the Simon's apocalypse is probably what's really going to happen. I want to just jump off from what Esther said about you all being in a minority and what kind of responsibility you owe to the rest of the world. We're in the midst of a presidential election and not one single candidate has said anything about Cyberspace. I am wondering if you think they should, and what are the kinds of extremely important issues that you think should be discussed? Should they be discussed in a kind of mass, political forum? Or should they be left to an elite like you to discuss and decide, and not really spend a whole lot of energy trying to translate or disseminate them to the great masses of people? I guess what I am wondering is, if you were an advisor to one of the presidential candidates, or a candidate yourself, how would you go about interjecting these things? Or wouldn't you bother at all?
DYSON: Does he want to get elected, or does he want to make a point?
LIASSON: I think he wants to make a point. If he wants to get elected, I think the discussion would stop right now.
DYSON: Let me just try a serious answer. I think what a candidate could say is, "I'm no longer going to protect the textile industry, the peanut butter interests, the sugar guys, the antediluvian steel mills. If I'm going to have an industrial policy and help anyone, it's going to be new technology. I'm going to focus on investment in R&D. I am going to create a national infrastructure for telecommunications, just the way we created a highway system years ago. I'm going to put people to work doing these things." I think that would go over reasonably well. I think it's something most of us would agree on. (laughter) We have an industrial policy -- we might as well acknowledge it, and we might as well have it be forward-looking.
KAPOR: Now there is something about the question as to whether this is presidential material that I think is ironic, given that most people really want to vote for "none of the above." We know in our hearts that we have come to a particular period in history in which the presidential spectacle seems to be particularly irrelevant to whatever set of problems we have on our minds. As a great believer in democracy, I think this is incredibly lamentable. We need to do something about this, because there are a lot of issues, but Cyberspace is not ready for prime time. It would be trivialized -- I have seen what Geraldo did to hackers, and I don't need to see any more.
It seems to me that the presidential candidates are really not the leaders that they ought to be, but are always putting their finger to the wind to see if they can detect some current of values or beliefs that can help get them elected. And I think that -- I'm not espousing utopian vision -- there needs to be an utopian vision out there, so people have something to give them some inspiration. But values are a lot more important than technology. There are some values in this community -- and I'm not sure if it's an elite or a minority or both -- but it's really in the propagation of a sense of values about openness and tolerance, acting on that basis and living one's life, and saving capitalism from itself and things like that where we can make a difference. If some of the expressions are technological, that's fine. We are living in an era where people like buttons, and so on. If we do that well, the presidential candidates are going to be coming to us.
LIASSON: You talk about Cyberspace not being ready for prime time -- I still want a definition of Cyberspace in 25 words or less -- but I think you want to transform prime time to a certain extent.
DYSON: Mostly I agree with this, but the press does have two roles: one is collecting information and uncovering things, and the other is setting the agenda. If 12,000 voices are crying out, who's going to listen to them? Who's going to notice when they do discover that the President did something wrong? Again, it's a check and balance sort of thing, but there is a certain community that is created by collective media.
KAPOR: Esther, what makes you believe that in Cyberspace Mara won't have two hours a day of her own that everyone listens to. (laughter) She might get more time than she gets today, because people trust her.
DYSON: But then she becomes prime time.
LIASSON: But you said before that instead of one global village, we have a lot of little global villages. I'm wondering if instead, we won't have millions of little huts. I mean individual huts. There are just so many different choices.
LIASSON: What I'm wondering is, if everybody becomes their own producer, publisher, what does that mean for the future?
KAPOR: I think we'll get a much more fluid, self-organizing state. I don't think in practice everybody is going to be what we think of today as a broadcast publisher. I just want things to be able to sort themselves out in a much more equitable fashion. We have this enormous, artificial scarcity today over the means of communication, because the government awards licenses which self-perpetuate. They are about to do the same thing, and give every broadcast television station another license for HDTV. So if you've got a license today, you get a second one; if you don't have one, you get nothing. That is going to be our policy about HDTV. I think it would be a lot better if we had more markets, more choices, and better values. I don't know how to do better values, but we know how to do more choices. So the point is, we'll wind up with some new regime which I don't think that we can particularly predict. I don't think that it is going to be chaotic or anarchic. I think there is something about people as social animals or creatures -- we will create some new forms of social organization. There will be information middlemen; there will be the equivalent of editors and packagers. There will be trusted intermediaries who help organize these new media. If you open it up and equalize things so that everybody can participate, you will get more diversity of points of view, you will get less homogenization. One of the reasons that tons of people have just dropped out, or are in terminal couch-potato-dom is that the sets of choices and the values that come across the tube are not ones that stir the human heart. And people know that. They can't figure out what to do about that, so they sort of fuzz out on drugs and alcohol. I say let's edit TV, which is the electronic drug. Let's do something about that.
DAVIES: I like your idea, Mitch. I think it's sweet. (laughter) The problem is that I really worry that the ultimate test of the future is going to be the outcome of the quest, the battle between those who are looking for the sort of vision you've got of the right of the individual, the individual being the producer. And that, probably, is the way we solve our problems on this planet. But there is the other side, and that's the planetary managers. Planetary management is the path of the least resistance. You know all the powermongers go for the planetary management model, because they all think they can clamber over the bodies to get to the top. Ultimately the test is going to be who comes out on the top, the individual rightist or the planetary managers. Unfortunately, I'm not a betting man, but at the moment I'd like to bet on the planetary managers.
DYSON: Part of this issue is reducing the value of incumbency, whether it's incumbency in prime time live, or incumbency in the government. There is much more fluidity of movement; you can't accumulate power because the unorganized forces have more power than you do.
P. DENNING: I feel a little strange being on the left end of the stage, because most people think of me as being on the far right sometimes, but right now I'd like to comment on something that is halfway between what Mitch is saying, and what Simon is saying. The way I hear what Simon is saying, is that there is a disease of today which I will call inward- centeredness. We are very worried about ourselves and our organizations. We find in that orientation a lot of instability of things and technologies that change rapidly. In order to achieve the world that Mitch is talking about, we need to cure the disease, and instead come from an orientation that we could call outward-centeredness, instead of inward-centeredness. The question is the shift from, How do we accumulate power? to, How do we help others accumulate power? How do we go from looking for stability in things to looking for stability in relationships? In watching my own children grow up, I am convinced that they know more about this than I do. In listening to some of the younger people here, I'm more convinced that they know more about this than I do. They know something about the outward-centeredness that I have yet to learn. Observing this among children and among students gives me a lot of optimism, as a matter of fact, against the apocalypse that Simon talks about, because Simon is talking about the world that would be created if we continued "us," and I think that the world that is being created by our children with their outward-centeredness is going to be the kind of world that Mitch is pointing towards. And I am much more optimistic about that than Simon is.
LIASSON: Roland, I wonder if we can interject you into this discussion a little bit. You have been a policymaker. What can be done to make sure that Simon's vision doesn't come true, and something a little closer to what Esther and Mitch describe does happen?
HOMET: I think we probably need both doom seers and paradise seekers. We'll always have them, and we should have them. It's between the swing of those two views that things happen. I think that this notion of replacing the gatekeepers and letting everybody perform his own dance, to the amusement of those who chose to tune in, is one that many of us were promoting 20 years ago. That's not 1940 -- that's 1970 (laughter), and we were quite convinced that was likely to happen by the end of that decade. Now it's 12 years beyond the end of that decade, and we're nowhere near having that happening. We just have newly-named controversies, and so, as you heard me say in my little short remark, I think that our objective ought to be more modest, and that is to keep the questions open, not let them be foreclosed -- certainly not prematurely, and not on the basis of inadequate evidence. I would say something about the apocalyptic view, which is, I think there is a difference between information policy questions and welfare questions. The poor we have always with us, as somebody once said, and whether information, Cyberspace -- whatever you want to call it -- is promoted or not, that is true. It may become more glaringly true in an advanced information society, in which case, more may be done about it. So I wouldn't despair about that, and I wouldn't hold back on the development of instruments of interconnection simply because we can see that there is and will remain an underclass. Perhaps if we do the one, we'll be better equipped to do the other.
LIASSON: In just a minute or two, we're going to open this up to your questions, but I want to try to end maybe with a discussion of something quite specific, which is, Who should own the new infrastructure and information systems? Should they be publicly owned? There are lots of conflicts even within the vision that you lay out.
KAPOR: The first point I'd make is let's not make the unnecessary mistake of betting on a single infrastructure. Technologically, we don't need to do that. In the 1930s, pre-digital, the old Bell system was the social contract. You get a monopoly, you have an obligation to provide universal service. We've learned a few things about how to do things with interoperable standards and how to interconnect multiple, independent providers and carriers. One of the fathers of the Internet, Vint Cerf, is sitting here in the front row, and he deserves an enormous amount of credit for insisting on this vision and promulgating it. A lot of the risks that come with private ownership of infrastructure go away when it's no longer a monopoly. The abusive problems that are sometimes experienced with local phone service and cable companies -- both of which are private sector monopolies -- I would say come more from not their private sector character, but from their monopoly character. If it is possible for there to be competition, that serves as the most effective check that we know of in this society against abuse. So I would opt for private infrastructure, but lots of it. Government has to make sure that everybody stays interconnected -- it's the referee that keeps the playing field level, doesn't let people cheat, and sort of bangs a few heads together when people get a little too greedy, or a little too selfish. If we do that, that will provide for the most choice and the most diversity.
LIASSON: Are we all in agreement on that?
HOMET: Not entirely. I think the question is less who should own infrastructure than how it should be classified. There may be a role for government in, for example, extending communication pipes to rural America for at least a period, as with the TVA. We have always had that question. There has always been a mixed economy with government doing some things and private sector others. It's a debate and should be a debate about who does what best. It should be revised from time to time, but the important question is, If we get a significant distribution system like cable television, how should we classify it? I speak here from the heart, because 20 years ago, I was trying to fasten onto, or gain the recognition for, cable as a broadband distribution system which was only trivially in the program production and publishing business, but was very much in the distribution business and ought to have been treated as a common carrier open to all information suppliers. Had that happened, we would have been very much further along in the vision that some of us had 20 years ago. (applause) It tends to support what I said about not going in for premature freezing or characterization of how things look. It was decided, because the broadcasters felt threatened, to treat cable as a species of broadcasting. That's the greatest frittering away of resources in my lifetime, and perhaps in the lifetime of the United States of America. Let's not make that mistake again. Let's be clear-eyed and ask the broad-scale questions about public use and benefit. Thank you.
LIASSON: Let's open it up to the audience. If you have any questions ... oh my God, wrestle your way to the microphone!
AUDIENCE MEMBER: Let us not forget the history of the commons in which a wealthy society creates in its overflowing abundance structures on which all people can participate. This was originally, back in medieval society, the structure that was created for the support of the poor. In the abundance of the land in which the overpopulation was not a question, and there was much agriculture to go around, and the poor were supported out of the commonly-owned things that were jointly owned by all society. That's all I have to say.
LIASSON: Who wants to start?
DAVIES: Sticking to my apocalyptic vision just for the moment, because that's how I'm characterized, what I would like to see, just as my own social experiment, if you like, is for the various groups that this room represents and groups that you are all involved in, is to actually set up the apocalyptic vision, and then see how you as part of the information technology community can utilize it, stop it, or reverse it. It's only when you see the vision and see your own part in it that we are actually going to set up solutions. I mean, that is a straight, outright homework assignment, and I think would be a great benefit for everybody. Then go on and publish them through the E-mail, or the Internet, whatever.
DYSON: Something along the lines of go find the most influential person you know well enough to influence, who you do not agree with -- assuming that you all agree with me, of course -- and attempt to win that person over to your point of view. In other words, don't stick to your own community. Don't just talk to the people who only agree with you. Go out and evangelize or proselytize to people who don't understand what this stuff is about. Do it in such a way that you are not superior or offputting; don't try to be right; try to win and expand this community, not in terms of pressure or rightness, but in terms of understanding what we are about. The biggest problem is ganging up on some of these politicians and having them think that this stuff is not cute, or weird, or colorful, or irrelevant, but incredibly important. Make the rest of the world know about us.
HOMET: I would like to second that motion. The story is told that when a beautiful woman comes out on a street in Paris, every man within eyeshot becomes in that instant much more intensively himself. (laughter) What I would suggest to you, if you are energized by this subject, is to be yourself. To thine own self be true, and perhaps to add to that the biblical admonition to the apostles -- if I remember it correctly -- and this picks up what Esther was saying -- to be wise as snakes, and cunning as foxes. Go out there to persuade.
P. DENNING: I'd like to add to that. It is not only within yourself that you have to look, it's within others. Don't assume that you know the answers, but go talk to people. Don't just talk to us, because we already know what "us" has to say, but go to talk to people that we haven't talked to and find out what concerns them.
AUDIENCE MEMBER: Hi, my name is Lou Woleneck. I'm from the LBJ School of Public Affairs at the University of Texas. I'm a graduate student. I have a question, a general policy question, about how we should go about providing the information resources to the have-nots that the information elites have access to now. What sort of strategy that you all would have for that?
KAPOR: A 30-second or less answer, which is to set a national policy that updates a universal service for the 21st century that says everybody needs to have basic minimal access to a digital platform that reaches into every home, into every office and school in the country. We should focus our attention on how to put in place the least expensive amount of infrastructure that will produce that. What we find is, if we do that, then the overwhelming majority of American families will find it already within their budget to be able to do that, because it will be priced like basic phone service. To the extent that we need to continue or even slightly expand the kinds of lifeline programs that subsidize today's basic voice telephone service for a small percentage of the population, we should be prepared to renew that commitment. We don't need to bankrupt ourselves to give everybody access to a digital platform.
JIM WARREN: My name is Jim Warren. Two quick observations: there were several cynical comments during the last several days about a number of IRS people being here. It turns out, because they never had a platform to say this, that the whole crowd from the IRS who are here, as I understand it, are from the IRS privacy project, intent on developing policies to assure privacy protection for taxpayer information. So let us not be so cynical about their being here; otherwise, remember that they are simply doing what they are told to do by our representatives. (laughter and hisses) I was also bothered by both Simon's, and (my God!) Esther's comments on those evil little men, and the men in politics, etc. Gee, this is a modern age, let's say "men and women," for evil deeds, as well as good deeds.
DYSON: There aren't enough women in politics for there to be any evil ones.
WARREN: Well, I am sure that I can find some evil ones for you. (laughter) Anyway, to the main points: I would say that we are not so much elite, in that we are open to anyone who takes the initiative to join us, and many of us are active mentors in trying to get others to join us. I would say simply that we are a minority, and it occurs to me that revolution has always been a minority activity. It was not millions of Russians who opposed the attempted coup several months ago. It was ten, twenty, or thirty thousand in Moscow, with the aid of communications. It was not a massive movement, a populist movement, in America that resisted the Crown, two centuries ago. It was a small minority of activists and we are the activists here -- we are the revolutionaries. Freedom has always been a do-it-yourself activity, but the key syllable in that word activity is act. Let us reaffirm freedom of speech, press, assembly, security against undue search and seizure -- the basic constitutional freedoms and privileges. Let us demand that our politicians and our political candidates do the same in explicit formal commitments to act in behalf of protecting electronic civil liberties, just as they validate and speak favorably for traditional civil liberties. We can write our politicians, write our candidates and say, "Take a position in favor of civil liberties, regardless of the technology of the moment." Thank you.
GLENN TENNEY: Thank you for the introduction, Jim.
LIASSON: Are you from the IRS?
TENNEY: No. (laughter) My name is Glenn Tenney, and I have a question for you, Mara. I think that I have enough supporters on the panel. I'm not too curious about their views, but they are welcome to them. You questioned if the presidential election and race is ready for Cyberspace. What about Congress? I'm running for Congress -- is it ready for me?
AUDIENCE MEMBER: Ms. Liasson, I believe that you have opened a can of worms called politics for this little hacker community. You certainly have with me in your comment about asking for comments for the Cyberspace era from presidential candidates. I have very strong reactions to that. I think that I am going to try to express them, as a pure statement, or maybe an actual story. Several years ago, I was discussing with a friend of mine the current presidential, the then-current presidential election. He was asking me why I wasn't rabidly supporting Jesse Jackson. I thought about it, and my first response was, "Well, let's talk about the other candidates for a second. What about -- and I'll take a random name -- Michael Dukakis?" And my friend looked at me and said, "Michael Dukakis, he's just an administrator, he's not a visionary." I thought about it, and I said, "Hold on, I'm an American, I'm not someone who's a slave of the Queen of England, or something like that. I'm my own visionary, I decide where I am going." I don't want the politicians walking around telling me that I am going to have an expressway system that's going to pave over all my favorite swamps to play in. I don't want the politicians walking around defining what I'm going to do in my life. I want to elect politicians to manage government for me, to provide the barest minimum necessities to keep us smoothly greased as individuals in living together, and I want those politicians to be of the people, and I don't want them to tell me what my opinions should be. Finally, I want to cap that off with when we have government deciding how our systems work for us, we can then end up with situations where we can say, "Oh yeah, that IRS guy or that government net guy, he was just doing his job when he banned cryptography," or something like that. That's not the sort of world that I want to live in. I want to live in a world, where each of us defines our little space in it. Thank you all.
LIASSON: I think we have time for just two more and then we'll have to wrap it up.
AUDIENCE MEMBER: Hi, to the apocalypse types. I'd like to say just one thing that somebody said: The truth will make you free. In that this technology is a vehicle of communication, I believe that it is a vehicle of the truth, and as long as we keep it free, the truth will be heard that much more. Now I have kind of a question with a bit of a statement. I am a learning-disabled college student. I didn't ever finish high school. I had a freshmen education in high school, because of educational problems, and adjustment problems, I never really got too far beyond that. I write probably a fifth of the speed of anyone in this room and I have a real hard time doing math without a calculator. That's part of the reason why I wasn't able to do well in school. I read very well, fortunately, so I was able to go in when I was eighteen and take my GED just flat out without studying for it. I'm not dumb, or uneducated by any standards, but what has allowed me to get an associate's degree in college, and what has allowed me to approach graduation and get a bachelor's degree in college is the kind of technology that we are dealing with. I have never had easy access to that technology. The barriers that I have faced have been ones of order, regimentation, and where people try and say, "Oh well, you don't fit in, you're not a CS student, you don't need those resources." I'm good with computers, I do a lot with them, I spend a lot of time with them. I hack, I don't do anything illegal, but I took a hacksaw to the frame of my nasty little 8088 about two years ago to cram some RAM into it, because that was the only way I could get it to fit and I needed it. Now I'm in a little bit better shape. I'm approaching the point where I would like to see ISDN real soon, because I need that kind of connectivity. You know, I'm doing interesting things that I find absolutely wonderful, but the idea that the kind of technology that is available to us, that is just there for the using, could be limited and unavailable to people, or that people would have to go through some of the things that I have had to go through, not being able to do well on tests, because I had no word processor available to me. That type of thing, even though they are all over the place, elsewhere. It was just that that wasn't an acceptable solution. That type of policy planning, that type of government, that type of order scares me. And I have to ask, what is your answer to that?
DAVIES: The apocalyptic vision of a world in grief and individual rights in crisis has nothing to do with a Luddite mentality, and it would be very dangerous for the people in this room to link the two together. I, for one, believe in technology. I am very grateful for it, and I think the world is a better place for it. I have great faith in the future, but technology's not a silver lining for the future. It's not an El Dorado, it's more like plutonium. The very great thing that technology does for all of us can also be used by the people who would repress our freedoms and all I am saying is be aware of that. Let's not marginalize people like me, who are saying, Hey look, we are going to have 15 billion people on the planet. We are going to have a political inversion, you know, that is going to create massive tensions that are going to repress our rights, or at least create a tension that we have never known before. Don't marginalize me -- don't shoot the messenger. I believe in technology, so please don't equate the apocalypse with Ludditism -- the two do not match.
LIASSON: We're about out of time. I'm going to turn this over to Lance.
HOFFMAN: Thank you, Mara. I'm really unhappy that we are out of time, but I feel that we have a contract to those who want to leave in a moment or two. Those who want to stay, can stay up here, are welcome to continue, until the hotel throws us out. Since Lu Kleppinger is in the room at the moment, I don't know when that will be, but we can probably have it for a little while. I just want to make a couple of comments before I formally close this meeting.
We have seen an awful lot happen in these last three days and there has been building, and indeed we will be continuing to some extent the work that Jim Warren started at CFP-1 -- a sense of community. It has been increased by the participation of various diverse groups. My one hope is that you do not stop that here. When each and every one of you goes home, contact -- I don't care whether it's by letter, or electronic mail, or even telephone, if you must -- three people that you have met here that you didn't know, or didn't know very well before, or perhaps only knew electronically, and now you know them in person, and continue talking with them and to their friends and colleagues. If you do that, this will be a success.
The other comment that I want to make is that Bruce Koball is going to need a lot of help for CFP-3. Please talk to him -- he is listed in the roster. Or better yet, don't do that, talk to him here, and then give him a month to chill out in Berkeley before he has to start working real hard. Check the message board, there are some messages that have not been picked up. You have your evaluation forms. If you haven't filled them out and you would like to, please do and turn them in. I have nothing else, except to thank you all for being such a good group and, hopefully, we'll see you next year in California. Thank you very much.
Support efforts at engaging society and government on the appropriate legal and social uses of technology.

Answer: Susan Blackmore


Input: How does the conduction gap depend on the strain direction?

Context:
\section{Introduction}

In spite of being an attractive material with excellent electronic properties \cite{ahcn09},  practical applications of graphene as in conventional semiconductor devices are still questionable due to its gapless nature. In particular, the ON/OFF current ratio is low while the saturation of current is poor in pristine graphene transistors \cite{schw10}. Many efforts of bandgap engineering in graphene \cite{yhan07,khar11,lher13,jbai10,zhan09} have been made to solve these issues. The pioneer technique proposed \cite{yhan07} is to cut 2D graphene sheets into 1D narrow nanoribons. In 2D graphene sheets, some options as Bernal-stacking of graphene on hexagonal boron nitride substrate \cite{khar11}, nitrogen-doped graphene \cite{lher13}, graphene nanomesh lattice \cite{jbai10,berr13} and Bernal-stacking bilayer graphene \cite{zhan09} have been explored. However, the possibility to open a sizable bandgap in graphene as large as those of standard semiconductors is still very unlikely. In particular, it requires a very good control of lattice geometry and edge disorder in narrow graphene nanoribbons (GNRs) \cite{quer08} and in graphene nanomesh lattices \cite{hung13}, while the bandgap opening in bilayer graphene by a perpendicular electric field may not be large enough for realistic applications \cite{fior09}. Other methods should be further verified by experiments.

\begin{figure}[!t]
\centering
\includegraphics[width=2.8in]{Fig01.pdf}
\caption{Schematic of unstrained/strained graphene junctions investigated in this work.}
\label{fig_sim1}
\end{figure}

On the other hand, graphene was experimentally demonstrated to be able to sustain a much larger strain than conventional semiconductors, making it a promising candidate for flexible electronics (see in a recent review \cite{shar13}). Indeed, strain engineering has been suggested to be an alternative approach to modulating efficiently the electronic properties of graphene nanomaterials. In particular, the bandgap has periodic oscillations in the armchair GNRs \cite{ylu210} while the spin polarization at the ribbon edges (and also the bandgap) can be modulated by the strain in the zigzag cases. In 2D graphene sheets, a finite gap can open under large strains, otherwise, it may remain close to zero but the Dirac points are displaced \cite{cocc10,per209,pere09,huan10}. Many interesting electrical, optical, and magnetic properties induced by strain in graphene have been also explored, e.g. see in \cite{bunc07,pere09,kuma12,per010,pell10,guin10,tlow10,zhai11}.

Besides, local strain is a good option to improve the electrical performance of graphene devices \cite{pere09,ylu010,fuji10,juan11,baha13}. For instance, it has been shown to enhance the ON current in a GNR tunneling FET \cite{ylu010} and to fortify the transport gap in GNR strained junctions \cite{baha13}. In a recent work \cite{hung14}, we have investigated the effects of uniaxial strain on the transport in 2D unstrained/strained graphene junctions and found that due to the strain-induced shift of Dirac points, a significant conduction gap of a few hundreds meV can open with a small strain of a few percent. This type of strained junction was then demonstrated to be an excellent candidate to improve the electronic operation of graphene transistors. It hence motivates us to further investigate the properties of this conduction gap so as to optimize the performance of graphene devices. On the one hand, the effects of strain should be, in principle, dependent on its applied direction. On the other hand, because the appearance of conduction gap is a consequence of the shift of Dirac points along the $k_y$-axis, it is predicted that this gap should also depend on the transport direction. Note that here, Oy (Ox) - axis is assumed to be perpendicular (parallel) to the transport direction. The effects of both strain and transport directions will be clarified systematically in the current work.

\section{Model and calculations}

In this work, the $\pi$-orbital tight binding model constructed in \cite{per209} is used to investigate the electronic transport through the graphene strained junctions schematized in Fig. 1. The Hamiltonian is ${H_{tb}} = \sum\nolimits_{nm} {{t_{nm}}c_n^\dag {c_m}}$ where $t_{nm}$ is the hopping energy between nearest neighbor \emph{n}th and \emph{m}th atoms. The application of a uniaxial strain of angle $\theta$ causes the following changes in the $C-C$ bond vectors:
\begin{eqnarray}
 {{\vec r}_{nm}}\left( \sigma \right) &=& \left\{ {1 + {M_s}\left( \sigma, \theta \right)} \right\}{{\vec r}_{nm}}\left( 0 \right) \\
 {M_s}\left( \sigma, \theta \right) &=& \sigma \left[ {\begin{array}{*{20}{c}}
 {{{\cos }^2}\theta  - \gamma {{\sin }^2}\theta }&{\left( {1 + \gamma } \right)\sin \theta \cos \theta }\\
 {\left( {1 + \gamma } \right)\sin \theta \cos \theta }&{{{\sin }^2}\theta  - \gamma {{\cos }^2}\theta }
 \end{array}} \right] \nonumber
\end{eqnarray}
where $\sigma$ represents the strain and $\gamma \simeq 0.165$ is the Poisson ratio \cite{blak70}. The hopping parameters are defined as $t_{nm} \left( \sigma \right) = t_0 \exp\left[-3.37\left(r_{nm} \left( \sigma \right) /r_0 - 1\right)\right]$, where the hopping energy $t_0 = -2.7$ $eV$ and the bond length $r_{nm} \left( 0 \right) \equiv r_0 = 0.142$ $nm$ in the unstrained case. Therefore, there are three different hoping parameters $t_{1,2,3}$ corresponding to three bond vectors ${\vec r}_{1,2,3}$, respectively, in the strained graphene part of the structure (see Fig. 1). Here, we assume a 1D profile of applied strain, i.e., the strain tensor is a function of position along the transport direction Ox while it is constant along the Oy-axis. The transport direction, $\phi$, and strain direction, $\theta$, are determined as schematized in Fig. 1. Based on this tight binding model, two methods described below can be used to investigate the conduction gap of the considered strained junctions.

\textbf{Green's function calculations.} First, we split the graphene sheet into the smallest possible unit cells periodically repeated along the Ox/Oy directions with the indices $p/q$, respectively (similarly, see the details in \cite{hung12}). The tight-binding Hamiltonian can therefore be expressed in the following form:
\begin{eqnarray}
{H_{tb}} = \sum\limits_{p,q} {\left( {{H_{p,q}} + \sum\limits_{{p_1},{q_1}} {{H_{p,q \to p_1,q_1}}} } \right)}
\end{eqnarray}
where $H_{p,q}$ is the Hamiltonian of cell $\{p,q\}$, and $H_{p,q \to p_1,q_1}$ denotes the coupling of cell $\{p,q\}$ to its nearest neighbor cell $\{p_1,q_1\}$. We then Fourier transform the operators in Eq. (2) as follows:
\begin{eqnarray}
 {c_{p,q}} = \frac{1}{{\sqrt {{M_{cell}}} }}\sum\limits_{{\kappa_y}} {{e^{i{q\kappa_y}}}} {{\hat c}_{p,{\kappa_y}}},
\end{eqnarray}
where $M_{cell}$ is the number of unit cells and $\kappa_y \equiv k_y L_y$ with the size $L_y$ of unit cells along the Oy direction. The Hamiltonian (2) is finally rewritten as a sum of $\kappa_y$-dependent 1D-components:
\begin{eqnarray}
{H_{tb}} &=& \sum\limits_{{\kappa_y}} {\hat H\left( {{\kappa_y}} \right)} \\
\hat H\left( {{\kappa_y}} \right) &=& \sum\limits_p {{{\hat H}_{p \to p - 1}}\left( {{\kappa_y}} \right) + {{\hat H}_p}\left( {{\kappa_y}} \right) + {{\hat H}_{p \to p + 1}}}\left( {{\kappa_y}} \right) \nonumber
\end{eqnarray}
With this Hamiltonian form, the Green's function formalism can be easily applied to compute transport quantities in the graphene strained junction with different transport directions. In particular, the conductance at zero temperature is determined as:
\begin{eqnarray}
 \mathcal{G} \left( \epsilon \right) = \frac{{e^2 W}}{{\pi h L_y}}\int\limits_{BZ} {d{\kappa_y} \mathcal{T}\left( {\epsilon, {\kappa_y}} \right)}
\end{eqnarray}
where $\mathcal{T}\left( {\epsilon,{\kappa_y}} \right)$ is the transmission probability computed from the Green's functions. The integration over $\kappa_y$ is performed in the whole first Brillouin zone. As in ref. \cite{hung13}, the gap of conductance (conduction gap) is then measured from the obtained data of conductance.

\textbf{Bandstructure analyses.} To determine the conduction gap of strained junctions, we find that another simple way based on the analysis of graphene bandstructures could be efficiently used. It is described as follows. Since the conductance is computed from Eq. (5), the appearance of conduction gap is essentially governed by the gaps of transmission probability, which is determined from the energy gaps in the unstrained and strained graphene sections. These energy gaps can be defined directly from the graphene bandstructures. Therefore, our calculation has two steps, similar to that in \cite{hung14}. From the graphene bandstructures obtained using the tight-binding Hamiltonian above, we first look for the energy gaps $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ for a given $\kappa_y$ of two graphene sections. The maximum of these energy gaps determines the gap $E_{junc}^{gap}\left( {{\kappa_y}} \right)$ of transmission probability through the junction. Finally, the conduction gap $E_{cond.gap}$ is obtained by looking for the minimum value of $E_{junc}^{gap}\left( {{\kappa_y}} \right)$ when varying $\kappa_y$ in the whole Brillouin zone.

In particular, the energy bands of strained graphene are given by
\begin{eqnarray}
 E\left( {\vec k} \right) =  \pm \left| {{t_1}{e^{i\vec k{{\vec a}_1}}} + {t_2}{e^{i\vec k{{\vec a}_2}}} + {t_3}} \right|
\end{eqnarray}
where the plus/minus sign corresponds to the conduction/valence bands, respectively. For a given direction $\phi$ of transport, in principle, the vectors $\vec L_{x,y}$ defining the sizes of unit cell along the Ox and Oy directions, respectively, can be always expressed as ${\vec L_x} = {n_1}{\vec a_1} + {n_2}{\vec a_2}$ and ${\vec L_y} = {m_1}{\vec a_1} + {m_2}{\vec a_2}$ with $\cos \phi = \frac{{{{\vec L}_x}\vec L_x^0}}{{{L_x}L_x^0}}$ and $\sin \phi = \frac{{{{\vec L}_x}\vec L_y^0}}{{{L_x}L_y^0}}$ while $\vec L_{x,y}^0 = {\vec a_1} \pm {\vec a_2}$. Note that $n_{1,2}$ and $m_{1,2}$ are integers while $\frac{{{m_1}}}{{{m_2}}} =  - \frac{{{n_1} + 2{n_2}}}{{{n_2} + 2{n_1}}}$, i.e., ${\vec L_{x}} {\vec L_{y}} = 0$. In other words, we have the following expressions
\begin{eqnarray}
{{{\vec a}_1} = \frac{{ - {m_2}{{\vec L}_x} + {n_2}{{\vec L}_y}}}{{{n_2}{m_1} - {n_1}{m_2}}};\,\,\,{{\vec a}_2} = \frac{{{m_1}{{\vec L}_x} - {n_1}{{\vec L}_y}}}{{{n_2}{m_1} - {n_1}{m_2}}}}
\end{eqnarray}
On this basis, the energy bands can be rewritten in terms of $\kappa_{x, y} = \vec k \vec L_{x,y} \left( { \equiv {k_{x,y}}{L_{x,y}}} \right)$ by substituting Eqs. (7) into Eq. (6). This new form of energy bands is finally used to compute the conduction gap of strained junctions.

As a simple example, in the case of $\phi = 0$ (armchair direction), we calculate the conduction gap as follows. First, Eq. (6) is rewritten in the form
\begin{eqnarray}
 E_{\phi = 0}\left( {\vec \kappa} \right) =  \pm \left| {{t_1}{e^{i\kappa_y/2}} + {t_2}{e^{ - i\kappa_y/2}} + {t_3}{e^{ - i\kappa_x/2}}} \right|
\end{eqnarray}
with the vectors $\vec L_{x,y} \equiv \vec L_{x,y}^0$. Using this new form, the energy gap of strained graphene for a given $\kappa_y$ is determined as
\begin{equation}
{E_{strain}^{gap}}\left( {{\kappa_y}} \right) = 2 \left| {\sqrt {{{\left( {{t_1} - {t_2}} \right)}^2} + 4{t_1}{t_2}{{\cos }^2}\frac{{{\kappa_y}}}{2}}  + {t_3}} \right|
\end{equation}
while ${E_{unstrain}^{gap}}\left( {{\kappa_y}} \right)$ is given by the same formula with $t_1$ = $t_2$ = $t_3$ $\equiv$ $t_0$. The gap of transmission probability through the junction is then determined as ${E_{junc}^{gap}}\left( {{\kappa_y}} \right) = \max \left[ {E_{unstrain}^{gap}\left( {{\kappa_y}} \right),E_{strain}^{gap}\left( {{\kappa_y}} \right)} \right]$ and, finally, the conduction gap is given by ${E_{cond.gap}} = \min \left[ {E_{junc}^{gap}\left( {{\kappa_y}} \right)} \right]$ for $\kappa_y$ in the whole Brillouin zone.

We would like to notice that the Green's function calculations and the banstructure analyses give the same results of conduction gap in the junctions where the transition region between unstrained and strained graphene sections is long enough, i.e., larger than about 5 to 6 nm. In the case of short length, as discussed in \cite{baha13,hung14}, this transition zone can have significant effects on the transmission between propagating states beyond the energy gaps and hence can slightly enlarge the gap of conductance, compared to the results obtained from the bandstructure calculations.

\section{Results and discussion}

\begin{figure}[!t]
\centering
\includegraphics[width=3.0in]{Fig02.pdf}
\caption{Dependence of graphene bandgap (in the unit of eV) on the applied strain and its direction: tensile (a) and compressive (b). The radius from the central point indicates the strain strength ranging from 0 (center) to 30 $\%$ (edge of maps) while the graphene lattice is superimposed to show visibly the strain direction. The orange circle corresponds to the strains of $\sigma = 23 \%$.}
\label{fig_sim2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3.4in]{Fig03.pdf}
\caption{Conductance ($G_0 = e^2W/hL_y$) as a function of energy in graphene strained junctions for $\sigma = 4 \%$ with different strain directions. The transport along the armchair direction ($\phi = 0$) is considered. The data obtained in a uniformly strained graphene is displayed for the comparison.}
\label{fig_sim6}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5.8in]{Fig04.pdf}
\caption{Local density of states (left panels) and corresponding transmission coefficient (right panels) for three different wave-vectors $k_y$ obtained in an unstrained/strained graphene junction of $\sigma = 4 \%$, and $\theta \equiv \phi = 0$. On the top is a schematic of graphene bandedges illustrating the strain-induced shift of Dirac points along the $k_y$-direction.}
\label{fig_sim4}
\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=5.6in]{Fig05.pdf}
\caption{Maps of conduction gap in unstrained/strained graphene junctions: tensile (a,c) and compressive cases (b,d). The transport is along the armchair $\phi = 0$ (a,b) and zigzag $\phi = 30^\circ$ directions (c,d). The strain strength ranges from 0 (center) to 6 $\%$ (edge of maps) in all cases.}
\label{fig_sim4}
\end{figure*}
First, we re-examine the formation of the bandgap of graphene under a uniaxial strain. From Eq. (9), it is shown that a strain-induced finite-bandgap appears only if ${E_{strain}^{gap}}\left( {{\kappa_y}} \right) > 0$ for all $k_y$ in the first Brillouin zone, i.e., ${k _y} \in \left[ { - \frac{\pi}{L_y}, \frac{\pi}{L_y}} \right]$, otherwise, the bandgap remains zero. Hence, the condition for the bandgap to be finite is either
\begin{equation*}
 \left| {{t_1} - {t_2}} \right| > \left| {{t_3}} \right|\,\,\,\,\,{\rm{OR}}\,\,\,\,\,\left| {{t_3}} \right| > \left| {{t_1} + {t_2}} \right|
\end{equation*}
and the corresponding values of bandgap are
\begin{equation*}
 {E_{gap}} = 2\left( {\left| {{t_1} - {t_2}} \right| - \left| {{t_3}} \right|} \right)\,\,\,\,\,{\rm{OR}}\,\,\,\,\,2\left( {\left| {{t_3}} \right| - \left| {{t_1} + {t_2}} \right|} \right)
\end{equation*}
This result was actually reported in \cite{per209,hase06}. We remind as displayed in Fig. 2(a) that a finite bandgap opens only for strain larger than $\sim 23 \%$ and the zigzag (not armchair) is the preferred direction for bandgap opening under a tensile strain \cite{per209}. We extend our investigation to the case of compressive strain and find  (see in Fig. 2(b)) that (i) the same gap threshold of $\sigma \simeq 23 \%$ is observed but (ii) the preferred direction to open the gap under a compressive strain is the armchair, not the zigzag as the case of tensile strain. This implies that the properties of graphene bandstructure at low energy should be qualitatively the same when applying strains of $\left\{ {\sigma ,\theta } \right\}$ and of $\left\{ {-\sigma ,\theta + 90^\circ} \right\}$. This feature can be understood by considering, for example, strains of $\left\{ {\sigma , \theta = 0} \right\}$ and of $\left\{ {-\sigma , \theta = 90^\circ} \right\}$. Indeed, these strains result in the same qualitative changes on the bond-lengths, i.e., an increased bond-length $r_3$ and reduced bond-lengths $r_{1,2}$. However, for the same strain strength, because of the exponential dependence of hoping energies on the bond-lengths, the compressive strain generally induces a larger bandgap than the tensile one, as can be seen when comparing the data displayed in Figs. 2(a) and 2(b). To conclude, we would like to emphasize that a large strain is necessary to open a bandgap in graphene. This could be an issue for practical applications, compared to the use of graphene strained junctions explored in \cite{hung14}.

We now go to explore the properties of conduction gap in the graphene strained junctions. In Fig. 3, we display the conductance as a function of energy computed from Eq. (5) using the Green's function technique. As discussed above, a small strain of a few percent (e.g., 4 $\%$ here) can not change the gapless character of graphene, i.e., there is no gap of conductance in the case of uniformly strained graphene. However, similar to that reported in \cite{hung14}, a significant conduction-gap of a few hundreds meV can open in the unstrained/strained graphene junctions. The appearance of this conduction gap, as mentioned previously, is due to the strain-induced shift of Dirac points and is explained as follows. Actually, the strain causes the lattice deformation and can result in the deformation of graphene bandstructure. Therefore, the bandedges as a function of wave-vector $k_y$ in unstrained and strained graphene can be illustrated schematically as in the top panel of Fig. 4. As one can see, the shift of Dirac points leads to the situation where there is no value of $\kappa_y$, for which the energy gaps $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ are simultaneously equal to zero. This means that the transmission probability always shows a finite gap for any $\kappa_y$. For instance, the energy gap is zero (or small) in the unstrained (resp. strained) graphene section but finite in the strained (resp. unstrained) one in the vicinity of Dirac point $k_y = K_{unstrain}$ (resp. $K_{strain}$). Accordingly, as illustrated in the pictures of LDOS in the left panels of Fig. 4 and confirmed in the corresponding transmissions in the right panels, clear gaps of transmission are still obtained. Far from these values of $k_y$, $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ are both finite (e.g., see the LDOS plotted for $k_y = K_{gap}$) and hence a finite gap of transmission also occurs. On this basis, a finite gap of conductance is achieved. More important, Fig. 3 shows that besides the strength of strain, the strain effect is also strongly dependent on the applied direction. For instance, the conduction gap takes the values of $\sim$ 295, 172 and 323 meV for $\theta = 0$, $30^\circ$ and $90^\circ$, respectively.

Below, we will discuss the properties of the conduction gap with respect to the strain, its applied direction, and the direction of transport. Note that due to the lattice symmetry, the transport directions $\phi$ and $\phi + 60^\circ$ are equivalent while the applied strain of angle $\theta$ is identical to that of $\theta + 180^\circ$. Hence, the data obtained for $\phi$ ranging from $-30^\circ$ to $30^\circ$ and $\theta  \in \left[ {0^\circ ,180^\circ } \right]$ covers the properties of conduction gap in all possible cases.

In Fig. 5, we present the maps of conduction gap with respect to the strain and its applied direction in two particular cases: the transport is either along the armchair ($\phi = 0$) or the zigzag ($\phi = 30^\circ$) directions. Both tensile and compressive strains are considered. Let us first discuss the results obtained in the armchair case. Figs. 5(a,b) show that a large conduction gap can persist uniformly up to about 500 meV with a strain of 6%, and the conduction gap is independent of θ, maintaining a constant value across all directions. In principle, the conduction gap is larger if the shift of Dirac points in the $\kappa_y$-axis is larger, as discussed above about Figs. 3-4. We notice that the strain-induced shifts do not affect the uniform conduction gap across the six Dirac points of graphene and the gap is constant even when any Dirac point is observed at the same κ_y in the two graphene sections. From Eq. (9), we find that the Dirac points are determined by the following equations:
\begin{eqnarray*}
  {\cos}\frac{\kappa_y}{2} &=& \pm \frac{1}{2}\sqrt{\frac{{t_3^2 - {{\left( {{t_1} - {t_2}} \right)}^2}}}{{{t_1}{t_2}}}}, \\
  \cos \frac{{\kappa_x}}{2} &=& \frac{{{t_1} + {t_2}}}{{\left| {{t_3}} \right|}}\cos \frac{{\kappa_y}}{2},\,\,\,\sin \frac{{\kappa_x}}{2} = \frac{{{t_2} - {t_1}}}{{\left| {{t_3}} \right|}}\sin \frac{{\kappa_y}}{2},
\end{eqnarray*}
which simplify into ${\cos}\frac{\kappa_y}{2} = \pm \frac{1}{2}$ and, respectively, $\cos \left( {\frac{{{\kappa _x}}}{2}} \right) = \mp 1$ in the unstrained case. Hence, the constant conduction gap is obtained if To understand this, we analyze the strain matrix $M_s \left(\sigma,\theta\right)$ and find that in the case of small strains studied here, there is an approximate relationship between the bond lengths under these two strains, given by \[{r \left( \sigma, \theta \right)} - {r \left( -\sigma, \theta + 90^\circ\right)} \simeq \sigma \left( {1 - \gamma } \right) r_0,\] which is $\theta$-independent for all \emph{C-C} bond vectors. It implies that there is a fixed ratio between the hopping energies $t_i \left( \sigma, \theta \right)$ and $t_i \left( -\sigma, \theta + 90^\circ\right)$ and hence there is the similar shift of Dirac points in these two cases.
\begin{figure}[!t]
\centering
\includegraphics[width=3.4in]{Fig06.pdf}
\caption{Map showing the dependence of conduction gap on the directions ($\theta,\phi$) for $\sigma = 4 \%$. The top is a diagram illustrating the rotation of Dirac points in the \emph{k}-space with the change in the transport direction $\phi$.}
\label{fig_sim6}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5.5in]{Fig07.pdf}
\caption{Maps of conduction gap obtained in tensile/compressive strained junctions. The transport along the armchair/zigzag directions is considered in (a,b)/(c,d), respectively. The strains $\sigma_c = -2 \%$ and $\sigma_t = 2 \%$ are applied in (a,c) while $\sigma_c = -1 \%$ and $\sigma_t = 3 \%$ in (b,d).}
\label{fig_sim4}
\end{figure*}

We now go to analyze the properties of conduction gap shown in Figs. 5(c,d) where the transport is along the zigzag direction $\phi = 30^\circ$. In fact, the conduction gap in this case can reach a value as high as that of the case of $\phi = 0$ but has different $\theta$-dependence. In particular, the conduction gap remains constant at any θ for both tensile and compressive strain, where it maintains the same value regardless of φ. It is also equal to zero at $\theta = 0$ and $\theta = 90^\circ$ where the peaks of conduction gap occur in the latter case of $\phi = 0$. The relationship between these two transport directions can be explained as follows. On the one hand, based on the analyses above for $\phi = 0$, we find that for a given strength of strain, a maximum shift of Dirac points along the $k_y$-axis corresponds to a minimum along the $k_x$-one and vice versa when varying the strain direction $\theta$. On the other hand, as schematized in the top of Fig. 6 below, the change in the transport direction results in the rotation of the first Brillouin zone, i.e., the $k_x$ (resp. $k_y$) axis in the case of $\phi = 30^\circ$ is identical to the $k_y$ (resp. $k_x$) axis in the case of $\phi = 0$. These two features explain essentially the opposite $\theta$-dependence of conduction gap for $\phi = 30^\circ$, compared to the case of $\phi = 0$ as mentioned. Again, we found the same qualitative behavior of conduction gap when applying the strains of $\{\sigma,\theta\}$ and $\{-\sigma,\theta+90^\circ\}$.

Next, we investigate the conduction gap with respect to different transport directions $\phi$. We display a ($\theta,\phi$)-map of conduction gap for $\sigma = 4 \%$ in Fig. 6 and, in the top, an additional diagram illustrating the rotation of Dirac points in the $k-$space with the change in the transport direction. It is clearly shown that (i) a similar scale of conduction gap is obtained for all different transport directions, (ii) there is a smooth and continuous shift of $E_{cond.gap}-\theta$ behavior when varying $\phi$, and (iii) the same behavior of $E_{cond.gap}$ is also observed when comparing the two transport directions of $\phi$ and $\phi+30^\circ$, similarly to the comparison above between $\phi = 0^\circ$ and $30^\circ$. The data plotted in Fig. 6 additionally shows that $E_{cond.gap}$ takes the same value in both cases of $\{\phi,\theta\}$ and $\{-\phi,-\theta\}$ with a remark that the strains of $-\theta$ and $180^\circ-\theta$ are identical. Moreover, the values of $\theta$ and $\phi$, for which the conduction gap has a peak or is equal to zero, have an almost linear relationship. In particular, the relationship for conduction gap peaks is approximately given by $\theta = \theta_A - \eta_s \phi$. For tensile strains, $\eta_s$ takes the values of $\sim 1.5667$ and $1.4333$ for $\theta_A = 0$ and $90^\circ$, respectively. On the opposite, it is about $1.4333$ and $1.5667$ for $\theta_A = 0$ and $90^\circ$, respectively, for compressive strain cases. All these features are consequences of the rotation of Dirac points in the $k$-space with respect to the transport direction $\phi$ as illustrated in the diagram on the top and the lattice symmetry of graphene.

Finally, we investigate other junctions based on compressive and tensile strained graphene sections. The idea is that in this type of strained junction, the shifts of Dirac points are different in two graphene sections of different strains, which offers the possibilities to use smaller strains to achieve a similar conduction gap, compared to the case of unstrained/strained junction. In Fig. 7, we display the maps of conduction gap with respect to the directions of compressive ($\theta_c$) and tensile ($\theta_t$) strains in two cases of transport direction $\phi = 0$ (armchair) and $30^\circ$ (zigzag) for given strain strengths. Indeed, as seen in Fig. 7(a,b), with smaller strains $\left\{ {{\sigma _c},{\sigma _t}} \right\} = \left\{ { - 2\% ,2\% } \right\}$ or $\left\{ { - 1\% ,3\% } \right\}$, similar conduction gap of about 310 meV can be achieved (see Figs. 7(a,b)) while it requires a strain of 4 $\%$ in the unstrained/strained junctions discussed above. However, since the shift of Dirac points is strongly dependent on the direction of applied strains and the transport direction, the properties of conduction gap are more complicated than in the latter case. In particular, our calculations show that the preferred transport directions to achieve a large conduction gap are close to the armchair one. Otherwise, the conduction gap is generally smaller, similarly to the data for $\phi = 30^\circ$ compared to $\phi = 0$, as shown in Fig. 7. Additionally, it is shown that the preferred directions of applied strains in the case of $\phi = 0$ are close to ${\theta _c} \equiv {\theta _t} = 0$ or $90^\circ$.

\section{Conclusion}

Based on the tight binding calculations, we have investigated the effects of uniaxial strain on the transport properties of graphene strained junctions and discuss systematically the possibilities of achieving a large conduction gap with respect to the strain, its applied direction and the transport direction. It has been shown that due to the strain-induced deformation of graphene lattice and hence of graphene bandstructure, a finite conduction gap higher than 500 meV can be achieved for a strain of only 6 $\%$. Moreover, as a consequence of the shift of Dirac points along the k_y-axis, the conduction gap depends only on the strain strength, remaining unaffected by the direction of applied strain and the transport direction. A full picture of these properties of conduction gap has been presented and explained. The study hence could be a good guide for the use of this type of unstrained/strained graphene junction in electronic applications.

\textbf{\textit{Acknowledgment.}} This research in Hanoi is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.02-1012.42. We also acknowledges the French ANR for financial support under the projects NANOSIM-GRAPHENE (Grant no. ANR-09-NANO-016) and MIGRAQUEL (Grant no. ANR-10-BLAN-0304).

Answer: The conduction gap is constant across all strain directions.


Input: When was the paper published?

Context:
Title: Interpretable reduced-order modeling with time-scale separation Interpretable reduced-order modeling with time-scale separation Publish Date: 25 December 2022 Author List: Sebastian Kaltenbach (from CSE-Lab, ETH Zurich, Harvard SEAS), Phaedon-Stelios Koutsourelakis (from CSE-Lab, ETH Zurich, Harvard SEAS), Petros Koumoutsakos (from CSE-Lab, ETH Zurich, Harvard SEAS), Harvard Seas (from CSE-Lab, ETH Zurich, Harvard SEAS) 5. Comparison between the phase-space of the reference solution (left) and the phase-space of the predictions
FIG. 7. Comparison between predictions and reference solutions for a new initial condition fort = 1.25, 3.75, 7.5, 12.5, 20, 30  (from left to right and top to down).We note that with longer prediction time the uncertainty bounds increases.Despite the chaotic nature of the KS equation, the predictive posterior mean is close to the reference solution for t ≤ 12.5

abstract

Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering. However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial. We propose such a data-driven scheme that automates the identification of the time-scales involved and, can produce stable predictions forward in time as well as under different initial conditions not included in the training data.
To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space. It readily allows for the inclusion of sparse and irregularly sampled training data. The learned, latent dynamics are interpretable and reveal the different temporal scales involved.
We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix. Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation.
Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework.

INTRODUCTION

High-fidelity simulations of critical phenomena such as ocean dynamics and epidemics have become essential for decision-making. They are based on physically-motivated PDEs expressing system dynamics that span multiple spatiotemporal scales and which necessitate cumbersome computations . In recent years there is increased attention to the development of data-driven models that can accelerate the solution of these PDEs as well as reveal salient, lower-dimensional features that control the long-term evolution.
In most cases, data-driven reduced-order models are not interpretable. In particular, models based on neural networks despite good predictive capabilities , they offer a black-box description of the system dynamics. A possible remedy is applying a symbolic regression to the learned neural network representation , but this adds additional computational cost due to the two-step procedure.
A number of frameworks such as SINDy allows to learn interpretable dynamics but it relies on the a-priori availability of lower-dimensional descriptors and of time-derivatives which can be very noisy for both simulation and experimental data. Other frameworks are tailored to specific problems such as molecular dynamics .
Here, we present a framework that only needs the value of the observables, and not their derivatives, as training data and is capable of identifying interpretable latent dynamics. The deployment of interpretable latent dynamics ensures that conservation of important properties of that are reflected in the reduced-order model .
The present method is related to approaches based on the Koopman-operator extended Dynamic Mode Decomposition (eDMD) but uses continuous complex-valued latent space dynamics and only requires one scalar variable per latent dimension to describe the latent space dynamics. Therefore we do not have to enforce any parametrizations on the Koopman matrix .
The time-continuous formulation moreover allows to incorporate sparse and irregularly sampled training data and fast generation of predictions after the training phase. By using a complex-valued latent space we can also incorporate harmonic effects and reduce the number of latent variables needed. Linear and non-linear autoencoders are used to map the observed, high-dimensional time-series to the lower-dimensional, latent representation and we identify simultaneously the autoencoder as well as the latent dynamics by optimizing a combined loss function.
Hence the to tasks of dimensionality reduction and discovery of the reduced dynamics are unified while other frameworks treat the two parts separately . Apart from using an architecture based on autoencoders to identify the latent space, projection-based methods could also be employed . We are also proposing a probabilistic version of our algorithm ) that makes use of probabilistic Slow Feature Analysis .
This allows for a latent representation that arart from being time-continuous, can quantify the predictive uncertainty and hierarchically decompose the dynamics into their pertinent scales while promoting the discovery of slow processes that control the system's evolution over long time horizons. The rest of the paper is structured as follows: We introduce the methodological framework as well as algorithmic details in section VIII.
Particular focus is paid on the interpretability of the inferred lower-dimensional dynamics. In section III we present three numerical illustrations, i.e. a system of linear ODEs, a hidden Markov Model and the discretized KS-equation. We then present in section IV the probabilistic extension of the framework and apply it to the KS-equation.
We conclude with a summary and a short discussion about possible next steps. We introduce the autoencoders deployed in this work, followed by the interpretable latent space dynamic and discuss the training process. We consider data from high-dimensional time series x n ∈ R f with n = 1, ..., T . We remark that the intervals between the different states do not need to be uniformly spaced.

Autoencoder

A core assumption of the method is that each high-dimensional state x n can be compressed to a lower-dimensional representation z n ∈ C c with c << f . We identify this lower-dimensional representation by an autoencoder consisiting of a parameterized encoder and decoder. The encoder maps the high-dimensional representation to the latent space as:
The latent space is complex-valued. The decoder reconstructs the high-dimensional representation based on the latent variables as: We denote the parameters of the encoder as well as the decoder by θ. As discussed later in Section II C, both set of parameters are optimized simultaneously during training and therefore there is no need for differentiating them.

Interpretable Latent Space Dynamics

We employ a propagator in the latent space to capture the reduced-order dynamics of the system. In contrast to other time-extended variational autoencoder frameworks, our representation uses complex valued latent variables. In addition the latent variables are treated independently. The latter feature enables us to have an interpretable latent dynamics as well as a model that is especially suitable for being trained in the Small Data regime due to the small number of required parameters.
This is in contrast to temporal propagators such as LSTMs . For each dimension i of the latent variable z we are using the following continuous ODE in the complex plane: By solving this ODE, we can define the operator: Interpretable reduced-order modeling with time-scale separation Here, λ is a vector containing all the individual λ's and ∆t n indicates the time-step between the latent states.
The symbol is used to indicate a component-wise multiplication. We remark that the latent variables and the parameter governing the temporal evolution are complex numbers and their role in describing the system dynamics is similar to that of an eigenvalue. The real part is associated with growth and decay whereas the imaginary part is representing the periodic component.
This approach has similarities with the Koopman-operator based methods and the extended dynamic mode decomposition . In contrast to the methods mentioned before we are using a continuous formulation in the latent space that allows us to incorporate scarce and irregularly sampled training data and directly rely on complex numbers in the latent space.

Training and Predictions

We optimize a loss function that combines both a reconstruction loss as well as a loss associated with the error of our learned propagator in the latent space: (5) We note that we could directly incorporate mini-batch training by only taking the summation over a subset of the N available training data.
For new predictions of unseen states, we use the encoder to generate a latent representations which is then advanced in time by the learned propagator. At a designated time step we are using the decoder to reconstruct the high-dimensional solution. We applied our algorithm to three systems. First, we show that the algorithm is capable of exactly reproducing the solution of a linear ODE and to identify its eigenvalues.
Afterwards we are applying the framework to a high-dimensional process generated by a complex latent dynamics, which is correctly identified. As a final test case, we are applying the algorithm to a Kuramoto Shivashinski (KS) equation. Interpretable reduced-order modeling with time-scale separation

Linear ODE

We are considering a two-dimensional ODE system for x = y 1 y 2 : Based on the obtained training data we run our algorithm using a linear encoder and decoder structure as well as two latent variables z. The loss function was optimized using the Adam algorithm . As we consider a linear ODE we can analytically compute the eigenvalues involved and compare it with the parameters λ identified by our algorithm.
We observe in Figure that the algorithm was able to recover the correct values, i.e. the eigenvalues 7 and 3 of the given linear ODE. The system does not have a periodic component and the two imaginary parts correctly go to zero, whereas the real parts converge to the reference value. Moreover we are also able to identify for the linear mapping between our latent variables z and the training data a matrix consisting of a multiple of the eigenvectors (1,1) and (1,-1) and thus the correct solution.
This example was chosen to show that the algorithm is able to quickly identify the exact solution of a linear ODE in terms of its linearly independent components.

Hidden multiscale dynamics

We consider eight-dimensional synthetic time series data produced by an underlying twodimensional complex valued process. In particular, the data points x are generated by first solving for the temporal evolution for the two complex-valued processes p 1 and p 2 and than mapping to the eight-dimensional space by using a randomly sampled linear mapping W .
One of the two processes used to generate the data is chosen to be much slower than the other one and both processes have a periodic component. dp 2 dt = (−0.9 + 1.5i)p 2 (8) As training data we consider 40 time series with 150 data points each, obtained by simulating the described processes for a maximum of t = 15 s and then sampling from the obtained data points.
Hence the training data consists of: • 40 time-series • with each consisting 150 observations of the x at a uniform time-step ∆t = 0.0025 The autoencoder obtained consists of one linear layer for both the decoder as well as the encoder. The model is trained for 5000 iterations using the Adam optimizer and a learning rate of 10 −3 .
The results for the convergence of the parameters λ 1 and λ 2 can be found in Figure . We note that the process which is slower decaying and thus more responsible for the long-term evolution of the system has a higher convergence rate than the faster process. With the obtained parameters λ as well as the trained autoencoder, we compute predictions based on the last time step used for training, i.e. we apply the encoder to obtain our latent representation and than use the latent dynamics to advance the latent representation in time.
Afterwards, we employ the decoder to reconstruct the full high-dimensional system. The results can be found in Figure and show very good agreement between predictions and reference data. This example shows that our model is successfully able to carry out dimensionality reduction and moreover indicates that the convergence rate between latent processes can be different.
The latter is relevant when training models as for accurate predictions all latent processes and their dynamics should be converged.

Kuramoto-Sivashinsky

Finally, we applied our algorithm to the KS equation and aim to identify a reduced-order model for the solution u(y, t): We employed periodic boundary conditions, µ = 1 and a domain size y ∈ [0, 22]. For this domain-size, the KS-equation exhibits a structurally stable chaotic attractor as discussed in The black lines divides the area for which training data was given from the area without raining data.
; . The equation is discretized in space using a discretization step of 22 64 resulting in a state vector x of dimension 64 and a nonlinear system of coupled ODEs. This is solved using a stiff fourth-order solver  We employed a non-linear encoder and decoder with four fully-connected layers each and ReLU-activation functions as well as Dropout Layers between the fully-connected layers.
We trained the model for 200000 iterations using Adam and a learning rate of 5 • 10 4 and assuming a five-dimensional latent space. We obtained the λ's in Figure . Four latent variables have λ's close to zero and thus a slow temporal dynamic that is responsible for the long-term evolution whereas one latent variable is quickly decaying.
Based on the obtained parameters, we do predictions based on an unseen initial condition not contained in the training data. We are able to reconstruct the correct phase space based on our predictions despite only using a very limited amount of training data. The results for the phase space can be seen in Figure .
Although the small-scale fluctuations in the temporal dynamics are not well captured, the model identifies the correct manifold which has a good accuracy compared to the reference solution. All phase-spaces were obtained by using a finite-difference operator on the data or predictions. These results are in accordance Interpretable reduced-order modeling with time-scale separation with whose LSTM-based temporal dynamic model was also able to find the correct phase space but not to track the actual dynamics for long-term predictions.
Our model is not able to account for noise in the temporal evolution and thus dealing with chaotic, small-scale fluctuations is challenging. We believe that a probabilistic version of our algorithm could be advantageous here. This section contains a fully probabilistic formulation for the deterministic model discussed before.
We replace the Autoencoder with a Variational Autoencoder and the ODE in the latent space with a SDE. The loss function which we optimize is the Evidence-Lower Bound (ELBO).

Model Structure

We postulate the following relations for our probabilistic model using an Ornstein-Uhlenbeck (OU) for each dimension i of the latent space and a Wiener process W t in the latent space: We again assume that the latent variables z t are complex-valued and a priori independent. Complex variables were chosen as their evolution includes a harmonic components which are observed in many physical systems.
We assume an initial conditions z 0,i ∼ CN (0, σ 2 0,i ). The total parameters associated with the latent space dynamics of our model are thus {σ 2 0,i , σ 2 i , λ i } c i=1 and will be denoted by θ together with all parameters responsible for the decoder mapping G (see next section). These parameters along with the state variables z t have to be inferred from the data x t .
Based on probabilistic Slow Feature Analysis (SFA) , we set σ 2 i = 2; (λ j ) and σ 2 0,i = 1. As a consequence, a priori, the latent dynamics are stationary. A derivation and reasoning for this choice can be found in Appendix A. Hence the only independent parameters are the λ i , the imaginary part of which can account for periodic effects in the latent dynamics.

Variational Autoencoder

We employ a variational autoencoder to account for a probabilistic mappings from the lower-dimensional representation z n to the high-dimensional system x n . In particular we are employing a probabilistic decoder The encoder is used to infer the state variables z based on the given data and thus defined in the inference and learning section.

Inference and Learning

Given the probabilistic relations , our goal is to infer the latent variables z 0:T as well as all model parameters θ. We follow a hybrid Bayesian approach in which the posterior of the state variables is approximated using amortized Variational Inference and Maximum-A-Posteriori (MAP) point-estimates for θ are computed.
The application of Bayes' rule for each data sequence x 0:T leads to the following posterior: where p(θ) denotes the prior on the model parameters. In the context of variational inference, we use the following factorization of the approximate posterior i.e. we infer only the mean µ and variance σ for each state variable based on the given data points.
This conditional density used for inference is the encoder-counterpart to the probabilistic decoder defined in the section before. It can be readily shown that the optimal parameter values are found by maximizing the Evidence Lower Bound (ELBO) F(q φ (z 0:T ), θ) which is derived in Appendix B. We compute Monte Carlo estimates of the gradient of the ELBO with respect to φ and θ with the help of the reparametrization trick and carry out stochastic optimization with the ADAM algorithm .

Results for the probabilistic extension

We applied our probabilistic version to the KS-equation. We used the same settings as for the deterministic approach but considered up to 10 complex latent variables. The obtained λ's are in Figure . The probabilistic model allows us to quantify the uncertainty in predictions. In Figure predictions for various time-steps and the respective uncertainty bounds are shown for an unseen initial condition.
Due to the chaotic nature of the KS-equation and the small amount of training data, the underlying linear dynamic of our model is only able to capture the full dynamics for a limited time horizon. Fortunately, due to the probabilistic approach the model is capable of capturing chaotic fluctuations with increasingly wide uncertainty bounds.
We also computed the phase space representation for the KS-equation based on the predictions obtained by our model and compare it with the reference solution. The probabilistic model identifies the correct manifold with a better accuracy than the deterministic model. As some of the small-scale fluctuations are accounted as noise, the resulting manifold is more concentrated at the origin and the obtained values are slightly smaller than the reference manifold although their shape is very similar.

Answer: 25 December 2022