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@@ -318,7 +318,7 @@ $$
 
 The Jacobian has a simple form that can be understood by imagining the protein backbone as a robot arm that is planted at $\scriptstyle { \pmb x } _ { 0 }$ (Figure 2B). Increasing or decreasing the bond length $b _ { i }$ extends or retracts all downstream coordinates along the bonds axis, moving a bond angle $a _ { i }$ drives circular motion of all downstream coordinates around the bond normal vector $\hat { \mathbf { \ b { n } } } _ { i }$ centered at $\mathbf { \delta } _ { \mathbf { \mathcal { X } } i - 1 }$ , and moving a dihedral angle $d _ { i }$ drives circular motion of downstream coordinate $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { j } }$ around bond vector $\hat { \pmb { u } } _ { i - 1 }$ centered at ${ \bf { \mathbf { \mathit { x } } } } _ { i - 1 }$ .
 
-Unconstrained representations Bond lengths and angles are subject to the constraints $b _ { i } > 0$ and $0 < a _ { i } < \pi$ . We enforce these constraints by representing these degrees of freedom in terms of fully unconstrained variables $\tilde { b } _ { i }$ and $\tilde { a } _ { i }$ via the transformations $b _ { i } = \log \biggl ( 1 + e ^ { \tilde { b } _ { i } } \biggr )$ and $\begin{array} { r } { a _ { i } = \frac { \pi } { 1 + e ^ { - \bar { a } _ { i } } } } \end{array}$ ⇡1+ea˜i . All references to the internal coordinates $_ { z }$ and Jacobians $\textstyle { \frac { \partial { \boldsymbol { x } } } { \partial z } }$ will refer to the use of fully unconstrained representations (Table 2).
+Unconstrained representations Bond lengths and angles are subject to the constraints $b _ { i } > 0$ and $0 < a _ { i } < \pi$ . We enforce these constraints by representing these degrees of freedom in terms of fully unconstrained variables $\tilde { b } _ { i }$ and $\tilde { a } _ { i }$ via the transformations $b _ { i } = \log \biggl ( 1 + e ^ { \tilde { b } _ { i } } \biggr )$ and $\begin{array} { r } { a _ { i } = \frac { \pi } { 1 + e ^ { - \bar { a } _ { i } } } } \end{array}$ ⇡1+ea˜i . All references to the internal coordinates $_ { z }$ and Jacobians $\textstyle { \frac { \partial { \boldsymbol { x } } } { \partial z } }$ will refer to the use of fully unconstrained representations (Table 2).
 
 # A.2 ENERGY FUNCTION
 
@@ -364,7 +364,7 @@ $$
 
 Transform integrator The need to rebuild Cartesian geometry $_ { \textbf { \em x } }$ from internal coordinates $_ z$ with $\mathcal F ( z )$ at every time step is one of the major costs of conformational sampling codes based on Internal coordinates (Parsons et al., 2005) because it is intrinsically sequential. Here we show how it is possible to bypass the need for geometry reconstruction at every step by instead computing on-the-fly geometry modification.
 
-Imagine following a change to the internal coordinates $\Delta z ^ { ( t ) }$ along a straight path from $z ^ { ( t ) }$ to $\tilde { \mathbf { \Lambda } } _ { Z } ( t + \bar { \mathbf { \Lambda } } \epsilon )$ and tracking the corresponding nonlinear path of the Cartesian coordinates from $\mathbf { \boldsymbol { x } } ^ { ( t ) }$ to $\mathbf { \boldsymbol { x } } ^ { ( t + \epsilon ) }$ . If this path is indexed by $u \in ( t , t + \epsilon )$ , then the dynamics of $\mathbf { x }$ with respect to $u$ are given by $\begin{array} { r } { \frac { \partial \pmb { x } } { \partial u } = \frac { \bar { \partial } \pmb { x } } { \partial z } \frac { \partial z } { \partial u } = \frac { \partial \pmb { x } } { \partial z } \frac { 1 } { \epsilon } \Delta z } \end{array}$ @x@z 1✏ z. Integrating the dynamics of x gives
+Imagine following a change to the internal coordinates $\Delta z ^ { ( t ) }$ along a straight path from $z ^ { ( t ) }$ to $\tilde { \mathbf { \Lambda } } _ { Z } ( t + \bar { \mathbf { \Lambda } } \epsilon )$ and tracking the corresponding nonlinear path of the Cartesian coordinates from $\mathbf { \boldsymbol { x } } ^ { ( t ) }$ to $\mathbf { \boldsymbol { x } } ^ { ( t + \epsilon ) }$ . If this path is indexed by $u \in ( t , t + \epsilon )$ , then the dynamics of $\mathbf { x }$ with respect to $u$ are given by $\begin{array} { r } { \frac { \partial \pmb { x } } { \partial u } = \frac { \bar { \partial } \pmb { x } } { \partial z } \frac { \partial z } { \partial u } = \frac { \partial \pmb { x } } { \partial z } \frac { 1 } { \epsilon } \Delta z } \end{array}$ @x@z 1✏ z. Integrating the dynamics of x gives
 
 $$
 \begin{array} { l } { \displaystyle \pmb { x } ^ { ( t + \epsilon ) } = \mathcal { F } \left( \pmb { z } ^ { ( t ) } + \Delta \pmb { z } ^ { ( t ) } \right) } \\ { \displaystyle = \pmb { x } ^ { ( t ) } + \int _ { t } ^ { t + \epsilon } \frac { 1 } { \epsilon } \frac { \partial \pmb { x } } { \partial \pmb { z } } ^ { ( u ) } \Delta \pmb { z } ^ { ( t ) } d u . } \end{array}
@@ -398,7 +398,7 @@ The translational component of motion is simply the average displacement across
 Input : Initial state $z ^ { ( 0 ) }$ , energy $U ( { \pmb x } )$ , time steps $\epsilon _ { x } , \epsilon _ { z }$ , total time $T$ , preconditioners $\mathbf { C } _ { \mathbf { \mathcal { X } } } , \mathbf { C } _ { z }$ ,   
 Output :Trajectory $\pmb { x } ^ { ( 0 ) } , \ldots , \pmb { x } ^ { ( T ) }$   
 Initialize $\pmb { x } ^ { ( 0 ) }  \mathcal { F } ( \pmb { z } ^ { ( 0 ) } )$ ;   
-while $t < T$ do $f _ { x } \gets \nabla _ { x } U$ ; x(Cart) CartesianStep $\begin{array} { r } { \langle \pmb { x } ^ { ( t ) } , \pmb { f } _ { \pmb { x } } , \epsilon _ { \pmb { x } } , \mathbf { C } _ { \pmb { x } } \rangle } \end{array}$ ; $\Delta \pmb { x } ^ { ( I n t ) }  \mathsf { C 1 }$ ippedInternalStep $( \pmb { x } + \Delta \pmb { x } ^ { ( C a r t ) } , \pmb { f _ { x } } , \epsilon _ { z } , \mathbf { C } _ { z } ) ;$ ; x x + Detrend $( \Delta \pmb { x } ^ { ( C a r t ) } + \Delta \pmb { x } ^ { ( I n t ) } )$ ; $t  t + \epsilon$ ;   
+while $t < T$ do $f _ { x } \gets \nabla _ { x } U$ ; x(Cart) CartesianStep $\begin{array} { r } { \langle \pmb { x } ^ { ( t ) } , \pmb { f } _ { \pmb { x } } , \epsilon _ { \pmb { x } } , \mathbf { C } _ { \pmb { x } } \rangle } \end{array}$ ; $\Delta \pmb { x } ^ { ( I n t ) }  \mathsf { C 1 }$ ippedInternalStep $( \pmb { x } + \Delta \pmb { x } ^ { ( C a r t ) } , \pmb { f _ { x } } , \epsilon _ { z } , \mathbf { C } _ { z } ) ;$ ; x x + Detrend $( \Delta \pmb { x } ^ { ( C a r t ) } + \Delta \pmb { x } ^ { ( I n t ) } )$ ; $t  t + \epsilon$ ;   
 end
 
 Speed clipping We found it helpful to stabilize the model by enforcing a speed limit on overall structural motions for the internal coordinate steps. This prevents small changes to the energy function during learning from causing extreme dynamics that in turn produce a non-informative learning signal. To accomplish this, we translationally and rotationally detrend the update of the predictor step $\Delta \mathbf { x }$ and compute a hypothetical time step $\hat { \epsilon } _ { z }$ that would limit the fastest motion to 2 Angstroms per iteration. We then compute modified predictor and corrector steps subject to this new, potentially slower, time step. While this breaks the asymptotics of Langevin dynamics, (i) it is unlikely on our timescales that we achieve stationarity and (ii) it can be avoided by regularizing the dynamics away from situations where clipping is necessary. In the future, considering non-Gaussian perturbations with kinetic energies similar to Relativistic Monte Carlo (Lu et al., 2017) might accomplish a similar goal in a more principled manner. The final integrator combining these ideas is presented in Figure 3.
@@ -455,7 +455,7 @@ $$
 
 is a measure of superposition quality between two protein structures on $[ 0 , 1 ]$ that was presented as an approximately length-independent alternative to RMSD. The TM-score is the best attainable value of the preceding quantity for all possible superpositions of two structures, where $D _ { i } \ =$ $| | \mathbf { \boldsymbol { x } } ^ { ( \mathrm { M o d e l } ) } - \mathbf { \bar { \boldsymbol { x } } } ^ { ( \mathrm { D a t a } ) } | |$ . This requires iterative optimization, which we implemented with a sign gradient descent with 100 iterations to optimally superimpose the model and target structure. We backpropagate through this unrolled optimization process as well as that of the simulator.
 
-Hydrogen bond loss We determine intra-backbone hydrogen bonds using the electrostatic model of DSSP (Kabsch & Sander, 1983). First, we place virtual hydrogens at 1 Angstroms along the negative angle bisector of the $C _ { i - 1 } - N _ { i } - C \alpha _ { i }$ bond angle. Second, we compute a putative energy $U _ { i j } ^ { \mathrm { { h - b o n d } } }$ ( $\mathrm { n \ k c a l / m o l } )$ ) for each potential hydrogen bond from an amide donor at $i$ to a carbonyl acceptor at $j$ as
+Hydrogen bond loss We determine intra-backbone hydrogen bonds using the electrostatic model of DSSP (Kabsch & Sander, 1983). First, we place virtual hydrogens at 1 Angstroms along the negative angle bisector of the $C _ { i - 1 } - N _ { i } - C \alpha _ { i }$ bond angle. Second, we compute a putative energy $U _ { i j } ^ { \mathrm { { h - b o n d } } }$ ( $\mathrm { n \ k c a l / m o l } )$ ) for each potential hydrogen bond from an amide donor at $i$ to a carbonyl acceptor at $j$ as
 
 $$
 \begin{array} { l } { { \displaystyle U _ { i j } ^ { \mathrm { h - b o n d } } ( { \bf X } ) = \left( \frac { q _ { N } q _ { O } } { D _ { N O } } + \frac { q _ { H } q _ { C } } { D _ { H C } } + \frac { q _ { H } q _ { O } } { D _ { H O } } + \frac { q _ { N } q _ { C } } { D _ { N C } } \right) ~ 3 3 2 } } \\ { { \displaystyle ~ = 0 . 0 8 4 \left( \frac { 1 } { D _ { N O } } + \frac { 1 } { D _ { H C } } - \frac { 1 } { D _ { H O } } - \frac { 1 } { D _ { N C } } \right) ~ 3 3 2 } } \end{array}
@@ -497,7 +497,7 @@ $$
 \frac { \partial \pmb { x } ^ { ( T ) } } { \partial \pmb { x } ^ { ( 0 ) } } = \frac { \partial \pmb { x } ^ { ( T ) } } { \partial \pmb { x } ^ { ( T - 1 ) } } \frac { \partial \pmb { x } ^ { ( T - 1 ) } } { \partial \pmb { x } ^ { ( T - 2 ) } } \cdot \cdot \cdot \frac { \partial \pmb { x } ^ { ( 1 ) } } { \partial \pmb { x } ^ { ( 0 ) } } .
 $$
 
-When the norms of the per time-step Jacobians @x(t)@x(t1) are typically larger than 1, the sensitivity $\big | \big | \frac { \partial \pmb { x } ^ { ( T ) } } { \partial \pmb { x } ^ { ( 0 ) } } \big | \big |$ will grow exponentially with $T$ . Ideally, we would keep these norms well-behaved which is the rationale recent work on stabilization of RNNs (Henaff et al., 2016; Chen et al., 2018b). Next we will offer a general-purpose regularizer to approximately enforce this goal for any differentiable computational iteration with continuous state.
+When the norms of the per time-step Jacobians @x(t)@x(t1) are typically larger than 1, the sensitivity $\big | \big | \frac { \partial \pmb { x } ^ { ( T ) } } { \partial \pmb { x } ^ { ( 0 ) } } \big | \big |$ will grow exponentially with $T$ . Ideally, we would keep these norms well-behaved which is the rationale recent work on stabilization of RNNs (Henaff et al., 2016; Chen et al., 2018b). Next we will offer a general-purpose regularizer to approximately enforce this goal for any differentiable computational iteration with continuous state.
 
 ![](images/ca7d5b1eb6e91174b340bdcfb50f31950203a916d4ab08d2b7044b70de45d1ab.jpg)  
 Figure 9: Chaos impedes meta-learning for gradient descent in a well. (a) Gradient descent of a particle in a well with initial conditions $x ^ { ( 0 ) }$ and step size $\alpha$ . (b) Orbit diagrams visualize long-term dynamics from iterations 1000 to 2000 of the position $x$ (top) and the gradient $\frac { d x ^ { ( t ) } } { d x ^ { ( 0 ) } }$ (bottom). When the step size $\alpha$ is small, these dynamics converge to a periodic orbit over $2 ^ { k }$ values where $0 \leq k < \infty$ . After some critical step size, the dynamics undergo a period-doubling bifurcation (Strogatz, 2018), become chaotic, and the gradients regularly diverge to huge numbers.
@@ -548,7 +548,7 @@ $$
 w _ { a } ^ { ( i ) } = \sigma \left( \log \frac { f _ { a } ^ { ( i ) } } { B _ { a } } \right)
 $$
 
-where (u) = 11+exp( u) is the logistic sigmoid and $B _ { a }$ is the average frequency of amino acid $a$ in UniProt (Apweiler et al., 2004).
+where (u) = 11+exp( u) is the logistic sigmoid and $B _ { a }$ is the average frequency of amino acid $a$ in UniProt (Apweiler et al., 2004).
 
 Table 4: Qualitative timings. †Results on CATH dataset and 2 M40 GPUs.