{
  "@context": {
    "@language": "en",
    "@vocab": "https://schema.org/",
    "conformsTo": "dct:conformsTo",
    "cr": "http://mlcommons.org/croissant/",
    "dataType": {
      "@id": "cr:dataType",
      "@type": "@vocab"
    },
    "dct": "http://purl.org/dc/terms/",
    "prov": "http://www.w3.org/ns/prov#",
    "rai": "http://mlcommons.org/croissant/RAI/",
    "sc": "https://schema.org/",
    "value": {
      "@id": "cr:value",
      "@type": "@json"
    },
    "recordSet": "cr:recordSet",
    "field": "cr:field",
    "includes": "cr:includes",
    "source": "cr:source",
    "fileSet": "cr:fileSet",
    "extract": "cr:extract",
    "column": "cr:column"
  },
  "@type": "sc:Dataset",
  "name": "Poisson_Multimode PDEFieldDataset",
  "description": "PosteriorBench v2.0 Hugging Face Arrow training dataset for the Poisson_Multimode inverse problem. Each of the 50,000 records contains an integer id, raw float32 forcing field a=f with shape [128, 128], raw float32 solution field u=phi with shape [128, 128], and scalar generation parameters alpha and tau. Runtime loaders concatenate a and u into channel-first [2, 128, 128] tensors and apply normalization from metadata.json.",
  "conformsTo": [
    "http://mlcommons.org/croissant/1.1",
    "http://mlcommons.org/croissant/RAI/1.0"
  ],
  "url": "https://huggingface.co/datasets/anonymousmay/PosteriorBench",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "creator": {
    "@type": "Organization",
    "name": "PosteriorBench / DDIS contributors",
    "url": "https://github.com/neuraloperator/FunDPS-Physics-Guidance"
  },
  "publisher": {
    "@type": "Organization",
    "name": "PosteriorBench / DDIS contributors",
    "url": "https://github.com/neuraloperator/FunDPS-Physics-Guidance"
  },
  "dateCreated": "2026-05-02",
  "datePublished": "2026-05-05",
  "dateModified": "2026-05-07",
  "version": "2.0",
  "keywords": [
    "inverse problems",
    "partial differential equation",
    "Poisson equation",
    "posterior sampling",
    "diffusion models",
    "scientific machine learning",
    "uncertainty quantification"
  ],
  "distribution": [
    {
      "@type": "cr:FileSet",
      "@id": "arrow-shards",
      "includes": "data-*.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream"
    },
    {
      "@type": "cr:FileObject",
      "@id": "data-00000-of-00014.arrow",
      "contentUrl": "data-00000-of-00014.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream",
      "sha256": "1cc3079b780150e67f626dd949f9797d831199791c1bd95e26e6d43d0135187f"
    },
    {
      "@type": "cr:FileObject",
      "@id": "data-00001-of-00014.arrow",
      "contentUrl": "data-00001-of-00014.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream",
      "sha256": "ae0c70111d9af7ab80374dd6dc233d563dc8d37098ac164e53a1bd9bcb3d4967"
    },
    {
      "@type": "cr:FileObject",
      "@id": "data-00002-of-00014.arrow",
      "contentUrl": "data-00002-of-00014.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream",
      "sha256": "20fa61d14065e2e5945ab420c5d42f30533c7e6148b76a3b004fda4d422e576e"
    },
    {
      "@type": "cr:FileObject",
      "@id": "data-00003-of-00014.arrow",
      "contentUrl": "data-00003-of-00014.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream",
      "sha256": "53693fa2b4e6dcb93c0dc49a167563b27c6452d35d970d1d8628132dd41f77c4"
    },
    {
      "@type": "cr:FileObject",
      "@id": "data-00004-of-00014.arrow",
      "contentUrl": "data-00004-of-00014.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream",
      "sha256": "bbf2e68b3f836076e91c14102d7a612efed653606ec6162081f026859a654cf0"
    },
    {
      "@type": "cr:FileObject",
      "@id": "data-00005-of-00014.arrow",
      "contentUrl": "data-00005-of-00014.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream",
      "sha256": "89eb2cbad51c5ab7c539136d7427103fe911d78fc5562232f60bb5563b42fbad"
    },
    {
      "@type": "cr:FileObject",
      "@id": "data-00006-of-00014.arrow",
      "contentUrl": "data-00006-of-00014.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream",
      "sha256": "26bade951e159f01b6973484a3a35476d627a281deea693cbc144ea323209997"
    },
    {
      "@type": "cr:FileObject",
      "@id": "data-00007-of-00014.arrow",
      "contentUrl": "data-00007-of-00014.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream",
      "sha256": "5da19aa088cb3bd5dc1f1d0b4413b59edf42c8f18785b7d0c269c97ff558e8a0"
    },
    {
      "@type": "cr:FileObject",
      "@id": "data-00008-of-00014.arrow",
      "contentUrl": "data-00008-of-00014.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream",
      "sha256": "61d9d69130e9a9395bded06b4f525fea9ff55da1d719839108073a0da3b761b6"
    },
    {
      "@type": "cr:FileObject",
      "@id": "data-00009-of-00014.arrow",
      "contentUrl": "data-00009-of-00014.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream",
      "sha256": "f68da5a1caea0ea611d6ed7e5a10def39c6ea7512c87cc708da324dc3e400339"
    },
    {
      "@type": "cr:FileObject",
      "@id": "data-00010-of-00014.arrow",
      "contentUrl": "data-00010-of-00014.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream",
      "sha256": "37393d44a83cef43ce80ced2c095018228100260f29cf13cfe6140e0a5dd69ed"
    },
    {
      "@type": "cr:FileObject",
      "@id": "data-00011-of-00014.arrow",
      "contentUrl": "data-00011-of-00014.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream",
      "sha256": "07fb3029387f50b809e54d27da6f39c36e1e8276d71a62e008303deed804cf8b"
    },
    {
      "@type": "cr:FileObject",
      "@id": "data-00012-of-00014.arrow",
      "contentUrl": "data-00012-of-00014.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream",
      "sha256": "7b77c240136bb363721ad19c2ed22cb27af4a10166a5d56d091aae45ed7e1e0f"
    },
    {
      "@type": "cr:FileObject",
      "@id": "data-00013-of-00014.arrow",
      "contentUrl": "data-00013-of-00014.arrow",
      "encodingFormat": "application/vnd.apache.arrow.stream",
      "sha256": "b45a97c9012cf40c3186f851a537779fca165a54dfdf500c76e977f4c8332bc7"
    },
    {
      "@type": "cr:FileObject",
      "@id": "dataset_info.json",
      "contentUrl": "dataset_info.json",
      "encodingFormat": "application/json",
      "sha256": "7add834c2c3129de2be7842105a023e737dfa16c318d3b5b4ea3243bb2ddf435"
    },
    {
      "@type": "cr:FileObject",
      "@id": "metadata.json",
      "contentUrl": "metadata.json",
      "encodingFormat": "application/json",
      "sha256": "dd7bfc10fa9616b382c2545d22b5c031ee38c9e2ecca9de76d4c9d04275081dc"
    },
    {
      "@type": "cr:FileObject",
      "@id": "state.json",
      "contentUrl": "state.json",
      "encodingFormat": "application/json",
      "sha256": "53771678512f2fa5cfd156de42fb5c3d985b8792c960fef80d439ca9bcdbaa86"
    }
  ],
  "rai:dataLimitations": "The dataset is synthetic Poisson PDE data at fixed 128x128 resolution and fixed alpha/tau parameter ranges. It should not be interpreted as physical measurements or as representative of all elliptic PDE inverse problems.",
  "rai:dataBiases": "The generation metadata records alpha in [1.5, 3.0] and tau in [2.0, 4.0]; behavior outside those ranges is not represented.",
  "rai:personalSensitiveInformation": "No personal or sensitive information is present.",
  "rai:dataUseCases": [
    "Training diffusion priors and neural operators for Poisson PDE inverse-problem research.",
    "Benchmarking posterior sampling, reconstruction, and scientific machine-learning methods on synthetic PDE fields.",
    "Not recommended for claims about real-world physical systems without validation against domain data."
  ],
  "rai:dataSocialImpact": "Positive impact is expected primarily through reproducible scientific machine-learning benchmarks for inverse problems. No significant negative impacts are anticipated.",
  "rai:hasSyntheticData": true,
  "prov:wasDerivedFrom": [
    {
      "@id": "urn:posteriorbench:source:poisson_multimode_params_h5",
      "@type": "prov:Entity",
      "name": "poisson_random_params.h5",
      "description": "Raw conversion source containing f, phi, alpha, and tau arrays."
    }
  ],
  "prov:wasGeneratedBy": [
    {
      "@id": "urn:posteriorbench:activity:poisson_multimode_data_collection",
      "@type": "prov:Activity",
      "name": "Synthetic data collection",
      "description": "The source fields were produced by a synthetic numerical Poisson PDE generator."
    },
    {
      "@id": "urn:posteriorbench:activity:poisson_multimode_hf_export",
      "@type": "prov:Activity",
      "name": "PosteriorBench v2.0 HF export",
      "description": "The conversion stores raw physical fields under canonical names a and u, stores alpha/tau in params, records normalization statistics in metadata.json, and saves 50,000 rows as a Hugging Face Dataset backed by Apache Arrow shards."
    },
    {
      "@id": "urn:posteriorbench:activity:poisson_multimode_annotation",
      "@type": "prov:Activity",
      "name": "Annotation and labeling",
      "description": "No human annotation was performed; semantic labels come from the numerical source field names."
    }
  ],
  "recordSet": [
    {
      "@type": "cr:RecordSet",
      "@id": "poisson_multimode_train",
      "field": [
        {
          "@type": "cr:Field",
          "@id": "poisson_multimode_train/split",
          "dataType": "sc:Text",
          "value": "train"
        },
        {
          "@type": "cr:Field",
          "@id": "poisson_multimode_train/id",
          "dataType": "sc:Integer",
          "source": {
            "fileSet": {
              "@id": "arrow-shards"
            },
            "extract": {
              "column": "id"
            }
          }
        },
        {
          "@type": "cr:Field",
          "@id": "poisson_multimode_train/a",
          "dataType": "cr:Float32",
          "description": "Raw Poisson forcing field f with shape [128, 128].",
          "source": {
            "fileSet": {
              "@id": "arrow-shards"
            },
            "extract": {
              "column": "a"
            }
          }
        },
        {
          "@type": "cr:Field",
          "@id": "poisson_multimode_train/u",
          "dataType": "cr:Float32",
          "description": "Raw Poisson solution field phi with shape [128, 128].",
          "source": {
            "fileSet": {
              "@id": "arrow-shards"
            },
            "extract": {
              "column": "u"
            }
          }
        },
        {
          "@type": "cr:Field",
          "@id": "poisson_multimode_train/params/alpha",
          "dataType": "cr:Float64",
          "description": "GRF smoothness parameter alpha.",
          "source": {
            "fileSet": {
              "@id": "arrow-shards"
            },
            "extract": {
              "column": "params.alpha"
            }
          }
        },
        {
          "@type": "cr:Field",
          "@id": "poisson_multimode_train/params/tau",
          "dataType": "cr:Float64",
          "description": "GRF scale parameter tau.",
          "source": {
            "fileSet": {
              "@id": "arrow-shards"
            },
            "extract": {
              "column": "params.tau"
            }
          }
        },
        {
          "@type": "cr:Field",
          "@id": "poisson_multimode_train/runtime_tensor_order",
          "dataType": "sc:Text",
          "value": [
            "a=f",
            "u=phi"
          ]
        },
        {
          "@type": "cr:Field",
          "@id": "poisson_multimode_train/runtime_tensor_shape",
          "dataType": "sc:Text",
          "value": "PDEFieldDataset returns normalized concat(a, u) with shape [2, 128, 128]."
        },
        {
          "@type": "cr:Field",
          "@id": "poisson_multimode_train/normalization",
          "dataType": "sc:Text",
          "value": "(x - mean) * (0.5 / std), with stats stored in metadata.json."
        }
      ]
    }
  ]
}