{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:ZMZMUYE7M5EODBD42MDT5BXMD7","short_pith_number":"pith:ZMZMUYE7","canonical_record":{"source":{"id":"2206.14862","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"cs.LG","submitted_at":"2022-06-29T19:03:10Z","cross_cats_sorted":["cs.AI"],"title_canon_sha256":"3a63f97ee32be0ce3fc1fbc56d77bae4eb615a4acd9b5a74dfaefb65015cf0cc","abstract_canon_sha256":"5f9921fe99032226b18776fb5671f843565fd20922042c1bd1049d47ea54ef83"},"schema_version":"1.0"},"canonical_sha256":"cb32ca609f6748e1847cd3073e86ec1fd8d818804d3ffe4049c35032429b93ee","source":{"kind":"arxiv","id":"2206.14862","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2206.14862","created_at":"2026-07-05T04:36:11Z"},{"alias_kind":"arxiv_version","alias_value":"2206.14862v1","created_at":"2026-07-05T04:36:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2206.14862","created_at":"2026-07-05T04:36:11Z"},{"alias_kind":"pith_short_12","alias_value":"ZMZMUYE7M5EO","created_at":"2026-07-05T04:36:11Z"},{"alias_kind":"pith_short_16","alias_value":"ZMZMUYE7M5EODBD4","created_at":"2026-07-05T04:36:11Z"},{"alias_kind":"pith_short_8","alias_value":"ZMZMUYE7","created_at":"2026-07-05T04:36:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:ZMZMUYE7M5EODBD42MDT5BXMD7","target":"record","payload":{"canonical_record":{"source":{"id":"2206.14862","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"cs.LG","submitted_at":"2022-06-29T19:03:10Z","cross_cats_sorted":["cs.AI"],"title_canon_sha256":"3a63f97ee32be0ce3fc1fbc56d77bae4eb615a4acd9b5a74dfaefb65015cf0cc","abstract_canon_sha256":"5f9921fe99032226b18776fb5671f843565fd20922042c1bd1049d47ea54ef83"},"schema_version":"1.0"},"canonical_sha256":"cb32ca609f6748e1847cd3073e86ec1fd8d818804d3ffe4049c35032429b93ee","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:36:11.069116Z","signature_b64":"RbkEquoI2UUGO70XPJFzOrjeHgu1DYrI+oBtWO6zF7fbC3dCxVt9QrkpK/j0gVSe51kY5Uu+U1rG82uiFV6MAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb32ca609f6748e1847cd3073e86ec1fd8d818804d3ffe4049c35032429b93ee","last_reissued_at":"2026-07-05T04:36:11.068733Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:36:11.068733Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2206.14862","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T04:36:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dVBMXimWRmaVuSeTtvL374ULQmL8upAvYNohqJoankgxuOPGphgWwFOba2BWGZtm3BSvrQEqxWR3U+XAk2JMAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T21:36:20.387391Z"},"content_sha256":"80bba8eea910a854cd2fdcf005a94c6da924af7ef53d96bf334b5e509aa69411","schema_version":"1.0","event_id":"sha256:80bba8eea910a854cd2fdcf005a94c6da924af7ef53d96bf334b5e509aa69411"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:ZMZMUYE7M5EODBD42MDT5BXMD7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Momentum Diminishes the Effect of Spectral Bias in Physics-Informed Neural Networks","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["cs.AI"],"primary_cat":"cs.LG","authors_text":"Alexander Kazachek, Boyu Wang, Ghazal Farhani","submitted_at":"2022-06-29T19:03:10Z","abstract_excerpt":"Physics-informed neural network (PINN) algorithms have shown promising results in solving a wide range of problems involving partial differential equations (PDEs). However, they often fail to converge to desirable solutions when the target function contains high-frequency features, due to a phenomenon known as spectral bias. In the present work, we exploit neural tangent kernels (NTKs) to investigate the training dynamics of PINNs evolving under stochastic gradient descent with momentum (SGDM). This demonstrates SGDM significantly reduces the effect of spectral bias. We have also examined why "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.14862","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2206.14862/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T04:36:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"InCLMyM4NY5h/1L1BgMXg4OB2x67rMNJLS2cx8chh/rMI04gcA6Qj90PLUmgfXlDSbi92TfyN1/S34BrTfSAAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T21:36:20.387902Z"},"content_sha256":"16f6331fe1dd9059a9b5ba1ee23eb40cf35b2c99ed0cc2b0db40dd01e97bdfb5","schema_version":"1.0","event_id":"sha256:16f6331fe1dd9059a9b5ba1ee23eb40cf35b2c99ed0cc2b0db40dd01e97bdfb5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZMZMUYE7M5EODBD42MDT5BXMD7/bundle.json","state_url":"https://pith.science/pith/ZMZMUYE7M5EODBD42MDT5BXMD7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZMZMUYE7M5EODBD42MDT5BXMD7/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-17T21:36:20Z","links":{"resolver":"https://pith.science/pith/ZMZMUYE7M5EODBD42MDT5BXMD7","bundle":"https://pith.science/pith/ZMZMUYE7M5EODBD42MDT5BXMD7/bundle.json","state":"https://pith.science/pith/ZMZMUYE7M5EODBD42MDT5BXMD7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZMZMUYE7M5EODBD42MDT5BXMD7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:ZMZMUYE7M5EODBD42MDT5BXMD7","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"5f9921fe99032226b18776fb5671f843565fd20922042c1bd1049d47ea54ef83","cross_cats_sorted":["cs.AI"],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"cs.LG","submitted_at":"2022-06-29T19:03:10Z","title_canon_sha256":"3a63f97ee32be0ce3fc1fbc56d77bae4eb615a4acd9b5a74dfaefb65015cf0cc"},"schema_version":"1.0","source":{"id":"2206.14862","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2206.14862","created_at":"2026-07-05T04:36:11Z"},{"alias_kind":"arxiv_version","alias_value":"2206.14862v1","created_at":"2026-07-05T04:36:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2206.14862","created_at":"2026-07-05T04:36:11Z"},{"alias_kind":"pith_short_12","alias_value":"ZMZMUYE7M5EO","created_at":"2026-07-05T04:36:11Z"},{"alias_kind":"pith_short_16","alias_value":"ZMZMUYE7M5EODBD4","created_at":"2026-07-05T04:36:11Z"},{"alias_kind":"pith_short_8","alias_value":"ZMZMUYE7","created_at":"2026-07-05T04:36:11Z"}],"graph_snapshots":[{"event_id":"sha256:16f6331fe1dd9059a9b5ba1ee23eb40cf35b2c99ed0cc2b0db40dd01e97bdfb5","target":"graph","created_at":"2026-07-05T04:36:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2206.14862/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Physics-informed neural network (PINN) algorithms have shown promising results in solving a wide range of problems involving partial differential equations (PDEs). However, they often fail to converge to desirable solutions when the target function contains high-frequency features, due to a phenomenon known as spectral bias. In the present work, we exploit neural tangent kernels (NTKs) to investigate the training dynamics of PINNs evolving under stochastic gradient descent with momentum (SGDM). This demonstrates SGDM significantly reduces the effect of spectral bias. We have also examined why ","authors_text":"Alexander Kazachek, Boyu Wang, Ghazal Farhani","cross_cats":["cs.AI"],"headline":"","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"cs.LG","submitted_at":"2022-06-29T19:03:10Z","title":"Momentum Diminishes the Effect of Spectral Bias in Physics-Informed Neural Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.14862","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:80bba8eea910a854cd2fdcf005a94c6da924af7ef53d96bf334b5e509aa69411","target":"record","created_at":"2026-07-05T04:36:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"5f9921fe99032226b18776fb5671f843565fd20922042c1bd1049d47ea54ef83","cross_cats_sorted":["cs.AI"],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"cs.LG","submitted_at":"2022-06-29T19:03:10Z","title_canon_sha256":"3a63f97ee32be0ce3fc1fbc56d77bae4eb615a4acd9b5a74dfaefb65015cf0cc"},"schema_version":"1.0","source":{"id":"2206.14862","kind":"arxiv","version":1}},"canonical_sha256":"cb32ca609f6748e1847cd3073e86ec1fd8d818804d3ffe4049c35032429b93ee","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cb32ca609f6748e1847cd3073e86ec1fd8d818804d3ffe4049c35032429b93ee","first_computed_at":"2026-07-05T04:36:11.068733Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:36:11.068733Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RbkEquoI2UUGO70XPJFzOrjeHgu1DYrI+oBtWO6zF7fbC3dCxVt9QrkpK/j0gVSe51kY5Uu+U1rG82uiFV6MAg==","signature_status":"signed_v1","signed_at":"2026-07-05T04:36:11.069116Z","signed_message":"canonical_sha256_bytes"},"source_id":"2206.14862","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:80bba8eea910a854cd2fdcf005a94c6da924af7ef53d96bf334b5e509aa69411","sha256:16f6331fe1dd9059a9b5ba1ee23eb40cf35b2c99ed0cc2b0db40dd01e97bdfb5"],"state_sha256":"0465434beb176be4abd60adb13b6f9f2b65b2ae7eb82ecb41625e1ad39e291f4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XBkfnawfU1R3qNj3fkNCHAdqyIk/sBUBY2n0rho+KAqx2IPPPGd9x7wDnMzgJlCorqicdbyx2OzlzP3teIGdBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-17T21:36:20.393413Z","bundle_sha256":"d28345af8cdecaaf6470ac8edc334fd31e5d62d3433a455c51b0e58efe1b711f"}}