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Recent hybrid methods promote physical correctness by targeting the PDE residual via gradient descent or Gauss--Newton steps, but inherit the compute cost and instability of the underlying classical optimizers. We show, theoretically and empirically, that numerically minimizing the PDE residual can be an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining why these methods often do not make accurate predictions despite achieving low residuals. We propose error-conditioned Neural Solvers (ENS), built on a different principle: rather than an optimization target, the PDE residual field is passed as a direct input to the network at each it","title":"Error-Conditioned Neural Solvers","url":"https://arxiv.org/abs/2606.27354","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.27354v1 Announce Type: cross \nAbstract: Neural surrogate models offer fast approximate mappings from PDE parameters to solutions, but they typically treat solving as a purely statistical task: once trained, they struggle to correct their own constraint violations and extrapolate beyond the training distribution. Recent hybrid methods promote physical correctness by targeting the PDE residual via gradient descent or Gauss--Newton steps, but inherit the compute cost and instability of the underlying classical optimizers. We show, theoretically and empirically, that numerically minimizing the PDE residual can be an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining why these methods often do not make accurate predictions despite achieving low residuals. We propose error-conditioned Neural Solvers (ENS), built on a different principle: rather than an optimization target, the PDE residual field is passed as a direct input to the network at each it","title":"Error-Conditioned Neural Solvers","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-26T04:43:58Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2606.27354"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:ad61113a71157f8dfd03bf196975da98194479c8e653f3837d2c2bc4bb45d50b754315dc54c577fe507cf49b2fd09b9d2262cead3a8e76fbe924b1ab77db9d0f","signer":"crovia.substrate","subject":{"observed_at":"2026-06-26T04:43:58Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2606.27354"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"70d229efe1844ac1be47a7ed478e8dc625ff07c25261d956d47137e509a76ed9","leaf_index":251192,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"0d3970824f051864adefa3a8d76a9a7ac62393b306e6153803663d35dd35b747","side":"right"},{"sibling":"af521edee789d0f177f38a7a7102898ec4ba4f7a466e2dc4ff1253b2a99c15ec","side":"right"},{"sibling":"40ae5be9664d93d323b68583cdf91c2df2ffc2d3bd5f76fa74933a3abddb58dc","side":"right"},{"sibling":"38c4fdd81eed7f9e2b5321776557efdded6d808fc0ac8d6fac725cd39e255885","side":"left"},{"sibling":"57782f777ca8465e79e296a48eaebca293ee1a6c49bc25d4f42b514f0ad7c576","side":"left"},{"sibling":"560f6bfbc80c60f9cd8b6d1801c24aad7b45471335f3a4eacdc44dcad4f9129a","side":"left"},{"sibling":"2f14dad8e0a8f8bab0ca581ed1b54b6a2b4cc27363a36e17359b2be932eb3fb9","side":"right"},{"sibling":"846fe23f5401fbe8ae1dd175eade4cb955db0a1adde862418a9a781a48f6283c","side":"right"},{"sibling":"d097e1d1d25acb37e74792d46ad69ba853039e2f2b33a092c9ea32a837aeed84","side":"left"},{"sibling":"b6709caadc8510310ee2ec0b66d1058fcad31c91c65cc2bad6f46a693d553580","side":"right"},{"sibling":"a72c3b8804a37d1a9d18e02e6fdb048bc8ea6b0746909cd2de10bdbabc793737","side":"left"},{"sibling":"803703dc2c50a646fa77b57c0056e9a5126611ba4bcde0d6013ccd6b2d44bdbf","side":"right"},{"sibling":"e78f244b1b8df6d5e3fdc6dd76b5c27d4e6fe3b93b8cd61355497b63d7e4cfe8","side":"left"},{"sibling":"6167cb552ed6871fbf0afcf3db01d1017af7d472b136fcbe5404d1df09f41cc1","side":"right"},{"sibling":"f29798d8bb6aa9900eab878992d9ff0c53266debd87472f31ab26a6a3fb55880","side":"left"},{"sibling":"a04392fb9f2a3a620840e3b3fecd93d12e6d224e481c162389d8ae64e7194599","side":"left"},{"sibling":"c300cf0154c136afc09b1702a0be98f4ba5b6dc5cf57e8cc714ec1eaf4196eff","side":"left"},{"sibling":"d841ad93efda0869e5eb97678f348f03f5caab4353e05ff4bf18f47fb945b822","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":251380,"merkle_root":"e042805d07cd8dc777d49695ad78b8d4ec9721df271ff0245d7706773c30b4a5","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260626T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-26T05:37:59Z","sig_algorithm":"ed25519","signature":"c68a6e827804771acd244208495c5e35c6f417307db3c76192f038dedaa5b5f86e019074a3bea353357ff7457924c4f7907832638bb9a4d3d18e7a7f50c6a10b","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_b45944a1dc392db4ce708a5b728f6fef7ca318436446368781f894ab7966a22a"}}