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We introduce \\EMNGDfull{}, a manifold optimization framework for physics-informed and variational neural PDE solvers whose parameters lie on a Riemannian manifold. EMNGD restricts the energy-induced quadratic model to feasible tangent directions and uses retractions to preserve parameter constraints throughout optimization. Under coercivity, we prove that the push-forward of the undamped EMNGD direction is the best feasible approximation to the function-space Newton vector in the energy metric. We establish coordinate invariance, exact reduction to ENGD in Euclidean space, global first-order convergence with Armijo backtracking, and robustness to inexact tangent solves. For quadratic residual energies and generalized Gauss--Newton pullbacks, the Woodbury identity transfers the tangent sys","title":"Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers","url":"https://arxiv.org/pdf/2607.22004v1","vendor":"arxiv"},"summary":"Energy natural gradient descent (ENGD) aligns parameter updates with the curvature of an underlying function-space energy, but existing formulations assume an unconstrained Euclidean parameter domain. We introduce \\EMNGDfull{}, a manifold optimization framework for physics-informed and variational neural PDE solvers whose parameters lie on a Riemannian manifold. EMNGD restricts the energy-induced quadratic model to feasible tangent directions and uses retractions to preserve parameter constraints throughout optimization. Under coercivity, we prove that the push-forward of the undamped EMNGD direction is the best feasible approximation to the function-space Newton vector in the energy metric. We establish coordinate invariance, exact reduction to ENGD in Euclidean space, global first-order convergence with Armijo backtracking, and robustness to inexact tangent solves. For quadratic residual energies and generalized Gauss--Newton pullbacks, the Woodbury identity transfers the tangent sys","title":"Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers","vendor":"arxiv"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-07-27T05:56:17Z","notes":"Spider arxiv_retraction (research) research.arxiv_retraction.v1","object":{"captured_by":"crovia.spider.arxiv_retraction","primary_source_url":"https://arxiv.org/pdf/2607.22004v1"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:727ffb9491d8e47613fcbbbcba69b67ae77312c4125a3ae191f4b9e409a3e861c73def6d025e9b3560f9a88fb71524ccaa78e06001452142f2eca4b98253db06","signer":"crovia.substrate","subject":{"observed_at":"2026-07-27T05:56:17Z","source_collector":"spider:arxiv_retraction","target_id":"https://arxiv.org/pdf/2607.22004v1"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/research/arxiv_retraction_v1.jsonl\",\"source_seal_merkle_root\":\"spider_arxiv_retraction_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"d43bf9892d293dabc1485cee74359e1625f1d7b16a0f6dda10667137bf4a17ff","leaf_index":357436,"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":"c27b15ebcce8367f5e49887f65f5fba5d7caed1488f94315bcfef3796880a35f","side":"right"},{"sibling":"59107d76dd5b7db474d3cf2269ecb27f7344b7d27f23ec9cf3b0491ca8b2087c","side":"right"},{"sibling":"1f0f1853d8d76fb1bb65dd445adee750432205d01d3107766365d6c22e90ce61","side":"left"},{"sibling":"73330a60308eb1a7e3fdbaeb700a4fb7b1077c8de55b8d2230ec42f79d2ee97a","side":"left"},{"sibling":"129e0353e2727a2664910fa0543852210c21cf050428c69456061298526b8281","side":"left"},{"sibling":"7d8af9d2aceab68773e6f7b0c40c08d6d69c8531757e2ea4d79296f457a099f1","side":"left"},{"sibling":"ff88ab03d7e151981930197a45f6a6612b1d7eda6895824f5c072caa3d122489","side":"right"},{"sibling":"22ceb40d903b8880d5c09ed9638c71b21d6d03eebcda241e0aab704202b9c613","side":"right"},{"sibling":"d41880931d7cbe937ba0cb1658acfa67f250dc2752c9e7d67b86d5c21bf2593e","side":"right"},{"sibling":"cbd60230bf12725472bc76daa70fb50954fa698144930d77c563ca2f789b0130","side":"right"},{"sibling":"4d2a614793acf8e504d5e123c9908b54ab1245ab361a50b2a7099b66a71e4054","side":"left"},{"sibling":"ff606ff69698eb0307c13a7eea3c1770e157c7f6e48126cba99fffea49911c16","side":"right"},{"sibling":"cff500acb83b14a8a7195a72d90dd4b7a6b8a28fc1069f8300b1191728718f90","side":"left"},{"sibling":"51ee2566d84aba54b9d07233fb460f9fab044b4edfa3fbe376fa1df0727acfa5","side":"left"},{"sibling":"f3e45bceed774d2402fa45d41ff5190f295823bd2f216eb90157884150034693","side":"left"},{"sibling":"45cba1486a7d165a2ae09b178ef9b1c8e19ca01d7b74df995f3ba064a2da052c","side":"right"},{"sibling":"77025bcb374a7ad74f520643e20a8ae1205a7ee78507b0beb93117760f1c29d3","side":"left"},{"sibling":"4db76682448080bc1df0d6843c02aa6a73a97528f17c7601dc5e2838ca54e19e","side":"right"},{"sibling":"1cecb7f447febd025aac272837c80de218aecc6485d2395a509b2a1f1b9c746e","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":357550,"merkle_root":"4e6200fa754eb40650f3a77f4386128e92f29acbce21e6b5fb421f7b062780b8","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260727T063701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-07-27T06:38:46Z","sig_algorithm":"ed25519","signature":"57812646c048fa6af15c1201d989cff4559e1b3b27548774f6bc584eec1df6e42b1a066c0cd026cb010043b9aae83ac376eca513479dad959efbe1fda2e93a05","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_e5cd33e4b94f824c4bd547056fd43fda25a756c1cec650b44d85dc88d6d56fdd"}}