{"_canonicalization":{"envelope_id":"axm_ + sha256(envelope minus {signature, axiom_id, anchors})","envelope_signature":"ed25519(envelope minus {signature, axiom_id})","json":"sort_keys=True, separators=(',',':'), ensure_ascii=False, allow_nan=False, utf-8","leaf_hash":"sha256(0x00 || canonical_json(envelope_full))","seal_signature":"ed25519(seal minus {signature, sig_algorithm})"},"axiom_id":"axm_2a1e31d695e485cc8670bd40831f6c201d6877dceb6eec83eff1498402971903","bitcoin_anchor":{"bitcoin_attestations":[],"calendar_attestations":[],"ots_url":"","stamped_at":"","status":"pending_next_stamp"},"envelope":{"anchors":[{"chain":"crovia.axiom_graph","height":0,"merkle_proof":"spider_vendor_press_v1","root_at_anchor":"spider_vendor_press_v1"}],"axiom_id":"axm_2a1e31d695e485cc8670bd40831f6c201d6877dceb6eec83eff1498402971903","axiom_type":"AX.OBS","body":{"axiom_subtype":"news.vendor_press.v1","category":"news","fingerprint":"a96c6f08c2ae8771051ce8be67a856b48d61bb12022e9c3673bbb3b9a634858a","published":"Tue, 16 Jun 2026 00:00:00 -0400","receipt_hash":"a96c6f08c2ae8771051ce8be67a856b48d61bb12022e9c3673bbb3b9a634858a","schema":"spider.news.vendor_press.v1","spider":"vendor_press","spider_record":{"axiom_subtype":"news.vendor_press.v1","category":"news","decision_hint":"POSITIVE","envelope_target":"AX.OBS","fingerprint":"a96c6f08c2ae8771051ce8be67a856b48d61bb12022e9c3673bbb3b9a634858a","observed_at":"2026-06-16T04:43:43.281320Z","parent_run_hash":"eb6edcf82c3507c59161a4ab46d2e904e507004f44677402bb24d106997ed7c2","published":"Tue, 16 Jun 2026 00:00:00 -0400","runtime_version":"0.1.0","schema":"spider.news.vendor_press.v1","source_status":200,"source_url":"https://export.arxiv.org/rss/cs.AI","spider":"vendor_press","summary_excerpt":"arXiv:2309.07401v2 Announce Type: replace-cross \nAbstract: Deep neural networks (DNNs) show great promise for solving partial differential equations (PDEs), but their deep architectures introduce complex, large-scale, non-convex optimization challenges. Nonlinear PDEs, like the viscous Burgers' equation, compound these difficulties due to steep gradients and shock-like solutions. To address this, we propose a two-stage multi-grade deep learning (TS-MGDL) method. In the first stage, shallow networks are trained progressively grade by grade to fit the target function from low- to high-frequency components; previously learned grades are frozen, and each new residual block is trained solely to minimize the remaining approximation error. The second stage unfreezes and retrains selected layers using the first-stage network as initialization, achieving an interpretable, stable hierarchical refinement while mitigating optimization complexity. Furthermore, we theoretically prove that each grade","title":"Multi-Grade Deep Learning for Partial Differential Equations with Applications to the Burgers Equation","url":"https://arxiv.org/abs/2309.07401","vendor":"arxiv_cs_ai"},"summary":"arXiv:2309.07401v2 Announce Type: replace-cross \nAbstract: Deep neural networks (DNNs) show great promise for solving partial differential equations (PDEs), but their deep architectures introduce complex, large-scale, non-convex optimization challenges. Nonlinear PDEs, like the viscous Burgers' equation, compound these difficulties due to steep gradients and shock-like solutions. To address this, we propose a two-stage multi-grade deep learning (TS-MGDL) method. In the first stage, shallow networks are trained progressively grade by grade to fit the target function from low- to high-frequency components; previously learned grades are frozen, and each new residual block is trained solely to minimize the remaining approximation error. The second stage unfreezes and retrains selected layers using the first-stage network as initialization, achieving an interpretable, stable hierarchical refinement while mitigating optimization complexity. Furthermore, we theoretically prove that each grade","title":"Multi-Grade Deep Learning for Partial Differential Equations with Applications to the Burgers Equation","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-16T04:43:43Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2309.07401"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:0f0f4a3e4fafe2bd92dec86c808cd0db6cb91e485dd4d18918f164a3bd25e6425407c3e1b6fcd636e815a509672957930fdcedca72175265af22ee726e8b270f","signer":"crovia.substrate","subject":{"observed_at":"2026-06-16T04:43:43Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2309.07401"},"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":"05cc4480eab5c53f31c0a886e1dd8bf821f930d31e0207b0b0457bb57817d580","leaf_index":230677,"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":"7112543b9e9ba1ca004603373358a5644e25ca6b6e6231018d4071379f74956a","side":"left"},{"sibling":"ef797b3c7d31ac7f70b8cd5daa54941efca118f93f4fb4b39323d16e53bab14e","side":"right"},{"sibling":"31c1f7817266446de14eff5385d23a259a6604523165cf46bf0358427fd537a2","side":"left"},{"sibling":"6adf0b9f3f86efe1ef5163440842a4aaec7a548f53858d78d155a87fae6c527b","side":"right"},{"sibling":"a5a9a8a666b7915d5c56709ca1e51c2e23b1240ae97220cc8effc279437f99eb","side":"left"},{"sibling":"120ca24907f6650fa144149c3bc36bfd6ae080480737291c8d1cbd326fd56651","side":"right"},{"sibling":"427fbf2f7401da34e38f276c49611c2b1cf6e816c0a116d36f0e04e54f7562d3","side":"right"},{"sibling":"7a295c86e2ca5719bf2e33d1fcc4bc631f05c6f31aeb907113e2c58401e3087c","side":"right"},{"sibling":"ee59602dad0bf74c97a32a93f0a1a19e7a12f2800791988a6fdf35611febe031","side":"left"},{"sibling":"0c5669692381d605223c74b8d30f70cd308e77e33d5e40ea84bb7b4f84f2d4d9","side":"right"},{"sibling":"d5b9f8b1a2c9f6a46e17982dfbe6ce1f3b5fa4e730220397f2253d114dcc8486","side":"left"},{"sibling":"d10d772a4984cae00e65ab24af21d1d260e475bbaae3f17871859eb705bd3999","side":"right"},{"sibling":"e79159853f2f35ddae8e3247d515e433c534277b287d65bbd77ae989aa4992fa","side":"right"},{"sibling":"3054319f1840cce0eaaf0bc4b1ae38e5bf8b6210927d924a750775cc7d77cca6","side":"right"},{"sibling":"0fd8b5059f279c4a4a6688de2472fdbc25543fee183df19b21dacca880354cff","side":"right"},{"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":232015,"merkle_root":"62bfb7809bb55667ad7eeebdb48267b9b2c1ee89bb808ea1e6d3a814a15aa402","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260617T133701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-17T13:39:59Z","sig_algorithm":"ed25519","signature":"930a3563c643cc7518d048b12a1f5392a96a5edce49533ba0a56f11b6cc69319bb1c800aacc46b52a6941c4d05dae255a45cac757d6093c971b96852c24f140e","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_2a1e31d695e485cc8670bd40831f6c201d6877dceb6eec83eff1498402971903"}}