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To mitigate this measure-induced bias, we propose M$^3$ (Multi-scale Morton Measure), a scalable framework that balances training measures by partitioning space according to physical variation and allocating supervision across multiple scales. Applied to three industrial-scale datasets with diverse discretizations, M$^3$ consistently improves predictions in the continuous physical domain, achieving up to 4.7$\\times$ lower error in large-scale volumetric cases. These gains persist under aggressive subsampling (160M $\\rightarrow$ 16M $\\rightarrow$ 1.6M points), where M$^3$-trained models outperform those trained on higher-resolution data, reducing physics-weighted relative $L_2$ error by 3--4$","title":"M$^3$: Reframing Training Measures for Discretized Physical Simulations","url":"https://arxiv.org/abs/2605.08843","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.08843v1 Announce Type: new \nAbstract: Neural surrogate models for physical simulations are trained on discretized samples of continuous domains, where the induced empirical measure leads to uneven supervision, biasing optimization and causing spatial inconsistencies in physical fidelity. To mitigate this measure-induced bias, we propose M$^3$ (Multi-scale Morton Measure), a scalable framework that balances training measures by partitioning space according to physical variation and allocating supervision across multiple scales. Applied to three industrial-scale datasets with diverse discretizations, M$^3$ consistently improves predictions in the continuous physical domain, achieving up to 4.7$\\times$ lower error in large-scale volumetric cases. These gains persist under aggressive subsampling (160M $\\rightarrow$ 16M $\\rightarrow$ 1.6M points), where M$^3$-trained models outperform those trained on higher-resolution data, reducing physics-weighted relative $L_2$ error by 3--4$","title":"M$^3$: Reframing Training Measures for Discretized Physical Simulations","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-12T04:43:42Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.08843"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:d98b30399a0272b3927eef4f33b765161a44102f2f8dbbc611815b11ddab1141f2345b82c6613474b2854de959c6dac491a4aaf8141766d5b682cd35c856420f","signer":"crovia.substrate","subject":{"observed_at":"2026-05-12T04:43:42Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.08843"},"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":"a8c278e2580d10675b6c0ec7c3839210b42b135ae8d397cd73d63b9e3487d5c3","leaf_index":128280,"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":"8d148a631914d5e3e69725fabbd0c874756a19dcc57800a1c25c7c2daa6b8a66","side":"right"},{"sibling":"74c8b22ba42d7ce463e95e499348bcd35a478daa60a8aa1200d29013e5c47e04","side":"right"},{"sibling":"fe852372d0ef52eaafdf2f5cc94359d163cbbef8e2f1bdb01c4439e808f65d6b","side":"right"},{"sibling":"8e09d1c31520589aa9d83b25af240fa691ce27a1625939b3a440e4c8596bf929","side":"left"},{"sibling":"16e8854317f2ecde494482b81c681b1014a97171afb656d588617eaa83c05153","side":"left"},{"sibling":"79887ccdee3cac201aa808169f330092829ec6bb893c35133d831e6deaaceaff","side":"right"},{"sibling":"36b72adf8f4027d1a6a05d7a02a6b5b374f1f82055a2144c22a6e8e86b961927","side":"right"},{"sibling":"606e431ab526a2729c7a10ae34c1890601117997719a643180d252ace34a222e","side":"right"},{"sibling":"aa3ddc60ad25fcdd4bf93520d8baeca26d807789d80fb0deffbb19c4f8882286","side":"left"},{"sibling":"077005bbaed7b42537914de0b512ed52d33243215eab288181aba807d37bf04e","side":"right"},{"sibling":"c6eaf7a4fcab2db96e9e9423acb6922c80f64882d0f3f50d09e53a4807d23084","side":"left"},{"sibling":"df12eaabc0a370aff5d0488478f48d2b3f90d025c903643f98ea804b017688ec","side":"right"},{"sibling":"ffc4d51379293bc3e1910c7d612f409dc610fd9acf8241793fb89f82e1bad4ef","side":"left"},{"sibling":"62ac6554017807bd83187f5a3e5f4f72d6c482616429c2780e9fff1f4845fa04","side":"left"},{"sibling":"3a5e69cf0803f4c91f3895ed7c9a95748fef240bec4422e167c05300f79f06c0","side":"left"},{"sibling":"f2817ab288b5324fe49770372c7a10f33f7cd11005f8d4c0a730316f5229dc98","side":"left"},{"sibling":"725fac972e772ca0dc598810ea1abc70df472f72d2d6ab8a0baee2b80e5d2f4c","side":"left"},{"sibling":"98fc57dfef8873b512edc8340f7181df57302bb96777625e072235c62d7c5895","side":"right"}]},"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":134292,"merkle_root":"77fc9c28fae777b81da5b495b3115474df6592dfac590333213d3bdf8b94a9b3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260515T023701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-15T02:37:25Z","sig_algorithm":"ed25519","signature":"68107a834b00b24f5d4501e5ec727445311f132a486567ecc4c72a4e6dff24c8c21f2de3105293353ba5fdbe370d032819af6aa70f694e2e39b6af6737507009","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_89099baa761180cf3a845260c867f6632908a214f22947029daf011e0a41e030"}}