{"paper":{"title":"The critical slowing down in diffusion models","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Two-layer networks with local score approximation reduce critical slowing down in diffusion models to logarithmic scaling.","cross_cats":["cond-mat.stat-mech","cs.AI","cs.LG","physics.comp-ph"],"primary_cat":"cond-mat.dis-nn","authors_text":"Giulio Biroli, Luca Maria Del Bono, Marylou Gabri\\'e, Patrick Charbonneau","submitted_at":"2026-05-12T18:00:02Z","abstract_excerpt":"Computational sampling has been central to the sciences since the mid-20th century. While machine-learning-based approaches have recently enabled major advances, their behavior remains poorly understood, with limited theoretical control over when and why they succeed. Here we provide such insight for diffusion models-a class of generative schemes highly effective in practice-by analyzing their application to the $O(n)$ model of statistical field theory in the Gaussian limit $n \\to \\infty$. In this analytically tractable setting, we show that training a score model with a one-layer network arch"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Using a two-layer architecture drastically reduces the critical slowing down, with the training time scaling logarithmically rather than quadratically with system size. By introducing a local score approximation we show that this acceleration in training time can be achieved without increasing the number of neural network parameters.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the Gaussian limit n→∞ of the O(n) model and the one-layer network exactly matching its score function are representative of the critical slowing down that occurs in practical diffusion models trained on finite-n or non-Gaussian systems.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Diffusion models on the Gaussian O(n) model exhibit critical slowing down with shallow networks that deeper local score approximations can reduce to logarithmic training-time scaling.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Two-layer networks with local score approximation reduce critical slowing down in diffusion models to logarithmic scaling.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"eca8e5f6475974899fc805810158feb2730e8961f735c8d038ee5012113f2e15"},"source":{"id":"2605.12597","kind":"arxiv","version":1},"verdict":{"id":"b46391af-3144-4687-bc59-462a9e20b203","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:26:53.131007Z","strongest_claim":"Using a two-layer architecture drastically reduces the critical slowing down, with the training time scaling logarithmically rather than quadratically with system size. By introducing a local score approximation we show that this acceleration in training time can be achieved without increasing the number of neural network parameters.","one_line_summary":"Diffusion models on the Gaussian O(n) model exhibit critical slowing down with shallow networks that deeper local score approximations can reduce to logarithmic training-time scaling.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the Gaussian limit n→∞ of the O(n) model and the one-layer network exactly matching its score function are representative of the critical slowing down that occurs in practical diffusion models trained on finite-n or non-Gaussian systems.","pith_extraction_headline":"Two-layer networks with local score approximation reduce critical slowing down in diffusion models to logarithmic scaling."},"references":{"count":103,"sample":[{"doi":"","year":null,"title":"(14) with the Fourier space kernel in Eq","work_id":"571ae045-fc3d-4825-b871-b32e3e243733","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"(22)—the generation dynamics Eq","work_id":"02d93bd8-e575-436b-9940-8634c344ab8a","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"MUR PON Ricerca e Innovazione 2014-2020","work_id":"0948c2cc-3eb4-43b1-a8c9-18b42faf7b23","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"G.Battimelli, G.Ciccotti, P.Greco,andG.Giobbi,Com- puter Meets Theoretical Physics: The New Frontier of Molecular Simulation(Springer, 2020)","work_id":"48304f98-11fa-4ee6-a2d2-991a32303c28","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1983,"title":"S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Optimiza- tion by simulated annealing, Science220, 671 (1983)","work_id":"ea890f0e-9948-4b57-b753-a2a2901fe660","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":103,"snapshot_sha256":"67e84c09a32c729eda40f42f6f125559590ca7328912d538469d607299ce4738","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"28cdbbc5a67b0181ef897c353969a1b221d878d9665d6815367a873122a97ab4"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}