{"paper":{"title":"A deep backward regression-based scheme for high-dimensional nonlinear partial differential equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The deep backward regression scheme solves high-dimensional nonlinear PDEs by turning stochastic losses into deterministic conditional expectations.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Qiang Han, Shaolin Ji, Yunzhang Li","submitted_at":"2026-03-16T01:54:09Z","abstract_excerpt":"A deep backward regression-based (DBR) scheme for solving high-dimensional nonlinear parabolic partial differential equations is proposed. Building upon the seminal DBDP method introduced by Hur\\'{e} , Pham and Warin [29], our algorithm introduces a novel reformulation of local loss functions optimized sequentially via backward induction. At the heart of this approach is the transformation of simulated backward stochastic difference equations into their conditional expectation representations, thereby recasting a projection-based stochastic optimization problem as a robust deterministic functi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Numerical experiments demonstrate that the DBR scheme consistently outperforms the DBDP1 method; notably, for complex unbounded PDEs, DBR maintains high accuracy in regimes where DBDP1 fails to converge beyond d=10. Theoretically, we derive rigorous upper error bounds and establish half-order convergence for the proposed scheme.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The transformation of simulated backward stochastic difference equations into their conditional expectation representations can be accurately approximated by neural networks without introducing bias that invalidates the error bounds or the observed stability gains.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A new DBR algorithm reformulates backward stochastic difference equations via conditional expectations to reduce variance and improve accuracy for high-dimensional nonlinear parabolic PDEs, outperforming DBDP1 beyond dimension 10.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The deep backward regression scheme solves high-dimensional nonlinear PDEs by turning stochastic losses into deterministic conditional expectations.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"42acae3b4198d63104e65f309545c195e70d76b662b88ede75975cfd5a03a012"},"source":{"id":"2603.14721","kind":"arxiv","version":2},"verdict":{"id":"e4b69606-f70e-443c-863b-95c996557255","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T10:55:02.166700Z","strongest_claim":"Numerical experiments demonstrate that the DBR scheme consistently outperforms the DBDP1 method; notably, for complex unbounded PDEs, DBR maintains high accuracy in regimes where DBDP1 fails to converge beyond d=10. Theoretically, we derive rigorous upper error bounds and establish half-order convergence for the proposed scheme.","one_line_summary":"A new DBR algorithm reformulates backward stochastic difference equations via conditional expectations to reduce variance and improve accuracy for high-dimensional nonlinear parabolic PDEs, outperforming DBDP1 beyond dimension 10.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The transformation of simulated backward stochastic difference equations into their conditional expectation representations can be accurately approximated by neural networks without introducing bias that invalidates the error bounds or the observed stability gains.","pith_extraction_headline":"The deep backward regression scheme solves high-dimensional nonlinear PDEs by turning stochastic losses into deterministic conditional expectations."},"references":{"count":38,"sample":[{"doi":"","year":2019,"title":"Anil, C., Lucas, J. and Grosse, R. (2019).Sorting out Lipschitz function approximation. In International conference on machine learning (pp. 291-301). PMLR","work_id":"22877ebb-f8f1-4e90-a8c3-594e21c26b35","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"Bouchard, B. and Touzi, N. (2004).Discrete-time approximation and Monte-Carlo simulation of back- ward stochastic differential equations. Stoch. Process. Their Appl., 111(2), 175-206","work_id":"bb2ad229-a014-4ef5-836e-1f3c0ad46a8f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2008,"title":"Bouchard, B. and Chassagneux, J. F. (2008).Discrete-time approximation for continuously and dis- cretely reflected BSDEs.Stoch. Process. Their Appl, 118(12), 2269-2293","work_id":"0154a4e8-867c-46a6-96c9-bf411f927476","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"Bungartz, H. J. and Griebel, M. (2004).Sparse grids. Acta Numer., 13, 147-269. 29","work_id":"a1a9e7ed-69de-46e7-bf9a-174e2cef8093","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Cai, W., Fang, S. and Zhou, T. (2025).SOC-MartNet: A martingale neural network for the Hamilton- Jacobi-Bellman equation without explicit in stochastic optimal controls. SIAM J. Sci. Comput., 47(4), C","work_id":"48b14135-315e-4e79-b7c0-96d4fde2eb15","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":38,"snapshot_sha256":"4d29898afab06e3e2f778dd480bc79904ba290c809f80379115b5b9ded2213c2","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"7c297c1f0ddd98b09678eacbf9fe72361a1d6144b4a7c0b3983fe456171110bf"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}