A deep backward regression-based scheme for high-dimensional nonlinear partial differential equations
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| Format: | Preprint |
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2026
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| _version_ | 1866918516022575104 |
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| author | Han, Qiang Ji, Shaolin Li, Yunzhang |
| author_facet | Han, Qiang Ji, Shaolin Li, Yunzhang |
| contents | We propose a deep backward regression-based (DBR) scheme for solving high-dimensional nonlinear parabolic partial differential equations. Building on the DBDP method of Huré, Pham, and Warin~\cite{HCPHWX20}, the proposed method reformulates the local backward losses through conditional expectations and trains the resulting regression problems sequentially in time. This conditional-expectation formulation replaces pathwise Brownian fluctuations in the Euler residual by their averaged effect and therefore provides an intrinsic variance-reduction mechanism before loss evaluation. In practice, the conditional expectations are approximated by local multi-path Monte Carlo averages, which leads to smoother training targets and improved numerical stability. Numerical experiments show that DBR performs competitively on standard high-dimensional benchmarks and is more stable than DBDP1 on the challenging unbounded benchmark considered in Example~2. Under an idealized population-loss minimization setting, we provide an error analysis and establish a half-order convergence result under suitable approximation and integrability assumptions. We also discuss an extension to variational inequalities. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_14721 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A deep backward regression-based scheme for high-dimensional nonlinear partial differential equations Han, Qiang Ji, Shaolin Li, Yunzhang Numerical Analysis 60H35, 65C20, 65M12 We propose a deep backward regression-based (DBR) scheme for solving high-dimensional nonlinear parabolic partial differential equations. Building on the DBDP method of Huré, Pham, and Warin~\cite{HCPHWX20}, the proposed method reformulates the local backward losses through conditional expectations and trains the resulting regression problems sequentially in time. This conditional-expectation formulation replaces pathwise Brownian fluctuations in the Euler residual by their averaged effect and therefore provides an intrinsic variance-reduction mechanism before loss evaluation. In practice, the conditional expectations are approximated by local multi-path Monte Carlo averages, which leads to smoother training targets and improved numerical stability. Numerical experiments show that DBR performs competitively on standard high-dimensional benchmarks and is more stable than DBDP1 on the challenging unbounded benchmark considered in Example~2. Under an idealized population-loss minimization setting, we provide an error analysis and establish a half-order convergence result under suitable approximation and integrability assumptions. We also discuss an extension to variational inequalities. |
| title | A deep backward regression-based scheme for high-dimensional nonlinear partial differential equations |
| topic | Numerical Analysis 60H35, 65C20, 65M12 |
| url | https://arxiv.org/abs/2603.14721 |