Unbiased and Second-Order-Free Training for High-Dimensional PDEs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916012561006592 |
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| author | Seo, Jaemin Lee, Surin Lee, Jae Yong |
| author_facet | Seo, Jaemin Lee, Surin Lee, Jae Yong |
| contents | Deep learning methods based on backward stochastic differential equations (BSDEs) have emerged as competitive alternatives to physics-informed neural networks (PINNs) for solving high-dimensional partial differential equations (PDEs). By leveraging probabilistic representations, BSDE approaches can avoid the curse of dimensionality and often admit second-order-free training objectives that do not require explicit Hessian evaluations. It has recently been established that the commonly used Euler-Maruyama (EM) time discretization induces an intrinsic bias in BSDE training losses. While high-order schemes such as Heun can fully eliminate this bias, such schemes re-introduce second-order spatial derivatives and incur substantial computational overhead. In this work, we provide a principled analysis of EM-induced loss bias and propose an unbiased, second-order-free training framework that preserves the computational advantages of BSDE methods. Our code is available at https://github.com/seojaemin22/Un-EM-BSDE. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14643 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unbiased and Second-Order-Free Training for High-Dimensional PDEs Seo, Jaemin Lee, Surin Lee, Jae Yong Machine Learning Numerical Analysis Optimization and Control 65C30, 68TO7 Deep learning methods based on backward stochastic differential equations (BSDEs) have emerged as competitive alternatives to physics-informed neural networks (PINNs) for solving high-dimensional partial differential equations (PDEs). By leveraging probabilistic representations, BSDE approaches can avoid the curse of dimensionality and often admit second-order-free training objectives that do not require explicit Hessian evaluations. It has recently been established that the commonly used Euler-Maruyama (EM) time discretization induces an intrinsic bias in BSDE training losses. While high-order schemes such as Heun can fully eliminate this bias, such schemes re-introduce second-order spatial derivatives and incur substantial computational overhead. In this work, we provide a principled analysis of EM-induced loss bias and propose an unbiased, second-order-free training framework that preserves the computational advantages of BSDE methods. Our code is available at https://github.com/seojaemin22/Un-EM-BSDE. |
| title | Unbiased and Second-Order-Free Training for High-Dimensional PDEs |
| topic | Machine Learning Numerical Analysis Optimization and Control 65C30, 68TO7 |
| url | https://arxiv.org/abs/2605.14643 |