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<p>In this paper, we demonstrate that a class of high-dimensional stochastic partial differential equations (SPDEs) can be approximated by residual neural networks (ResNet). Under our numerical scheme, the complexity of our proposed ResNet is controllable in the case of high dimensions, growing at most polynomially. We use the splitting-up method to separate the SPDE into deterministic partial differential equation (PDE) and stochastic differential equation (SDE). In each time interval, we associate the solution of PDE with SDE by Feynman–Kac formula. Since the right endpoint of PDE and SDE for each time interval in the splitting-up method are generated by the two ResNets, the ResNet constructed by us in the proof requires repeated cross-iteration, but the complexity is still well controlled.</p>

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Numerical analysis of residual neural networks for stochastic partial differential equations

  • 期刊论文

  • 发表日期：

    2026年8月3日

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  • Juan Yang,
  • Qianlu Pan

摘要

In this paper, we demonstrate that a class of high-dimensional stochastic partial differential equations (SPDEs) can be approximated by residual neural networks (ResNet). Under our numerical scheme, the complexity of our proposed ResNet is controllable in the case of high dimensions, growing at most polynomially. We use the splitting-up method to separate the SPDE into deterministic partial differential equation (PDE) and stochastic differential equation (SDE). In each time interval, we associate the solution of PDE with SDE by Feynman–Kac formula. Since the right endpoint of PDE and SDE for each time interval in the splitting-up method are generated by the two ResNets, the ResNet constructed by us in the proof requires repeated cross-iteration, but the complexity is still well controlled.