| Abstract: |
| The curse of dimensionality makes it difficult to solve high-dimensional partial differential equations (PDEs) numerically. The nonlinear Feynman-Kac formula gives a probabilistic link between these PDEs and forward-backward stochastic differential equations (FBSDEs), but classical deep-learning solvers often become unstable for strongly coupled problems. We introduce a Deep Truncated FBSDE method for high-dimensional quasilinear PDEs that overcomes these limitations. The method uses the gradient truncation and applies iterative decoupling to separate the forward and backward parts, reducing unstable feedback between them. We also rewrite the PDE as equivalent coupled subsystems that are easier to optimize and enforce pathwise consistency through repeated refinement to control errors. Theoretical results and numerical tests show this method improves stability, convergence, and accuracy, offering a scalable and reliable tool for high-dimensional problems. |
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