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Deep Feynman-Kac Methods for High-dimensional Semilinear Parabolic Equations: Revisit

Xiaotao Zheng, Xingye Yue, Jiyang Shi

TL;DR

This work tackles the challenge of solving high-dimensional semilinear parabolic PDEs by leveraging a BSDE–Feynman–Kac framework and neural networks. It introduces two global-training schemes, DS-GT and DFK-GT, along with a data-pair remodeling strategy that aligns with Monte Carlo behavior in the linear case. Across 100-dimensional tests on HJB, Allen–Cahn, and differential-rate pricing problems, the proposed methods, especially DFK-GT, demonstrate superior accuracy and computational efficiency compared with Deep BSDE and traditional Deep Splitting. The approach thus offers a scalable, accurate tool for complex high-dimensional PDEs with potential impact in finance, physics, and engineering.

Abstract

Deep Feynman-Kac method was first introduced to solve parabolic partial differential equations(PDE) by Beck et al. (SISC, V.43, 2021), named Deep Splitting method since they trained the Neural Networks step by step in the time direction. In this paper, we propose a new training approach with two different features. Firstly, neural networks are trained at all time steps globally, instead of step by step. Secondly, the training data are generated in a new way, in which the method is consistent with a direct Monte Carlo scheme when dealing with a linear parabolic PDE. Numerical examples show that our method has significant improvement both in efficiency and accuracy.

Deep Feynman-Kac Methods for High-dimensional Semilinear Parabolic Equations: Revisit

TL;DR

This work tackles the challenge of solving high-dimensional semilinear parabolic PDEs by leveraging a BSDE–Feynman–Kac framework and neural networks. It introduces two global-training schemes, DS-GT and DFK-GT, along with a data-pair remodeling strategy that aligns with Monte Carlo behavior in the linear case. Across 100-dimensional tests on HJB, Allen–Cahn, and differential-rate pricing problems, the proposed methods, especially DFK-GT, demonstrate superior accuracy and computational efficiency compared with Deep BSDE and traditional Deep Splitting. The approach thus offers a scalable, accurate tool for complex high-dimensional PDEs with potential impact in finance, physics, and engineering.

Abstract

Deep Feynman-Kac method was first introduced to solve parabolic partial differential equations(PDE) by Beck et al. (SISC, V.43, 2021), named Deep Splitting method since they trained the Neural Networks step by step in the time direction. In this paper, we propose a new training approach with two different features. Firstly, neural networks are trained at all time steps globally, instead of step by step. Secondly, the training data are generated in a new way, in which the method is consistent with a direct Monte Carlo scheme when dealing with a linear parabolic PDE. Numerical examples show that our method has significant improvement both in efficiency and accuracy.

Paper Structure

This paper contains 15 sections, 30 equations, 6 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Backward Stochastic Differential Equations and Feynman-Kac Formula
  3. Semilinear Parabolic Partial Differential Equations
  4. Backward Stochastic Differential Equation (BSDE)
  5. Feynman-Kac Formula
  6. Deep Feynman-Kac Method
  7. Deep Splitting (DS) Method
  8. Deep Splitting method under Global Trainning (DS-GT)
  9. Deep Feynman-Kac method under Global Training (DFK-GT)
  10. Numerical Results
  11. Hamilton-Jacobi-Bellman (HJB) equation
  12. Allen-Cahn equation
  13. Pricing of European financial derivatives with different interest rates for borrowing and lending(PricingDiffrate) equation
  14. Conclusions
  15. Appendix: Deep BSDE Method

Figures (6)

  • Figure 1: Deep Feynman-Kac Algorithm under Global Training. Here, we also fully connect $N-1$ sub-networks and train them as a whole. The structure of the sub-networks is the same as that in the Deep BSDE method. However, unlike that, the network here directly approximates the value function $u^\theta (t_n,X_{t_n}^m)$ instead of its gradient $\nabla u^\theta (t_n,X_{t_n}^m)$. As a result, we reduce two network parameters, namely ${\theta _{{u_0}}}$ and ${\theta _{\nabla {u_0}}}$.
  • Figure 2: Comparison of Two Methods for Solving the HJB Equation
  • Figure 3: Comparison of Two Methods for Solving the Allen-Cahn Equation
  • Figure 4: Comparison of Two methods for Solving the PricingDiffrate Equation
  • Figure 5: Comparison of Runtime and Accuracy on Different Numerical Examples (2000 Iterations)
  • ...and 1 more figures

Theorems & Definitions (1)

  • Remark 1