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XNet-Enhanced Deep BSDE Method and Numerical Analysis

Xiaotao Zheng, Zhihong Xia, Xin Li, Xingye Yue

TL;DR

This work tackles the curse of dimensionality in solving high-dimensional semilinear parabolic PDEs by enhancing the Deep BSDE framework with a novel XNet architecture. XNet, based on a Cauchy-approximation mechanism, achieves arbitrary-order approximation using O(L) parameters, offering improved accuracy and computational efficiency over traditional two-layer networks. The paper provides theoretical error decompositions for DBSDE (approximation, generalization, and optimization) and demonstrates substantial empirical gains on 100-dimensional Allen–Cahn and PricingDiffrate PDEs in both discrete-time and continuous-time settings. By reducing network-approximation and optimization errors and enabling scalable high-dimensional solvers, XNet potentially sets a new standard for DBSDE-based PDE computation in finance and physics.

Abstract

Solving high-dimensional semilinear parabolic partial differential equations (PDEs) challenges traditional numerical methods due to the "curse of dimensionality." Deep learning, particularly through the Deep BSDE method, offers a promising alternative by leveraging neural networks' capability to approximate high-dimensional functions. This paper introduces a novel network architecture, XNet, which significantly enhances the computational efficiency and accuracy of the Deep BSDE method. XNet demonstrates superior approximation capabilities with fewer parameters, addressing the trade-off between approximation and optimization errors found in existing methods. We detail the implementation of XNet within the Deep BSDE framework and present results that show marked improvements in solving high-dimensional PDEs, potentially setting a new standard for such computations.

XNet-Enhanced Deep BSDE Method and Numerical Analysis

TL;DR

This work tackles the curse of dimensionality in solving high-dimensional semilinear parabolic PDEs by enhancing the Deep BSDE framework with a novel XNet architecture. XNet, based on a Cauchy-approximation mechanism, achieves arbitrary-order approximation using O(L) parameters, offering improved accuracy and computational efficiency over traditional two-layer networks. The paper provides theoretical error decompositions for DBSDE (approximation, generalization, and optimization) and demonstrates substantial empirical gains on 100-dimensional Allen–Cahn and PricingDiffrate PDEs in both discrete-time and continuous-time settings. By reducing network-approximation and optimization errors and enabling scalable high-dimensional solvers, XNet potentially sets a new standard for DBSDE-based PDE computation in finance and physics.

Abstract

Solving high-dimensional semilinear parabolic partial differential equations (PDEs) challenges traditional numerical methods due to the "curse of dimensionality." Deep learning, particularly through the Deep BSDE method, offers a promising alternative by leveraging neural networks' capability to approximate high-dimensional functions. This paper introduces a novel network architecture, XNet, which significantly enhances the computational efficiency and accuracy of the Deep BSDE method. XNet demonstrates superior approximation capabilities with fewer parameters, addressing the trade-off between approximation and optimization errors found in existing methods. We detail the implementation of XNet within the Deep BSDE framework and present results that show marked improvements in solving high-dimensional PDEs, potentially setting a new standard for such computations.

Paper Structure

This paper contains 14 sections, 6 theorems, 29 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Deep BSDE Method
  3. Forward Backward Stochastic Differential Equation (FBSDE)
  4. Implementation of the Deep BSDE Method
  5. Network Architecture for the Deep BSDE Method: Two-layer Net and XNet
  6. Approximation errors, Generalization errors, and Optimization errors of the Deep BSDE method for coupled FBSDEs
  7. Approximation Errors for Deep BSDE method
  8. Generalization Errors and Optimazation Errors for Deep BSDE method
  9. Discrete time models
  10. Allen-Cahn Equation
  11. Pricing of European financial derivatives with different interest rates for borrowing and lending(PricingDiffrate) equation
  12. Continuous time models
  13. Allen-Cahn Equation
  14. Pricing of European financial derivatives with different interest rates for borrowing and lending(PricingDiffrate) equation

Key Result

Lemma 1

Assume $(\Omega,\mathcal{F},\mathbb{P})$ represents a probability space, and let $W=\left(W^{(1)}, \ldots, W^{(d)}\right):[0, T] \times \Omega \rightarrow \mathbb{R}^d$ be a standard Brownian motion, with $\mathcal{F}_t$ being a non-decreasing filtration generated by $W$. Let $X=\left(X^{(1)}, \ldot Under certain conditions, the BSDE is well-posed and corresponds to the PDE $($bxxpwxpde$)$. Specif

Figures (7)

  • Figure 1: The neural network architecture for Deep BSDE method. The network consists of multiple ($N-1$) sub-networks, with each sub-network corresponding to a time interval. Each sub-network has $H$ network parameters. It should be noted that in addition to these, ${\theta _{{u_0}}}$ and ${\theta _{\nabla {u_0}}}$ are also network parameters that need to be optimized.
  • Figure 2: Comparison of Two Network Architectures for Solving the Allen-Cahn Equation under 20-step-time Discretization
  • Figure 3: Comparison of Two Network Architectures for Solving the Allen-Cahn Equation under 80-step-time discretization
  • Figure 4: Comparison of Two Network Architectures for Solving the PricingDiffrate Equation under 20-step-time Discretization and 80-step-time Discretization
  • Figure 5: Results of solving the Allen-Cahn Equation using the Deep BSDE method by XNet under $N$-time-step discretization, with $N=10$, $40$, and $160$.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • Lemma 4
  • Definition 1