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Deep Learning Based on Randomized Quasi-Monte Carlo Method for Solving Linear Kolmogorov Partial Differential Equation

Jichang Xiao, Fengjiang Fu, Xiaoqun Wang

TL;DR

This work addresses solving high-dimensional linear Kolmogorov PDEs via deep learning by replacing MC-based data sampling with randomized quasi-Monte Carlo (RQMC) sampling in the empirical risk minimization framework. The authors derive a decomposition of the learning error into approximation and generalization components and prove that, under boundary-growth conditions, the mean generalization error decays as $O(n^{-1+ε})$ for RQMC and as $O(n^{-1/2+ε})$ for MC, with $ε>0$ arbitrarily small. They validate the theory through numerical experiments on the heat equation and Black–Scholes PDEs, showing that RQMC consistently yields smaller relative $L^{2}$ errors and more stable training, especially in lower dimensions. The results suggest that using scrambled digital nets in DL-based PDE solvers can substantially improve accuracy and efficiency for high-dimensional problems, though gains diminish with increasing dimension.

Abstract

Deep learning algorithms have been widely used to solve linear Kolmogorov partial differential equations~(PDEs) in high dimensions, where the loss function is defined as a mathematical expectation. We propose to use the randomized quasi-Monte Carlo (RQMC) method instead of the Monte Carlo (MC) method for computing the loss function. In theory, we decompose the error from empirical risk minimization~(ERM) into the generalization error and the approximation error. Notably, the approximation error is independent of the sampling methods. We prove that the convergence order of the mean generalization error for the RQMC method is $O(n^{-1+ε})$ for arbitrarily small $ε>0$, while for the MC method it is $O(n^{-1/2+ε})$ for arbitrarily small $ε>0$. Consequently, we find that the overall error for the RQMC method is asymptotically smaller than that for the MC method as $n$ increases. Our numerical experiments show that the algorithm based on the RQMC method consistently achieves smaller relative $L^{2}$ error than that based on the MC method.

Deep Learning Based on Randomized Quasi-Monte Carlo Method for Solving Linear Kolmogorov Partial Differential Equation

TL;DR

This work addresses solving high-dimensional linear Kolmogorov PDEs via deep learning by replacing MC-based data sampling with randomized quasi-Monte Carlo (RQMC) sampling in the empirical risk minimization framework. The authors derive a decomposition of the learning error into approximation and generalization components and prove that, under boundary-growth conditions, the mean generalization error decays as for RQMC and as for MC, with arbitrarily small. They validate the theory through numerical experiments on the heat equation and Black–Scholes PDEs, showing that RQMC consistently yields smaller relative errors and more stable training, especially in lower dimensions. The results suggest that using scrambled digital nets in DL-based PDE solvers can substantially improve accuracy and efficiency for high-dimensional problems, though gains diminish with increasing dimension.

Abstract

Deep learning algorithms have been widely used to solve linear Kolmogorov partial differential equations~(PDEs) in high dimensions, where the loss function is defined as a mathematical expectation. We propose to use the randomized quasi-Monte Carlo (RQMC) method instead of the Monte Carlo (MC) method for computing the loss function. In theory, we decompose the error from empirical risk minimization~(ERM) into the generalization error and the approximation error. Notably, the approximation error is independent of the sampling methods. We prove that the convergence order of the mean generalization error for the RQMC method is for arbitrarily small , while for the MC method it is for arbitrarily small . Consequently, we find that the overall error for the RQMC method is asymptotically smaller than that for the MC method as increases. Our numerical experiments show that the algorithm based on the RQMC method consistently achieves smaller relative error than that based on the MC method.
Paper Structure (15 sections, 15 theorems, 106 equations, 6 figures, 2 tables)

This paper contains 15 sections, 15 theorems, 106 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General deep learning framework for solving linear Kolmogorov PDEs
  4. Randomized quasi-Monte Carlo methods
  5. Analysis of the estimation error
  6. Main Results
  7. RQMC-based convergence rate
  8. MC-based convergence rate
  9. Applications for linear Kolmogorov PDE with affine drift and diffusion
  10. Numerical Experiments
  11. Heat equation
  12. Black-Scholes model
  13. Conclusion
  14. Low variation extension
  15. Rademacher complexity

Key Result

Lemma 2.1

Let $d \in \mathbb{N}$, $T,L \in (0,\infty)$, $a\in \mathbb{R}$, $b \in (a,\infty)$, let $\mu(x)$ and $\sigma(x)$ in eq:target satisfy for every $x, y \in \mathbb{R}^d$ that where $\left\Vert\cdot\right\Vert_{2}$ is the Euclidean norm and $\left\Vert\cdot\right\Vert_{HS}$ is the Hilbert-Schmidt norm. Let the function $u^*(t,x) \in C^{1,2}([0,T]\times\mathbb{R}^{d},\mathbb{R})$ be the solution of

Figures (6)

  • Figure 1: Average relative $L^{2}$ error vs. batchsize for solving heat equation in dimension 5 and 50.
  • Figure 2: The relative $L^{2}$ error of the training process for solving heat equation in dimension 5 and 50, the batchsize is chosen to be $2^{18}$.
  • Figure 3: The projection on $(0,1)^{2}$ with $x_{i} = 1/2,i=3,\dots,50$. The exact solution and the absolute error are calculated point-wisely on a $50\times 50$ uniform grid.
  • Figure 4: Average relative $L^{2}$ error vs. batchsize for solving Black-Scholes PDE in dimension 5 and 50.
  • Figure 5: The relative $L^{2}$ error of the training process for solving Black-Scholes PDE in dimension 5 and 50, the batchsize is chosen to be $2^{18}$.
  • ...and 1 more figures

Theorems & Definitions (25)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 15 more