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Deep random difference method for high-dimensional quasilinear parabolic partial differential equations

Wei Cai, Shuixin Fang, Tao Zhou

TL;DR

This work introduces the Deep Random Difference Method (DRDM) for solving high-dimensional quasilinear parabolic PDEs and Hamilton–Jacobi–Bellman (HJB) equations by approximating the convection–diffusion operator with first-order differences, thereby avoiding Hessian computations. It combines a Galerkin weak formulation to reduce variance with forward SDE sampling to explore the solution space, and proves a first-order convergence in the time step $h$. DRDM unifies derivative-free PINN-like approaches with martingale/FBSDE-based methods without requiring stochastic calculus, delivering efficient, parallelizable training. Numerical experiments demonstrate scalable performance up to $10^5$ dimensions for quasilinear PDEs and $10^4$ dimensions for HJB problems, with favorable memory and runtime characteristics compared to traditional PINN-based methods.

Abstract

Solving high-dimensional parabolic partial differential equations (PDEs) with deep learning methods is often computationally and memory intensive, primarily due to the need for automatic differentiation (AD) to compute large Hessian matrices in the PDE. In this work, we propose a deep random difference method (DRDM) that addresses these issues by approximating the convection-diffusion operator using only first-order differences and the solution by deep neural networks, thus avoiding Hessian and other derivative computations. The DRDM is implemented within a Galerkin framework to reduce sampling variance, and the solution space is explored using stochastic differential equations (SDEs) to capture the dynamics of the convection-diffusion operator. The approach is then extended to solve Hamilton-Jacobi-Bellman (HJB) equations, which recovers existing martingale deep learning methods for PDEs [{\it SIAM J. Sci. Comput.}, 47 (2025), pp. C795-C819], without using stochastic calculus. The proposed method offers two main advantages: it avoids the need to compute derivatives in PDEs and enables parallel computation of the loss function in both time and space. Moreover, a rigorous error estimate is proven for the quasi-linear parabolic equation, showing first-order accuracy in $h$, the time step used in the discretization of the SDE paths by the Euler-Maruyama scheme. Numerical experiments demonstrate that the method can efficiently and accurately solve quasilinear parabolic PDEs and HJB equations in dimensions up to $10^5$ and $10^4$, respectively.

Deep random difference method for high-dimensional quasilinear parabolic partial differential equations

TL;DR

This work introduces the Deep Random Difference Method (DRDM) for solving high-dimensional quasilinear parabolic PDEs and Hamilton–Jacobi–Bellman (HJB) equations by approximating the convection–diffusion operator with first-order differences, thereby avoiding Hessian computations. It combines a Galerkin weak formulation to reduce variance with forward SDE sampling to explore the solution space, and proves a first-order convergence in the time step . DRDM unifies derivative-free PINN-like approaches with martingale/FBSDE-based methods without requiring stochastic calculus, delivering efficient, parallelizable training. Numerical experiments demonstrate scalable performance up to dimensions for quasilinear PDEs and dimensions for HJB problems, with favorable memory and runtime characteristics compared to traditional PINN-based methods.

Abstract

Solving high-dimensional parabolic partial differential equations (PDEs) with deep learning methods is often computationally and memory intensive, primarily due to the need for automatic differentiation (AD) to compute large Hessian matrices in the PDE. In this work, we propose a deep random difference method (DRDM) that addresses these issues by approximating the convection-diffusion operator using only first-order differences and the solution by deep neural networks, thus avoiding Hessian and other derivative computations. The DRDM is implemented within a Galerkin framework to reduce sampling variance, and the solution space is explored using stochastic differential equations (SDEs) to capture the dynamics of the convection-diffusion operator. The approach is then extended to solve Hamilton-Jacobi-Bellman (HJB) equations, which recovers existing martingale deep learning methods for PDEs [{\it SIAM J. Sci. Comput.}, 47 (2025), pp. C795-C819], without using stochastic calculus. The proposed method offers two main advantages: it avoids the need to compute derivatives in PDEs and enables parallel computation of the loss function in both time and space. Moreover, a rigorous error estimate is proven for the quasi-linear parabolic equation, showing first-order accuracy in , the time step used in the discretization of the SDE paths by the Euler-Maruyama scheme. Numerical experiments demonstrate that the method can efficiently and accurately solve quasilinear parabolic PDEs and HJB equations in dimensions up to and , respectively.

Paper Structure

This paper contains 33 sections, 4 theorems, 171 equations, 6 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Random difference method
  3. Basic expansion
  4. RDM for convection-diffusion operator
  5. RDM for solving quasilinear parabolic PDEs
  6. Spatial sampling rule based on error propagation analysis
  7. Error equation
  8. Error propagation from residuals
  9. Sampling method
  10. Connection to martingale methods
  11. Connection to the Euler scheme for FBSDEs
  12. Deep neural network with RDM (DRDM)
  13. RDM with a Galerkin method and reduced variance
  14. Mini-batch estimation of the loss function
  15. Adversarial learning
  16. ...and 18 more sections

Key Result

Theorem 1

Let assum_polygrow hold. For any $(t, x) \in [0, T) \times \mathbb{R}^d$ and any step size $h$ with $0 < h < \min\{1,\, T - t\}$, it holds that where with $\hat{\mu} := \mu(t, x, \hat{v}(t, x))$ and $\hat{\sigma} := \sigma(t, x, \hat{v}(t, x))$. The constant $C_{\mathrm{loc}} > 0$ is independent of $\hat{v}$, $t$, $h$, $T$, and $x$; see eq_defCD for its explicit form.

Figures (6)

  • Figure 1: Numerical results of \ref{['alg_pde']} for the convection–diffusion equation \ref{['eq_linearPDE']} with $d = 10^3$. The "Loss" in the second column refers to $|L^{\top}(v_{\theta}, \rho_{\eta}; \mathbb{A}_1)\,L(v_{\theta}, \rho_{\eta}; \mathbb{A}_2)|$ in \ref{['alg_pde']}. The shaded regions in the second and third columns show the mean $\pm 2$ SD of the relative errors over 5 independent runs.
  • Figure 2: Numerical results of \ref{['alg_pde']} for QLP-1, QLP-2a, and QLP-2b in \ref{['sec_qlpde']} for $d = 10^4$ and $10^5$. The shaded regions in the third and fourth columns represent the mean $\pm 2 \times \text{SD}$ of the REs over 5 independent runs, where all the REs are computed at $t = 0$. The mean and the SD of the REs, and the RT, are given in \ref{['tab_RESQLP']}.
  • Figure 3: Numerical results of \ref{['alg_amnet']} for HJB-1a and -1b from \ref{['sec_hjb']} with $d = 10^4$. The shaded regions in the third and fourth columns represent the mean $\pm 2 \times \text{SD}$ of the REs over 5 independent runs. The REs in the fourth column are computed at $t = 0$. The mean and the SD of the REs, and the RT, are given in \ref{['tab_ch_hjb']}.
  • Figure 4: Numerical results of \ref{['alg_amnet']} for HJB-2 from \ref{['sec_hjb']} with $d = 10^4$. The RE$_{\infty}$ in (c) is evaluated at $t = 0$. The RT is 2953 seconds for Standard DNN and 2026 seconds for MscaleDNN, respectively.
  • Figure 5: Numerical results of \ref{['alg_pde']} for the linear parabolic PDE \ref{['eq_linearPDE']} with $d = 1$. In the first column, the scatter plots show the spatial samples, where the color indicates the exact solution $v(t, x)$. The second column shows the landscape of the exact solution $v(t, x)$ along the sampling path $t \mapsto X_t$. The third column presents the numerical results for $x \mapsto v(t, x)$ at $t = 0$.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Remark 1
  • Remark 2: Regularity condition
  • Remark 3: Comparison to randomized smoothing PINNs (RS-PINNs)
  • Remark 4: Connection to SDE-based approaches
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Theorem 1
  • proof
  • ...and 7 more