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Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations

Maziar Raissi

TL;DR

The work tackles the curse of dimensionality in solving high-dimensional parabolic PDEs by leveraging the FBSDE-PDE connection and learning a single neural network u(t,x) whose gradients are computed via automatic differentiation. Training uses Euler-Maruyama discretization over Brownian paths to form a loss that enforces the FBSDE dynamics and terminal condition, enabling evaluation of the full space-time solution after one training run. The authors demonstrate the approach on 100D Black-Scholes-Barenblatt and Hamilton-Jacobi-Bellman problems, and a 20D Allen-Cahn problem, achieving accurate results comparable to existing methods while reducing parameter complexity. They emphasize broad applicability to stochastic control and financial problems and provide plans for public code release.

Abstract

Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for solving forward and inverse problems associated with partial differential equations, we circumvent the tyranny of numerical discretization by devising an algorithm that is scalable to high-dimensions. In particular, we approximate the unknown solution by a deep neural network which essentially enables us to benefit from the merits of automatic differentiation. To train the aforementioned neural network we leverage the well-known connection between high-dimensional partial differential equations and forward-backward stochastic differential equations. In fact, independent realizations of a standard Brownian motion will act as training data. We test the effectiveness of our approach for a couple of benchmark problems spanning a number of scientific domains including Black-Scholes-Barenblatt and Hamilton-Jacobi-Bellman equations, both in 100-dimensions.

Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations

TL;DR

The work tackles the curse of dimensionality in solving high-dimensional parabolic PDEs by leveraging the FBSDE-PDE connection and learning a single neural network u(t,x) whose gradients are computed via automatic differentiation. Training uses Euler-Maruyama discretization over Brownian paths to form a loss that enforces the FBSDE dynamics and terminal condition, enabling evaluation of the full space-time solution after one training run. The authors demonstrate the approach on 100D Black-Scholes-Barenblatt and Hamilton-Jacobi-Bellman problems, and a 20D Allen-Cahn problem, achieving accurate results comparable to existing methods while reducing parameter complexity. They emphasize broad applicability to stochastic control and financial problems and provide plans for public code release.

Abstract

Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for solving forward and inverse problems associated with partial differential equations, we circumvent the tyranny of numerical discretization by devising an algorithm that is scalable to high-dimensions. In particular, we approximate the unknown solution by a deep neural network which essentially enables us to benefit from the merits of automatic differentiation. To train the aforementioned neural network we leverage the well-known connection between high-dimensional partial differential equations and forward-backward stochastic differential equations. In fact, independent realizations of a standard Brownian motion will act as training data. We test the effectiveness of our approach for a couple of benchmark problems spanning a number of scientific domains including Black-Scholes-Barenblatt and Hamilton-Jacobi-Bellman equations, both in 100-dimensions.

Paper Structure

This paper contains 8 sections, 22 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Setup and Solution methodology
  3. Related Work
  4. Results
  5. Black-Scholes-Barenblatt Equation in 100D
  6. Hamilton-Jacobi-Bellman Equation in 100D
  7. Allen-Cahn Equation in 20D
  8. Summary and Discussion

Figures (5)

  • Figure 1: Black-Scholes-Barenblatt Equation in 100D: Evaluations of the learned solution $Y_t = u(t,X_t)$ at representative realizations of the underlying high-dimensional process $X_t$. It should be highlighted that the state of the art algorithms beck2017machineweinan2017deephan2017overcoming can only approximate $Y_0 = u(0,X_0)$ at time $0$ and at the initial spatial point $X_0=\xi$.
  • Figure 2: Black-Scholes-Barenblatt Equation in 100D: Mean and mean plus two standard deviations of the relative errors between model predictions and the exact solution computed based on $100$ realizations of the underlying Brownian motion.
  • Figure 3: Hamilton-Jacobi-Bellman Equation in 100D: Evaluation of the learned solution $Y_t = u(t,X_t)$ at a representative realization of the underlying high-dimensional process $X_t$. It should be highlighted that the state of the art algorithms beck2017machineweinan2017deephan2017overcoming can only approximate $Y_0 = u(0,X_0)$ at time $0$ and at the initial spatial point $X_0=\xi$.
  • Figure 4:
  • Figure 5: Allen-Cahn Equation in 20D: Evaluation of the learned solution $Y_t = u(t,X_t)$ at representative realizations of the underlying high-dimensional process $X_t$. It should be highlighted that the state of the art algorithms beck2017machineweinan2017deephan2017overcoming can only approximate $Y_0 = u(0,X_0)$ at time $0$ and at the initial spatial point $X_0=\xi$.