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Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations
Maziar Raissi
TL;DR
The paper addresses the dimensionality limitations of grid-based PDE solvers by approximating PDE solutions with deep neural networks. It trains this representation through the connection between PDEs and forward-backward stochastic differential equations, using Brownian-motion realizations and automatic differentiation. The framework is applied to diverse high-dimensional benchmarks, including 100-dimensional Black-Scholes-Barenblatt and Hamilton-Jacobi-Bellman equations, with promising results.
Problem
Classical PDE methods suffer from the curse of dimensionality because they rely on spatio-temporal grids.
Method
The method approximates the unknown PDE solution with a deep neural network and trains it using forward-backward stochastic differential equations, Brownian-motion realizations, and automatic differentiation.
Results
The framework is demonstrated on diverse high-dimensional benchmark problems, including 100-dimensional Black-Scholes-Barenblatt and Hamilton-Jacobi-Bellman equations, with promising results.
Takeaways & Limitations
The methodology is presented as applicable across scientific domains including stochastic control, theoretical economics, and mathematical finance.
Takeaways & Limitations
More iterations of the Adam optimizer are needed to obtain more accurate estimates of Y_t = u(t, X_t) at later times t > 0.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper motivates neural-network approximations for PDE solutions by connecting forward-backward stochastic differential equations with deterministic parabolic PDEs and avoiding grid-based dimensionality limits.
- Forward-backward stochastic differential equations connect stochastic processes with deterministic functions that solve parabolic partial differential equations under suitable regularity assumptions.The forward equation supplies the state-dependent randomness in the backward equation’s terminal condition.
- Forward-backward stochastic differential equations are uncoupled when the backward solution does not enter forward dynamics and coupled when it does.
- Classical finite-element, finite-difference, and spectral methods suffer from the curse of dimensionality because they rely on spatio-temporal grids.
- The proposed approach approximates the unknown deterministic function of time and space with a deep neural network, using ideas from modern PDE solution methods.
2. Problem Setup and Solution methodology
The methodology represents the PDE solution with a neural network and exploits its stochastic differential-equation formulation to train the network through discretized forward-backward dynamics and automatic differentiation.
- The stochastic processes satisfy forward and backward dynamics driven by drift, diffusion, and Brownian-motion increments.The discretization uses Euler-Maruyama with ∆t_n = T/N and ∆W_n distributed normally with mean 0 and standard deviation determined by ∆t_n.
- Coupled forward-backward stochastic differential equations provide the stochastic formulation underlying the high-dimensional PDE solution method.Their solution consists of stochastic processes X_t, Y_t, and Z_t and is related to a quasi-linear PDE.
- Ito’s formula establishes the relationship between solutions of the coupled stochastic equations and the corresponding PDE.The PDE notation uses Du and D2u for the gradient vector and Hessian matrix of u.
- The unknown solution u(t, x) is approximated by a deep neural network, while automatic differentiation supplies the gradient Du(t, x).The gradient is obtained through the chain rule for differentiating compositions of functions.
- The network parameters are learned by minimizing a loss function obtained by discretizing the forward-backward stochastic differential equations with the Euler-Maruyama scheme.
3. Related Work1
Earlier deep-learning FBSDE methods approximate gradients with time-specific neural networks and target only the initial value, while the current method directly approximates u(t, x) with one network.
- Earlier FBSDE methods: Prior methods approximate Du(t_n, x) with N−1 distinct neural networks after Euler-Maruyama discretization.Their parameters include Y0, Z0, and the parameters of the gradient networks.
- Earlier FBSDE methods: These methods are designed to approximate only Y0 = u(0, X0) = u(0, ξ), requiring retraining to obtain values at later times.The restriction applies to approximations of Yt = u(t, Xt) for t > 0.
- Earlier FBSDE methods: Their parameter count grows with the number of time-discretization points N, making long integrations or smaller time steps potentially prohibitive.Smaller ∆t may be needed to improve Euler-Maruyama accuracy.
- Earlier FBSDE methods: A further concern is the ad-hoc combination of parameterized Y0 and Z0 with N−1 separate, non-sharing gradient networks.The cited discussion identifies this as a third major drawback.
- Current method: The current method places a neural network directly on u(t, x), with parameter count independent of the number of time-discretization points N.This follows the basis-function perspective of classical PDE methods and neural-network or Gaussian-process approaches for PDE problems.
- Second-order extensions: For fully nonlinear second-order PDEs, related prior work introduces neural networks for D2u(t_n, x) and LDu(t_n, x) at each intermediate time.That construction uses 2(N−1) distinct networks and retains terminal-condition matching as its loss.
4. Results
The framework is applied to high-dimensional Black-Scholes-Barenblatt, Hamilton-Jacobi-Bellman, and Allen-Cahn equations using neural-network approximations trained from forward-backward stochastic differential equations. The reported experiments evaluate learned solutions along Brownian-motion realizations and compare predictions with exact or Monte-Carlo reference solutions.
- Hamilton-Jacobi-Bellman Equation in 100D: A 5-layer network with 256 neurons per hidden layer is evaluated for the 100D Hamilton-Jacobi-Bellman equation using 10^5 Monte-Carlo samples as the reference solution.The time domain is partitioned into N = 50 equally spaced intervals, and the model uses 100 Brownian-motion realizations per optimizer iteration.
- Hamilton-Jacobi-Bellman Equation in 100D: 7.3 × 10^-3 relative error is achieved after 100 training steps for the Hamilton-Jacobi-Bellman equation, with accuracy comparable to earlier results.More Adam iterations are needed for more accurate estimates at later times t > 0.
- Allen-Cahn Equation in 20D: The Allen-Cahn experiment evaluates a 5-layer, 256-neuron network in 20 dimensions over N = 15 time intervals using 100 Brownian-motion realizations per mini-batch.The learned solution is evaluated at five representative realizations not seen during training.
5. Summary and Discussion
The work presents a deep learning approach for coupled forward-backward stochastic differential equations and corresponding high-dimensional partial differential equations, with promising benchmark results. The framework also points toward applications in stochastic control, theoretical economics, and mathematical finance, and extension to second-order backward stochastic differential equations.
- The authors propose a deep learning approach for coupled forward-backward stochastic differential equations and corresponding high-dimensional partial differential equations.
- The methodology produces promising results across a diverse collection of benchmark problems.
- The framework may benefit practitioners in stochastic control, theoretical economics, and mathematical finance.
- The proposed framework could be extended to second-order backward stochastic differential equations.