Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems

Abstract

Solving high-dimensional partial differential equations is a recurrent challenge in economics, science and engineering. In recent years, a great number of computational approaches have been developed, most of them relying on a combination of Monte Carlo sampling and deep learning based approximation. For elliptic and parabolic problems, existing methods can broadly be classified into those resting on reformulations in terms of backward stochastic differential equations (BSDEs) and those aiming to minimize a regression-type L 2 -error (physics-informed neural networks, PINNs). In this paper, we review the literature and suggest a methodology based on the novel diffusion loss that interpolates between BSDEs and PINNs. Our contribution opens the door towards a unified understanding of numerical approaches for high-dimensional PDEs, as well as for implementations that combine the strengths of BSDEs and PINNs. We also provide generalizations to eigenvalue problems and perform extensive numerical studies, including calculations of the ground state for nonlinear Schrödinger operators and committor functions relevant in molecular dynamics.

Stochastic Deep-Ritz for Parametric Uncertainty Quantification

“…In contrast, the deep-BSDE methods exploit the intrinsic connection between PDEs and BSDEs in the form of the nonlinear Feynman-Kac formula to recast s PDE as a stochastic control problem that is solved by reinforcement learning techniques [18,19,20,21,22]. See [23] for a recent work on interpolating between PINNs and deep BSDEs. Finally, energy methods construct the loss function by taking advantage of the variational formulation of elliptic PDEs [24,25,26,27].…”

Differential learning methods for solving fully nonlinear PDEs

“…In the last years, significant progress has been achieved towards these challenges with several numerical methods using techniques from deep learning, see the recent surveys by [3] and [8]: A first class of approximation algorithms, called Physics Informed Neural Network (PINNs) [28], also known as Deep Galerkin method (DGM) [30], directly approximates the solution to the PDE by a neural network, and its partial derivatives by automatic differentiation, by minimizing the loss function arising from the residual of the PDE evaluated on a random grid in the spacetime domain. A second class of algorithms relies on the backward stochastic differential representation of the PDE in the semi-linear case by minimizing either a global loss function (see [6], and extensions in [2], [17], [24]), or sequence of loss functions from backward recursion (see [16], and variations-extensions in [25], [1], [8]).…”