Repository: Freie Universität Berlin, Math Department

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

Nüsken, Nikolas and Richter, Lorenz (2021) Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems. ArXive . pp. 1-25. (Submitted)

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Official URL: https://arxiv.org/pdf/2112.03749.pdf

Abstract

Solving high-dimensional partial differential equations is a recurrent challenge in economics, science and en- gineering. 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 L2-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 implementa- tions 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.

Item Type:Article
Subjects:Mathematical and Computer Sciences > Mathematics > Applied Mathematics
ID Code:2799
Deposited By: Monika Drueck
Deposited On:14 Mar 2022 14:27
Last Modified:14 Mar 2022 14:27

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