TY - JOUR
T1 - Interpolating Between BSDEs and PINNs: Deep Learning for Elliptic and Parabolic Boundary Value Problems
AU - Nüsken , Nikolas
AU - Richter , Lorenz
JO - Journal of Machine Learning
VL - 1
SP - 31
EP - 64
PY - 2023
DA - 2023/03
SN - 2
DO - http://doi.org/10.4208/jml.220416
UR - https://global-sci.org/intro/article_detail/jml/21512.html
KW - Numerical methods for PDEs, Backward stochastic differential equations, Physics informed neural networks, Deep learning.
AB - <p style="text-align: justify;">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. The diffusion loss furthermore bears close similarities to (least squares) temporal difference objectives found in reinforcement learning.
We also discuss eigenvalue problems and perform extensive numerical studies, including calculations of the
ground state for nonlinear Schrödinger operators and committor functions relevant in molecular dynamics.</p>