Convergence Rates for a Deep Learning Algorithm for Semilinear PDEs
39 Pages Posted: 16 Feb 2022
Date Written: December 9, 2021
We derive convergence rates for a deep learning algorithm for semilinear partial differential equations which is based on a Feynman-Kac representation in terms of an uncoupled forward-backward stochastic differential equation and a discretization in time of the stochastic equation. We show that the error of the deep learning algorithm is bounded in terms of its loss functional, hence yielding a direct measure to judge the quality of the deep solver in numerical applications, and that the loss functional converges sufficiently fast to zero to guarantee that the error of the deep learning algorithm vanishes in the limit. As a consequence of these results, we argue that the deep solver has a strong convergence rate of order 1/2.
Keywords: semilinear PDE, forward-backward SDE, deep solver, strong convergence rate
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