Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations

Abstract

One of the most challenging issues in applied mathematics is to develop and analyze algorithms which are able to approximately compute solutions of high-dimensional nonlinear partial differential equations (PDEs). In particular, it is very hard to develop approximation algorithms which do not suffer under the curse of dimensionality in the sense that the number of computational operations needed by the algorithm to compute an approximation of accuracy $\epsilon >0$ grows at most polynomially in both the reciprocal $1/\epsilon$ of the required accuracy and the dimension $d\in \mathbb{N}$ of the PDE. Recently, a new approximation method, the so-called full history recursive multilevel Picard (MLP) approximation method, has been introduced and, until today, this approximation scheme is the only approximation method in the scientific literature which has been proven to overcome the curse of dimensionality in the numerical approximation of semilinear PDEs with general time horizons. It is a key contribution of this article to extend the MLP approximation method to systems of semilinear PDEs and to numerically test it on several example PDEs. More specifically, we apply the proposed MLP approximation method in the case of Allen-Cahn PDEs, Sine-Gordon-type PDEs, systems of coupled semilinear heat PDEs, and semilinear Black-Scholes PDEs in up to 1000 dimensions. The presented numerical simulation results suggest in the case of each of these example PDEs that the proposed MLP approximation method produces very accurate results in short runtimes and, in particular, the presented numerical simulation results indicate that the proposed MLP approximation scheme significantly outperforms certain deep learning based approximation methods for high-dimensional semilinear PDEs.

21 pages

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Cited by in corpus (11)

• Tackling the Curse of Dimensionality with Physics-Informed Neural Networks
• Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning
• An overview on deep learning-based approximation methods for partial differential equations
• Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks
• Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
• Solving non-linear Kolmogorov equations in large dimensions by using deep learning: a numerical comparison of discretization schemes
• Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities
• Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations
• Full history recursive multilevel Picard approximations for ordinary differential equations with expectations
• A Machine-Learning Method for Time-Dependent Wave Equations over Unbounded Domains
• Strong $L^p$-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations