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 $ε> 0$ grows at most polynomially in both the reciprocal $1/ε$ 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.
Full history recursive multilevel Picard approximations suffer from the curse of dimensionality for the Hamilton-Jacobi-Bellman equation of a stochastic control problem
Multilevel Picard approximations overcome the curse of dimensionality when approximating semilinear heat equations with gradient-dependent nonlinearities in $L^p$-sense
Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore（数学科学系，物理与数学科学学院，南洋理工大学，新加坡）；Faculty of Mathematics, Bielefeld University, Germany（数学学院，比勒菲尔德大学，德国）
A hybrid recursive multilevel incomplete factorization preconditioner for solving general linear systems
Johann Bernoulli Institute for Mathematics and Computer Science - University of Groningen（格罗宁根大学）；School of Mathematical Sciences, University of Electronic Science and Technology of China（电子科技大学）
Multilevel Picard approximations overcome the curse of dimensionality in the numerical approximation of general semilinear PDEs with gradient-dependent nonlinearities
Institute of Mathematics, Academy of Sciences of the Czech Republic（捷克科学院数学研究所）；Department of Mathematical and Statistical Sciences, University of Colorado Denver（科罗拉多大学丹佛分校数学与统计科学系）；Department of Aerospace and Mechanical Engineering, University of Southern California（南加州大学航空航天与机械工程系）；Department of Mathematics, Faculty of Mechanical Engineering, Czech Technical University（捷克理工大学机械工程学院数学系）
Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions
Swiss Federal Research Institute for Forest, Snow and Landscape (WSL)（瑞士联邦森林、雪与景观研究所（WSL））；ETH Zürich（苏黎世联邦理工学院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）；University of Münster（明斯特大学）
