High-Dimensional Partial Differential Equations

In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality and has been proven to be so for a class of nonlinear Monte Carlo methods.

The recent review paper has reviewed these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the Deep BSDE method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying partial differential equations as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near future.

We also collect a few code repositories in this website to present the advancement of the field and promote the exploration of new problems.

Solving high-dimensional partial differential equations using deep learning

Adaptive deep learning for high-dimensional Hamilton-Jacobi-Bellman equations

Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space

Numerically solving parametric families of high-dimensional Kolmogorov partial differential equations via deep learning

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