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.

An incomplete list of algorithms for high-dimensional PDEs (nonexclusive by category). Suggestions and contributions are welcome to this list.

BSDE/Feynman-Kac related:

Control and HJB equation:


Many-electron Schrödinger equation:

Commitor function:


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