High Dimensional PDEs: Theory and Numerical Algorithms

Weinan E
Princeton University

In physics, economics, and control theory, we often encounter PDEs in very
high dimensions. This has been a notoriously difficult problem due to the
curse of dimensionality.

In recent years, two classes of algorithms have emerged for solving
nonlinear parabolic PDEs in high dimension, with a complexity that scales
algebraically (linear or quadratic) in the dimension: the multi-level
Picard method and the deep learning based methods.
These algorithms have opened up new possibilities for attacking control
and many other problems in hundreds and thousands of dimensions.
They have also triggered questions about understanding PDEs in high dimensons.
In this talk, I will discuss what we have achieved and understood so far
about these problems.

Back to Workshop I: High Dimensional Hamilton-Jacobi Methods in Control and Differential Games