A neural multilevel method for high-dimensional parametric PDEs

Cosmas Heiß, Ingo Gühring, Martin Eigel

Keywords: scientific machine learning, deep learning, partial differential equations, multigrid methods, uncertainty quantification

Abstract:

In scientific machine learning, neural networks recently have become a popular tool for learning the solutions of differential equations.
However, practical results often conflict the existing theoretical predictions in that observed convergence stagnates early. A substantial improvement can be achieved by the presented multilevel scheme which decomposes the considered problem into easier to train sub-problems, resulting in a sequence of neural networks. The efficacy of the approach is demonstrated for high-dimensional parametric elliptic PDEs that are common benchmark problems in uncertainty quantification. Moreover, a theoretical analysis of the expressivity of the developed neural networks is devised.

Publication Status: This work is unpublished.