Multi-level neural networks for high-dimensional parametric obstacle problems
Authors
- Eigel, Martin
ORCID: 0000-0003-2687-4497 - Heiß, Cosmas
- Schütte, Janina
ORCID: 0009-0000-9924-3229
2020 Mathematics Subject Classification
- 65N22 68T07 65N30 65N55
Keywords
- Obstacle problems, multilevel, convolutional neural networks, expressivity theory
DOI
Abstract
A new method to solve computationally challenging (random) parametric obstacle problems is developed and analyzed, where the parameters can influence the related partial differential equation (PDE) and determine the position and surface structure of the obstacle. As governing equation, a stationary elliptic diffusion problem is assumed. The high-dimensional solution of the obstacle problem is approximated by a specifically constructed convolutional neural network (CNN). This novel algorithm is inspired by a finite element constrained multigrid algorithm to represent the parameter to solution map. This has two benefits: First, it allows for efficient practical computations since multi-level data is used as an explicit output of the NN thanks to an appropriate data preprocessing. This improves the efficacy of the training process and subsequently leads to small errors in the natural energy norm. Second, the comparison of the CNN to a multigrid algorithm provides means to carry out a complete a priori convergence and complexity analysis of the proposed NN architecture. Numerical experiments illustrate a state-of-the-art performance for this challenging problem.
Download Documents