Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks

Geist, Moritz; Petersen, Philipp; Raslan, Mones; Schneider, Reinhold und Kutyniok, Gitta (2021): Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks. In: Journal of Scientific Computing, Vol. 88, 22

Abstract

We perform a comprehensive numerical study of the effect of approximation-theoretical results for neural networks on practical learning problems in the context of numerical analysis. As the underlying model, we study the machine-learning-based solution of parametric partial differential equations. Here, approximation theory for fully-connected neural networks predicts that the performance of the model should depend only very mildly on the dimension of the parameter space and is determined by the intrinsic dimension of the solution manifold of the parametric partial differential equation. We use various methods to establish comparability between test-cases by minimizing the effect of the choice of test-cases on the optimization and sampling aspects of the learning problem. We find strong support for the hypothesis that approximation-theoretical effects heavily influence the practical behavior of learning problems in numerical analysis. Turning to practically more successful and modern architectures, at the end of this study we derive improved error bounds by focusing on convolutional neural networks.

• Item Type: Journal article
• Faculties: Mathematics, Computer Science and Statistics > Mathematics > Bavarian Chair for Mathematical Foundations of Artificial Intelligence
• Subjects: 500 Science > 510 Mathematics
• ISSN: 0885-7474
• Language: English