A Theoretical Analysis of Deep Neural Networks and Parametric PDEs

• Published in: Constructive approximation Vol. 55; no. 1; pp. 73 - 125
• Main Authors: Kutyniok, Gitta, Petersen, Philipp, Raslan, Mones, Schneider, Reinhold
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2022; Springer Nature B.V; Springer
• Subjects: Analysis; Approximation; Artificial neural networks; Fysikk: 430; Matematikk og Naturvitenskap: 400; Mathematical analysis; Mathematics; Mathematics and natural science: 400; Mathematics and Statistics; Neural networks; Numerical Analysis; Partial differential equations; Physics: 430; Upper bounds; VDP; 41A46; Deep neural networks; 41A25; 68T05; 35A35; Parametric PDEs; Approximation rates; Reduced basis method; 65N30; 35J99
• ISSN: 0176-4276; 1432-0940; 1432-0940
• DOI: 10.1007/s00365-021-09551-4

We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent low dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical neural network approximation results. Concretely, we use the existence of a small reduced basis to construct, for a large variety of parametric partial differential equations, neural networks that yield approximations of the parametric solution maps in such a way that the sizes of these networks essentially only depend on the size of the reduced basis.