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Solving Partial Differential Equations with Sampled Neural Networks by Datar, Chinmay; Kapoor, Taniya; Chandra, Abhishek; Sun, Qing; Burak, Iryna; Bolager, Erik Lien; Veselovska, Anna; Fornasier, Massimo; Dietrich, Felix is a scholarly article available to read on EtoBox.

What is Solving Partial Differential Equations with Sampled Neural Networks about?

Approximation of solutions to partial differential equations (PDE) is an important problem in computational science and engineering. Using neural networks as an ansatz for the solution has proven a challenge in terms of training time and approximation accuracy. In this contribution, we discuss how sampling the hidden weights and biases of the ansatz network from data-agnostic and data-dependent probability distributions allows us to progress on both challenges. In most examples, the random sampling schemes outperform iterative, gradient-based optimization of physics-informed neural networks regarding training time and accuracy by several orders of magnitude. For time-dependent PDE, we construct neural basis functions only in the spatial domain and then solve the associated ordinary differential equation with classical methods from scientific computing over a long time horizon. This alleviates one of the greatest challenges for neural PDE solvers because it does not require us to parameterize the solution in time. For second-order elliptic PDE in Barron spaces, we prove the existence of sampled networks with $L^2$ convergence to the solution. We demonstrate our approach on several t

Author

Datar, Chinmay; Kapoor, Taniya; Chandra, Abhishek; Sun, Qing; Burak, Iryna; Bolager, Erik Lien; Veselovska, Anna; Fornasier, Massimo; Dietrich, Felix

Published

2024

Language

EN