Good Lattice Training: Physics-Informed Neural Networks Accelerated by Number Theory

• URL: http://arxiv.org/abs/2307.13869v1
• Date: Wed, 26 Jul 2023 00:01:21 GMT
• Title: Good Lattice Training: Physics-Informed Neural Networks Accelerated by Number Theory
• Authors: Takashi Matsubara, Takaharu Yaguchi
• Abstract summary: We propose a new technique called good lattice training (GLT) for PINNs. GLT offers a set of collocation points that are effective even with a small number of points and for multi-dimensional spaces. Our experiments demonstrate that GLT requires 2--20 times fewer collocation points than uniformly random sampling or Latin hypercube sampling.
• Score: 7.462336024223669
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Physics-informed neural networks (PINNs) offer a novel and efficient approach to solving partial differential equations (PDEs). Their success lies in the physics-informed loss, which trains a neural network to satisfy a given PDE at specific points and to approximate the solution. However, the solutions to PDEs are inherently infinite-dimensional, and the distance between the output and the solution is defined by an integral over the domain. Therefore, the physics-informed loss only provides a finite approximation, and selecting appropriate collocation points becomes crucial to suppress the discretization errors, although this aspect has often been overlooked. In this paper, we propose a new technique called good lattice training (GLT) for PINNs, inspired by number theoretic methods for numerical analysis. GLT offers a set of collocation points that are effective even with a small number of points and for multi-dimensional spaces. Our experiments demonstrate that GLT requires 2--20 times fewer collocation points (resulting in lower computational cost) than uniformly random sampling or Latin hypercube sampling, while achieving competitive performance.

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