Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations

TL;DR

This paper introduces Neural Galerkin schemes with active learning to efficiently solve high-dimensional partial differential equations by adaptively generating training data based on the dynamics, improving accuracy and applicability.

Contribution

The work develops a novel Neural Galerkin approach that incorporates active learning guided by the PDE dynamics, enabling better high-dimensional problem solving.

Findings

Active data collection enhances neural network expressiveness in high dimensions.

Neural Galerkin schemes outperform traditional methods in high-dimensional wave and particle systems.

The approach effectively captures local features in evolving solutions.

Abstract

Deep neural networks have been shown to provide accurate function approximations in high dimensions. However, fitting network parameters requires informative training data that are often challenging to collect in science and engineering applications. This work proposes Neural Galerkin schemes based on deep learning that generate training data with active learning for numerically solving high-dimensional partial differential equations. Neural Galerkin schemes build on the Dirac-Frenkel variational principle to train networks by minimizing the residual sequentially over time, which enables adaptively collecting new training data in a self-informed manner that is guided by the dynamics described by the partial differential equations. This is in contrast to other machine learning methods that aim to fit network parameters globally in time without taking into account training data acquisition. Our finding is that the active form of gathering training data of the proposed Neural Galerkin schemes is key for numerically realizing the expressive power of networks in high dimensions. Numerical experiments demonstrate that Neural Galerkin schemes have the potential to enable simulating phenomena and processes with many variables for which traditional and other deep-learning-based solvers fail, especially when features of the solutions evolve locally such as in high-dimensional wave propagation problems and interacting particle systems described by Fokker-Planck and kinetic equations.