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

Deep neural networks have been shown to provide accurate function approximations in high dimensions. However, fitting network parameters requires training data that may not be available beforehand, which is particularly challenging in science and engineering applications where often it is even unclear how to collect new informative training data in the first place. This work proposes Neural Galerkin schemes based on deep learning that generate training data samples with active learning for numerically solving high-dimensional partial differential equations. Neural Galerkin schemes 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, which is in stark contrast to many 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.