Deep Adaptive Basis Galerkin Method for High-Dimensional Evolution Equations With Oscillatory Solutions

In this paper, we study deep neural networks (DNNs) for solving high-dimensional evolution equations with oscillatory solutions. Different from deep least-squares methods that deal with time and space variables simultaneously, we propose a deep adaptive basis Galerkin (DABG) method which employs the spectral-Galerkin method for time variable by a tensor-product basis for oscillatory solutions and the deep neural network method for high-dimensional space variables. The proposed method can lead to a linear system of differential equations having unknown DNNs that can be trained via the loss function. We establish a posteriori estimates of the solution error which is bounded by the minimal loss function and the term O(N^−m), where N is the number of basis functions and m characterizes the regularity of the equation, and show that if the true solution is a Barron-type function, the error bound converges to zero as M = O(N^p) approaches to infinity where M is the width of the used networks and p is a positive constant. Numerical examples, including high-dimensional linear parabolic and hyperbolic equations, and a nonlinear Allen–Cahn equation are presented to demonstrate that the performance of the proposed DABG method is better than that of existing DNNs.