Semi-Discrete in Time Method for Time-Dependent Equations by Random Neural Basis

Abstract: Neural network-based solvers for partial differential equations (PDEs) have attracted considerable attention, yet they often face challenges in accuracy and computational efficiency. In this work, we focus on time-dependent PDEs and observe that coupling space and time in a single network can increase the difficulty of approximation. To address this, we propose a semi-discrete in time method (SDTM) which leverages classical numerical time integrators and random neural basis (RNB). Additional adaptive operations are introduced to enhance the network's ability to capture features across scales to ensure uniform approximation accuracy for multi-scale PDEs. Numerical experiments demonstrate the framework's effectiveness and confirm the convergence of the temporal integrator as well as the network's approximation performance.