A New Method for Solving Nonlinear Partial Differential Equations Based on Liquid Time-Constant Networks

Abstract

In this paper, physics-informed liquid networks (PILNs) are proposed based on liquid time-constant networks (LTC) for solving nonlinear partial differential equations (PDEs). In this approach, the network state is controlled via ordinary differential equations (ODEs). The significant advantage is that neurons controlled by ODEs are more expressive compared to simple activation functions. In addition, the PILNs use difference schemes instead of automatic differentiation to construct the residuals of PDEs, which avoid information loss in the neighborhood of sampling points. As this method draws on both the traveling wave method and physics-informed neural networks (PINNs), it has a better physical interpretation. Finally, the KdV equation and the nonlinear Schrödinger equation are solved to test the generalization ability of the PILNs. To the best of the authors’ knowledge, this is the first deep learning method that uses ODEs to simulate the numerical solutions of PDEs.