Two-stage initial-value iterative physics-informed neural networks for simulating solitary waves of nonlinear wave equations

TL;DR

This paper introduces a novel two-stage physics-informed neural network approach for simulating solitary waves in nonlinear wave equations, eliminating the need for boundary data and demonstrating broad applicability and efficiency.

Contribution

The paper presents a new two-stage iterative neural network framework that effectively computes solitary wave solutions without requiring boundary conditions, supported by theoretical guarantees.

Findings

Successfully applied to various nonlinear wave equations including NLS, KdV, and KP.

Demonstrated efficiency and accuracy compared to traditional numerical methods.

Validated the method's effectiveness across multiple dimensions and potential types.

Abstract

We propose a new two-stage initial-value iterative neural network (IINN) algorithm for solitary wave computations of nonlinear wave equations based on traditional numerical iterative methods and physics-informed neural networks (PINNs). Specifically, the IINN framework consists of two subnetworks, one of which is used to fit a given initial value, and the other incorporates physical information and continues training on the basis of the first subnetwork. Importantly, the IINN method does not require any additional data information including boundary conditions, apart from the given initial value. Corresponding theoretical guarantees are provided to demonstrate the effectiveness of our IINN method. The proposed IINN method is efficiently applied to learn some types of solutions in different nonlinear wave equations, including the one-dimensional (1D) nonlinear Schr\"odinger equations (NLS) equation (with and without potentials), the 1D saturable NLS equation with PT -symmetric optical lattices, the 1D focusing-defocusing coupled NLS equations, the KdV equation, the two-dimensional (2D) NLS equation with potentials, the 2D amended GP equation with a potential, the (2+1)-dimensional KP equation, and the 3D NLS equation with a potential. These applications serve as evidence for the efficacy of our method. Finally, by comparing with the traditional methods, we demonstrate the advantages of the proposed IINN method.