Physics-Informed Neural Networks for the Korteweg-de Vries Equation for Internal Solitary Wave Problem: Forward Simulation and Inverse Parameter Estimation

TL;DR

This paper presents a physics-informed neural network framework for accurately simulating and estimating parameters of internal solitary waves governed by the Korteweg-de Vries equation, addressing challenges in operator learning and inverse problems.

Contribution

It introduces a unified PINN-based approach for both forward simulation and inverse parameter estimation of internal solitary waves, overcoming traditional computational limitations.

Findings

Prediction errors as low as 10^{-4} with 1000 training points

Successful inversion of nonlinear coefficients from noisy data

Effective modeling of internal waves in stratified fluids

Abstract

Physics-informed neural networks (PINNs) have emerged as a transformative framework for addressing operator learning and inverse problems involving the Korteweg-de Vries (KdV) equation for internal solitary waves. By integrating physical constraints with data-driven optimization, PINNs overcome the critical challenges of parameter unmeasurability in the KdV equation for internal solitary waves in two-layer fluid systems. This work addresses two problems: (1) Operator learning constructs a mapping from parameters to solutions, enabling wave evolution predictions from unknown parameters. Comparative studies demonstrate prediction errors as low as $10^{-4}$ when using 1000 training points. (2) Inverse problem solving leverages sparse and potentially noisy observational data with physics-regularized constraints to invert nonlinear coefficients successfully. Compared to conventional approaches, this end-to-end differentiable paradigm unifies operator learning and inverse problem-solving while overcoming mesh discretization errors and high-dimensional parameter space iteration costs. The method shows effectiveness for internal wave problems in stratified fluids, providing both accurate forward modeling and robust parameter inversion capabilities, even under noise.