ALM-PINNs Algorithms for Solving Nonlinear PDEs and Parameter Inversion Problems

TL;DR

The paper presents ALM-PINNs, a framework that augments physics-informed neural networks (PINNs) with an augmented Lagrangian approach to solve nonlinear PDEs and perform parameter inversion. It demonstrates higher accuracy than standard PINNs under the same TensorFlow 2.0-based architecture by balancing data, physics, and inversion terms, and by leveraging probabilistic priors for random errors in loss construction. A key contribution is the systematic analysis of loss function design, providing a theoretical basis for improving algorithm performance. The work offers a practical pathway for more reliable and accurate solving of nonlinear PDEs and associated inverse problems in scientific computing.

Abstract

This paper focuses on the PINNs algorithm by proposing the ALM-PINNs computational framework to solve various nonlinear partial differential equations and corresponding parameters identification problems. The numerical solutions obtained by the ALM-PINNs algorithm are compared with both the exact solutions and the numerical solutions implemented from the PINNs algorithm. This demonstrates that under the same machine learning framework (TensorFlow 2.0) and neural network architecture, the ALM-PINNs algorithm achieves higher accuracy compared to the standard PINNs algorithm. Additionally, this paper systematically analyzes the construction principles of the loss function by introducing the probability distribution of random errors as prior information, and provides a theoretical basis for algorithm improvement.