A two-stage physics-informed neural network method based on conserved quantities and applications in localized wave solutions

2021/07/02 by Shuning Lin, Yong Chen · 176 citations
Engineering · Mathematics · Physics and Astronomy · #Applied mathematics #Artificial intelligence #Artificial neural network #Classical mechanics #Computer science #Conserved quantity #Fluid Dynamics and Turbulent Flows #Fluid Dynamics and Vibration Analysis #Generalization #Integrable system #Mathematical analysis #Mathematics #Model Reduction and Neural Networks #Nonlinear system #Partial differential equation #Physics #Quantum mechanics #Soliton #Stage (stratigraphy) #nlin.PS #nlin.SI

paper · pdf · doi:10.1016/j.jcp.2022.111053

published in Journal of Computational Physics 457, 111053 (Elsevier BV)

arxiv created 2021/07/02 · openalex publication_date 2022/02/14 · arxiv updated 2022/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

With the advantages of fast calculating speed and high precision, the physics-informed neural network method opens up a new approach for numerically solving nonlinear partial differential equations. Based on conserved quantities, we devise a two-stage PINN method which is tailored to the nature of equations by introducing features of physical systems into neural networks. Its remarkable advantage lies in that it can impose physical constraints from a global perspective. In stage one, the original PINN is applied. In stage two, we additionally introduce the measurement of conserved quantities into mean squared error loss to train neural networks. This two-stage PINN method is utilized to simulate abundant localized wave solutions of integrable equations. We mainly study the Sawada-Kotera equation as well as the coupled equations: the classical Boussinesq-Burgers equations and acquire the data-driven soliton molecule, M-shape double-peak soliton, plateau soliton, interaction solution, etc. Numerical results illustrate that abundant dynamic behaviors of these solutions can be well reproduced and the two-stage PINN method can remarkably improve prediction accuracy and enhance the ability of generalization compared to the original PINN method.

Citations

• On the limited memory BFGS method for large scale optimization
• A Higher-Order Water-Wave Equation and the Method for Solving It
• Deep Hidden Physics Models: Deep Learning of Nonlinear Partial\n Differential Equations
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Variational Physics-Informed Neural Networks For Solving Partial Differential Equations
• Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations
• NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations
• B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data
• hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition
• PINN deep learning method for the Chen–Lee–Liu equation: Rogue wave on the periodic background
• Large Sample Properties of Simulations Using Latin Hypercube Sampling
• Solving localized wave solutions of the derivative nonlinear Schrodinger equation using an improved PINN method