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

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.

References in corpus (5)

• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Variational Physics-Informed Neural Networks For Solving Partial Differential Equations
• PINN deep learning for the Chen-Lee-Liu equation: Rogue wave on the periodic background
• Soliton Molecules In Dipolar Bose-Einstein Condensates
• Solving localized wave solutions of the derivative nonlinear Schrodinger equation using an improved PINN method

Cited by in corpus (12)

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• Physics-informed neural network methods based on Miura transformations and discovery of new localized wave solutions
• Physics-informed neural networks method in high-dimensional integrable systems
• Complex dynamics on the one-dimensional quantum droplets via time piecewise PINNs
• Data-driven discoveries of Bäcklund transforms and soliton evolution equations via deep neural network learning schemes
• FFV-PINN: A Fast Physics-Informed Neural Network with Simplified Finite Volume Discretization and Residual Correction
• A deep learning method for solving high-order nonlinear soliton equation
• Invariant deep neural networks under the finite group for solving partial differential equations
• Modified physics-informed neural network method based on the conservation law constraint and its prediction of optical solitons
• Discovery of Quasi-Integrable Equations from traveling-wave data using the Physics-Informed Neural Networks