Source-linked AI summary
A two-stage physics-informed neural network method based on conserved quantities and applications in localized wave solutions
Shuning Lin, Yong Chen
TL;DR
The paper addresses the challenge of improving PINNs for nonlinear partial differential equations by incorporating equation-specific physical information and global constraints. It proposes a two-stage conserved-quantity PINN and applies it to localized waves in integrable equations, reporting improved prediction accuracy and generalization over the original PINN.
Problem
The original PINN imposes local constraints at selected points, motivating a more targeted approach that can incorporate global physical information for nonlinear integrable equations.
Method
Stage one applies the original PINN, while stage two adds measurements of conserved quantities to the mean squared error loss.
Results
The method reproduces localized-wave dynamics including soliton molecules, M-shape double-peak solitons, plateau solitons, single-peak solitons, and interaction solutions.
Takeaways & Limitations
The two-stage method improves prediction accuracy and generalization while introducing integrable-system features into PINNs for more targeted equation solving.
Takeaways & Limitations
The proposed method increases training cost while improving accuracy, motivating future work on preserving efficiency.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper proposes a two-stage PINN that incorporates conserved quantities to impose global physical constraints and better match nonlinear integrable equations. It applies the method to reproduce localized-wave dynamics in several Boussinesq-Burgers and Sawada-Kotera systems.
- Proposed method: The authors develop a two-stage PINN that introduces conserved-quantity measurements into the mean squared error during a second training stage.Stage one uses the original PINN; stage two adds conserved quantities to the loss.
- Motivation: The method targets integrable systems by incorporating equation-specific physical information, including conserved quantities, into neural networks.The approach is motivated by integrable systems’ exact solutions, symmetries, and conservation laws.
- Proposed method: Global constraints from conserved quantities complement the original PINN’s local constraints at selected points.This design aims to impose physical constraints from a global perspective.
- Applications: The improved PINN is applied to nonlinear integrable equations, including Boussinesq-Burgers, classical Boussinesq-Burgers, and Sawada-Kotera equations.The applications focus on localized-wave solutions and their dynamic behaviors.
- Applications: The study reproduces localized-wave behaviors such as interaction solutions, soliton molecules, and M-shape double-peak solitons.The paper also considers plateau solitons and related solution types across the studied equations.
2. Methodology
The methodology reviews PINNs for nonlinear PDEs and introduces a two-stage variant that combines local equation and data constraints with globally measured conserved quantities. The method first trains an original PINN, then further optimizes it using conserved-quantity information to improve the numerical solution.
- Original PINN: The network maps input coordinates to a predicted solution through affine transformations and nonlinear activation functions across weighted layers.The architecture contains one input layer, hidden layers, and one output layer with trainable weights and biases.
- Original PINN: PINNs approximate nonlinear PDE solutions by minimizing mean squared errors that combine initial-boundary data with residuals of the governing equation.Automatic differentiation supplies the required derivatives, and network parameters are optimized against the resulting loss.
- Conserved quantities: Conserved quantities are time-independent integrals of conserved densities, and integrable systems can possess infinitely many such quantities.Their associated conservation laws relate conserved densities to fluxes and hold for all solutions of the governing equation.
- Two-stage PINN: The two-stage method applies the original PINN first, then introduces conserved-quantity measurements into the loss to impose constraints from a global perspective.The conserved-quantity calculation uses numerical integration by summation, complementing the local constraints imposed at selected points.
- Two-stage PINN: Directly optimizing a loss containing conserved-quantity error can converge to an undesirable local optimum, whereas the two-stage procedure further optimizes the first-stage solution.The authors report that the staged procedure avoids other non-ideal local optima and improves the numerical results.
3. Data-driven one-soliton solution of the Boussinesq-Burgers equations and interaction solution of the classical Boussinesq-Burgers equations
The two-stage PINN reproduces one-soliton and soliton–resonance interaction solutions for coupled Boussinesq-Burgers equations, while conserved-quantity training lowers prediction errors relative to the original PINN.
- Method: The two-stage PINN uses the original PINN in stage one and adds conserved-quantity measurements to the mean squared error in stage two.The method is applied to coupled Boussinesq-Burgers and classical Boussinesq-Burgers equations.
- One-soliton solution: The method obtains a data-driven one-soliton solution whose predicted u and v solutions closely reproduce the exact solutions and propagation behavior.Both fields propagate in the positive x direction, and v is identified as a dark soliton solution.
- One-soliton solution: 18.09% and 20.49% are the error reduction rates for u and v, respectively, when conserved-quantity training is compared with the original PINN.Stage-two relative L2 errors are 7.343612e-04 for u and 3.776971e-02 for v.
- Interaction solution: The method successfully simulates the classical Boussinesq-Burgers interaction between a soliton and a resonance.The predicted solution shows two peaks converging into one higher-amplitude peak in v and propagating toward negative x.
- Interaction solution: 22.06% and 22.04% are the error reduction rates for u and v, respectively, in the interaction-solution experiment.Stage-two relative L2 errors are 2.756669e-04 for u and 2.576679e-03 for v.
4. Data-driven soliton molecule and new types of solitons of the Sawada-Kotera equation
The two-stage PINN reproduces soliton molecules and several new Sawada-Kotera wave structures. Adding conserved quantities improves numerical accuracy across the examined localized solutions.
- Scope and motivation: The Sawada-Kotera equation is studied for data-driven soliton molecules and new soliton types, addressing limited prior PINN study of soliton molecules.The reproduced structures include soliton molecules, kink-antikink molecules, M-shape double-peak solitons, plateau solitons, and single-peak solitons.
- Problem setup: The Sawada-Kotera problem uses Dirichlet boundary conditions on a finite spatiotemporal domain, with the governing fifth-order nonlinear equation enforced by the PINN.The equation is posed on x ∈ [x0, x1] and t ∈ [t0, t1], together with initial and boundary values.
- Method and implementation: The soliton-molecule model is trained with a 9-layer, 40-neuron-per-hidden-layer network, automatic differentiation, and two-stage losses incorporating conserved-quantity measurements.The numerical setup uses 513 × 201 discretization points, 100 initial-boundary points, 2000 collocation points, and L-BFGS optimization.
- Soliton molecule: 4.336727e-03 relative L2 error is obtained for the soliton molecule in stage two, compared with 4.841810e-03 for the original PINN.Stage two required 7107 iterations and about 3987.8433 seconds, versus 10736 iterations and about 5551.9234 seconds in stage one.
- New soliton types: 8.148663e-04 and 3.678135e-04 relative L2 errors are reported for the M-shape double-peak soliton across the two numerical experiments.The corresponding stage-one errors are 1.557140e-03 and 1.766738e-03; the method also reproduces plateau and single-peak solitons with lower errors in stage two.
- Accuracy comparison: The plateau soliton reaches an error reduction rate of 79.18%, while three-dimensional plots show reproduction of four distinct dynamic behaviors.The reported comparison identifies conserved-quantity training as improving prediction accuracy across the examined soliton structures.
5. Conclusion
The paper proposes a conserved-quantity two-stage PINN to impose global physical constraints and applies it to localized waves in integrable equations. The reported experiments show improved accuracy and generalization, with increased training cost remaining a limitation.
- Method: The method applies the original PINN first, then adds conserved-quantity measurements to the mean squared error for further optimization.This design targets equation-specific physical information and global rather than only pointwise constraints.
- Applications: The method is demonstrated on one-soliton and interaction solutions of Boussinesq-Burgers systems and four localized Sawada-Kotera wave structures.The Sawada-Kotera examples include soliton molecule, M-shape double-peak, plateau, and single-peak solitons.
- Findings: Error-reduction comparisons indicate that the two-stage PINN improves prediction accuracy and enhances generalization relative to the original PINN.The comparison is based on relative L2 errors and reported error reduction rates.
- Limitation: The proposed method increases training cost while improving accuracy, motivating future work on efficiency-preserving improvements.The authors identify avoiding this efficiency trade-off as a future research goal.