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Data-driven localized waves and parameter discovery in the massive Thirring model via extended physics-informed neural networks with interface zones

arXiv 2309.17240 ↗First posted 29 Sep 2023nlin.PS cs.LG cs.NAcited by 0

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Abstract (arXiv)

In this paper, we study data-driven localized wave solutions and parameter discovery in the massive Thirring (MT) model via the deep learning in the framework of physics-informed neural networks (PINNs) algorithm. Abundant data-driven solutions including soliton of bright/dark type, breather and rogue wave are simulated accurately and analyzed contrastively with relative and absolute errors. For higher-order localized wave solutions, we employ the extended PINNs (XPINNs) with domain decomposition to capture the complete pictures of dynamic behaviors such as soliton collisions, breather oscillations and rogue-wave superposition. In particular, we modify the interface line in domain decomposition of XPINNs into a small interface zone and introduce the pseudo initial, residual and gradient conditions as interface conditions linked adjacently with individual neural networks. Then this modified approach is applied successfully to various solutions ranging from bright-bright soliton, dark-dark soliton, dark-antidark soliton, general breather, Kuznetsov-Ma breather and second-order rogue wave. Experimental results show that this improved version of XPINNs reduce the complexity of computation with faster convergence rate and keep the quality of learned solutions with smoother stitching performance as well. For the inverse problems, the unknown coefficient parameters of linear and nonlinear terms in the MT model are identified accurately with and without noise by using the classical PINNs algorithm.

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References · 20

  1. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  2. 2020Universal Differential Equations for Scientific Machine Learningcited by 353
  3. 2021Physics-informed neural networks with hard constraints for inverse designcited by 280
  4. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  5. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 147
  6. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 118
  7. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  8. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 102
  9. 2022Physics-informed neural networks for inverse problems in supersonic flowscited by 76
  10. 2019Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networkscited by 64
  11. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  12. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 58
8 more
  1. 2021Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functionscited by 49
  2. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  3. 2022Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodologycited by 44
  4. 2022How important are activation functions in regression and classification? A survey, performance comparison, and future directionscited by 29
  5. 2020Data-driven rogue waves and parameter discovery in the defocusing NLS equation with a potential using the PINN deep learningcited by 12
  6. 2021Data-driven peakon and periodic peakon travelling wave solutions of some nonlinear dispersive equations via deep learningcited by 6
  7. 2021Data-driven discoveries of B"acklund transforms and soliton evolution equations via deep neural network learning schemescited by 1
  8. 2022Data-driven soliton mappings for integrable fractional nonlinear wave equations via deep learning with Fourier neural operatorcited by 1

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  1. 2025Wave-PDE Nets: Trainable Wave-Equation Layers as an Alternative to Attention
  2. 2026Windowed Fourier Propagator: A Frequency-Local Neural Operator for Wave Equations in Inhomogeneous Media
  3. 2023LatentPINNs: Generative physics-informed neural networks via a latent representation learning
  4. 2026Oscillatory State-Space Models as Inductive Biases for Physics-Informed Neural PDE Solvers
  5. 2025Sub-Sequential Physics-Informed Learning with State Space Model
  6. 2024Partition of Unity Physics-Informed Neural Networks (POU-PINNs): An Unsupervised Framework for Physics-Informed Domain Decomposition and Mixtures of Experts
  7. 2024Domain-decoupled Physics-informed Neural Networks with Closed-form Gradients for Fast Model Learning of Dynamical Systems
  8. 2025DiffPINN: Generative diffusion-initialized physics-informed neural networks for accelerating seismic wavefield representation