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Two-stage initial-value iterative physics-informed neural networks for simulating solitary waves of nonlinear wave equations

Jin Song, Ming Zhong, George Em Karniadakis, Zhenya Yan

TL;DR

The paper tackles solitary-wave computations for nonlinear wave equations where boundary data are unavailable or insufficient. It introduces a two-stage initial-value iterative neural network (IINN) that combines an initial-value fitting network with a PDE-residual enforcing network via transfer learning, and provides theoretical guarantees for convergence under appropriate initialization. Through extensive 1D, 2D, and 3D examples—including NLS, KdV, KP, and GP-type models—the method achieves high accuracy with relative $L_2$ errors around $10^{-4}$ to $10^{-3}$ and demonstrates robustness across multiple soliton states. Compared with traditional soliton methods and standard PINNs, IINN offers a mesh-free, data-light framework that can realize multiple solitary-wave branches by choosing suitable initial values and leveraging two-stage training.

Abstract

We propose a new two-stage initial-value iterative neural network (IINN) algorithm for solitary wave computations of nonlinear wave equations based on traditional numerical iterative methods and physics-informed neural networks (PINNs). Specifically, the IINN framework consists of two subnetworks, one of which is used to fit a given initial value, and the other incorporates physical information and continues training on the basis of the first subnetwork. Importantly, the IINN method does not require any additional data information including boundary conditions, apart from the given initial value. Corresponding theoretical guarantees are provided to demonstrate the effectiveness of our IINN method. The proposed IINN method is efficiently applied to learn some types of solutions in different nonlinear wave equations, including the one-dimensional (1D) nonlinear Schrödinger equations (NLS) equation (with and without potentials), the 1D saturable NLS equation with PT -symmetric optical lattices, the 1D focusing-defocusing coupled NLS equations, the KdV equation, the two-dimensional (2D) NLS equation with potentials, the 2D amended GP equation with a potential, the (2+1)-dimensional KP equation, and the 3D NLS equation with a potential. These applications serve as evidence for the efficacy of our method. Finally, by comparing with the traditional methods, we demonstrate the advantages of the proposed IINN method.

Two-stage initial-value iterative physics-informed neural networks for simulating solitary waves of nonlinear wave equations

TL;DR

The paper tackles solitary-wave computations for nonlinear wave equations where boundary data are unavailable or insufficient. It introduces a two-stage initial-value iterative neural network (IINN) that combines an initial-value fitting network with a PDE-residual enforcing network via transfer learning, and provides theoretical guarantees for convergence under appropriate initialization. Through extensive 1D, 2D, and 3D examples—including NLS, KdV, KP, and GP-type models—the method achieves high accuracy with relative errors around to and demonstrates robustness across multiple soliton states. Compared with traditional soliton methods and standard PINNs, IINN offers a mesh-free, data-light framework that can realize multiple solitary-wave branches by choosing suitable initial values and leveraging two-stage training.

Abstract

We propose a new two-stage initial-value iterative neural network (IINN) algorithm for solitary wave computations of nonlinear wave equations based on traditional numerical iterative methods and physics-informed neural networks (PINNs). Specifically, the IINN framework consists of two subnetworks, one of which is used to fit a given initial value, and the other incorporates physical information and continues training on the basis of the first subnetwork. Importantly, the IINN method does not require any additional data information including boundary conditions, apart from the given initial value. Corresponding theoretical guarantees are provided to demonstrate the effectiveness of our IINN method. The proposed IINN method is efficiently applied to learn some types of solutions in different nonlinear wave equations, including the one-dimensional (1D) nonlinear Schrödinger equations (NLS) equation (with and without potentials), the 1D saturable NLS equation with PT -symmetric optical lattices, the 1D focusing-defocusing coupled NLS equations, the KdV equation, the two-dimensional (2D) NLS equation with potentials, the 2D amended GP equation with a potential, the (2+1)-dimensional KP equation, and the 3D NLS equation with a potential. These applications serve as evidence for the efficacy of our method. Finally, by comparing with the traditional methods, we demonstrate the advantages of the proposed IINN method.
Paper Structure (13 sections, 3 theorems, 82 equations, 17 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 3 theorems, 82 equations, 17 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some notations and definitions
  4. Problem statement
  5. Traditional numerical methods
  6. The PINNs method
  7. Methodology and applications
  8. Methodology: the IINN framework
  9. Examples
  10. Comparison between IINN method and traditional method
  11. Limitations of PINNs method
  12. Advantages over traditional numerical methods
  13. Summary

Key Result

Lemma 1

Let $\Lambda=\bigcup_{i=1}^{N} \Lambda_i$, where $\Lambda_i=\left\{\theta_i\,|\, \mathbf{L_0}\hat{\mathbf{u}}(\theta_i)=0\right\}$ and for any $\theta_i^m$, $\theta_i^n\in\Lambda_i$, $\|\hat{\mathbf{u}}(\theta_i^m)-\hat{\mathbf{u}}(\theta_i^n)\|_2=0$, and $N$ is the number of distinct solitary wave

Figures (17)

  • Figure 1: The schematic diagram of IINN method.
  • Figure 2: The soliton solutions $u(x)$ of 1D NLS equation (\ref{['NLS1d']}) in free space ($V=0$). (a1) The learned bright soliton solution and exact one at $\mu=-2$ in self-focusing case. (b1) The learned dark soliton solution and exact one at $\mu=2$ in self-defocusing case. (a2, b2) The loss-iteration plot of $\mathrm{NN}_2$, where the vertical axis represents $\log_{10}\mathcal{L}_2$ (the same hereinafter). (a3) The conserved quantities $\int_{\mathbf{R}}\omega\mathrm{d} x$ versus iteration for bright soliton, where $K_1$ and $K_2$ denote $\omega_1=UU^*$ and $\omega_2=UU_x^*$, respectively. (a4) The conserved quantity error versus iteration, where $E^i=\log_{10}|K_i-K_i^*|$ and $K_i^*$ is the true value of conserved quantity.
  • Figure 3: The ground state and dipole mode of 1D NLS equation (\ref{['NLS1d']}) with HG potential (\ref{['HO']}). (a1) The learned and exact ground state solution at $V_0=-4$, $\mu=1$ and $g=-1$. (b1) The learned and exact dipole mode at $V_0=-4$, $\mu=1$ and $g=-1$. (a2, b2) The loss-iteration plot of $\mathrm{NN}_2$ for ground state and dipole mode, respectively. (a3, b3) The conserved quantities $\int_{\mathbf{R}}\omega\mathrm{d} x$ versus iteration for ground state and dipole mode respectively, where $K_1$ and $K_2$ denote $\omega_1=UU^*$ and $\omega_2=UU_x^*$, respectively. (a4, b4) The conserved quantity error versus iteration for ground state and dipole mode respectively, where $E^i=\log_{10}|K_i-K_i^*|$ and $K_i^*$ is the true value of conserved quantity.
  • Figure 4: The dipole mode of 1D NLS equation (\ref{['NLS1d']}) with HG potential (\ref{['HO']}) using the eigenmodes in the linear regime as the initial value. (a1) The initial value $u_0$ (\ref{['hgeu0']}) and $u_{0e}=A\psi$ with the first excited state $\psi$ in the linear regime, where $A=8$. (a2) The learned and exact dipole mode at $V_0=-4$, $\mu=1$ and $g=-1$. (a3) The loss-iteration plot of $\mathrm{NN}_2$.
  • Figure 5: The complex solution $u(x)$ of 1D NLS equation (\ref{['NLS1d']}) with Scarf-II potential (\ref{['scarf']}) in self-focusing and self-defocusing cases. In self-focusing case: (a1, a2, a3) The real part, imaginary part and intensity diagrams of learned solution and exact one at $V_0=-1$, $W_0=-1$, $\mu=-1$ and $g=-1$. In self-defocusing case: (b1, b2, b3) The real part, imaginary part and intensity diagrams of learned solution and exact one at $V_0=-3$, $W_0=-1$, $\mu=-1$ and $g=1$. (a4, b4) The loss-iteration plot of $\mathrm{NN}_2$.
  • ...and 12 more figures

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Remark 2
  • Lemma 1
  • Proof
  • Theorem 1
  • Theorem 2
  • Proof
  • ...and 4 more