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Machine Learning of Nonlinear Waves: Data-Driven Methods for Computer-Assisted Discovery of Equations, Symmetries, Conservation Laws, and Integrability

Jimmie Adriazola, Panayotis G. Kevrekidis, Vassilis Koukouloyannis, Wei Zhu

TL;DR

The article surveys how data-driven and machine-learning methods can augment the nonlinear waves field, focusing on physics-informed neural networks (PINNs), operator learning, and sparse/symbolic regression to discover equations, symmetries, conservation laws, and integrability. It highlights concrete applications to lattice dynamics, soliton interactions, and Hamiltonian structure, and introduces structure-preserving variants (S-PINNs, Lax-pair learning, GENERIC-informed models) to maintain physical invariants and long-term stability. The work emphasizes synergistic use of ML with classical theory, detailing limitations, strategies for robustness, and a roadmap toward data-driven integrability, reduced-order modeling, and symbolic discovery. The practical impact lies in enabling rapid discovery of governing laws, reduced models, and hidden structures in nonlinear wave systems, with potential applications in BECs, optical lattices, and dispersive hydrodynamics. It points to future directions such as soliton-gas dynamics, Whitham theory, and Koopman/operator-based approaches to broaden the reach of data-driven nonlinear-wave science while preserving interpretability and physical consistency.

Abstract

The purpose of this article is to provide a perspective - admittedly, a rather subjective one - of recent developments at the interface of machine learning/data-driven methods and nonlinear wave studies. We review some recent pillars of the rapidly evolving landscape of scientific machine learning, including deep learning, data-driven equation discovery, and operator learning, among others. We then showcase these methods in applications ranging from learning lattice dynamical models and reduced order modeling of effective dynamics to discovery of conservation laws and potential identification of integrability of ODE and PDE models. Our intention is to make clear that these machine learning methods are complementary to the preexisting powerful tools of the nonlinear waves community, and should be integrated into this toolkit to augment and enable mathematical discoveries and computational capabilities in the age of data.

Machine Learning of Nonlinear Waves: Data-Driven Methods for Computer-Assisted Discovery of Equations, Symmetries, Conservation Laws, and Integrability

TL;DR

The article surveys how data-driven and machine-learning methods can augment the nonlinear waves field, focusing on physics-informed neural networks (PINNs), operator learning, and sparse/symbolic regression to discover equations, symmetries, conservation laws, and integrability. It highlights concrete applications to lattice dynamics, soliton interactions, and Hamiltonian structure, and introduces structure-preserving variants (S-PINNs, Lax-pair learning, GENERIC-informed models) to maintain physical invariants and long-term stability. The work emphasizes synergistic use of ML with classical theory, detailing limitations, strategies for robustness, and a roadmap toward data-driven integrability, reduced-order modeling, and symbolic discovery. The practical impact lies in enabling rapid discovery of governing laws, reduced models, and hidden structures in nonlinear wave systems, with potential applications in BECs, optical lattices, and dispersive hydrodynamics. It points to future directions such as soliton-gas dynamics, Whitham theory, and Koopman/operator-based approaches to broaden the reach of data-driven nonlinear-wave science while preserving interpretability and physical consistency.

Abstract

The purpose of this article is to provide a perspective - admittedly, a rather subjective one - of recent developments at the interface of machine learning/data-driven methods and nonlinear wave studies. We review some recent pillars of the rapidly evolving landscape of scientific machine learning, including deep learning, data-driven equation discovery, and operator learning, among others. We then showcase these methods in applications ranging from learning lattice dynamical models and reduced order modeling of effective dynamics to discovery of conservation laws and potential identification of integrability of ODE and PDE models. Our intention is to make clear that these machine learning methods are complementary to the preexisting powerful tools of the nonlinear waves community, and should be integrated into this toolkit to augment and enable mathematical discoveries and computational capabilities in the age of data.
Paper Structure (26 sections, 2 theorems, 147 equations, 12 figures)

This paper contains 26 sections, 2 theorems, 147 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Recent Pillars of Scientific Machine Learning
  3. Deep Learning
  4. Neural Network Architectures.
  5. Learning Functions through Physics-Informed Neural Networks
  6. Deep Operator Learning.
  7. Sparse and Symbolic Regression
  8. Applications of Physics-Informed Neural Networks
  9. PINNs for Learning Nonlinear Dynamics on Lattices
  10. Graph Neural Networks Learn Short and Long Range Interactions on Lattices
  11. Structure-Preserving PINNs
  12. PINNs for Scale- and Translational-Invariant Wave Solution Identification: the Burgers Equation Case Example
  13. Lax Pair Informed Neural Networks
  14. Discovering Dynamical Systems and Reduced Order Models from Dispersive PDE
  15. Use of SINDy to Develop Moment Equation Reductions Stemming from Nonlinear Dispersive PDEs
  16. ...and 11 more sections

Key Result

Theorem 1

Let $K \subset C(D)$ be a compact set, and let $\mathcal{G}: K \to \mathbb{R}$ be a continuous operator. Then, there exists a DeepONet $\mathcal{G}_\theta$ of the form above such that for any $\varepsilon > 0$ and fixed $y \in Y$.

Figures (12)

  • Figure 1: Discrete $\phi^{4}$ model numerical results [cf. Eq. \ref{['dphi4']}] for a coupling of $C=2$, adapted from Ref. SAQLAIN2023107498. The library $\mathrm{Lib}^{(1)}$ of Eq. \ref{['dphi4_lib1']} was considered in panel (a) with the solid blue, red, green and yellow lines corresponding to the discrete representation of the second and first derivative, as well as $u$, and $u^{3}$, respectively. The numerical results obtained by using the library $\mathrm{Lib}^{(2)}$ [cf. Eq. \ref{['dphi4_lib2']} are presented in panel (b) where solid blue, red, green, and yellow depict the $u_{n+1}$, $u_{n-1}$, $u_{n}$, and $u_{n}^{3}$, respectively. Panels (c) and (d) utilized the libraries of Eqs. \ref{['dphi4_lib3']} and \ref{['dphi4_lib4']}, respectively (with the same notational conventions). The solid black lines therein correspond to (c) the terms $u_{n\pm2}$, and (d) to all the other cubic terms. It is relevant to highlight that the dashed lines represent the reference values for the coefficients.
  • Figure 2: This figure, adapted from PGKGeng, illustrates evolution over time of the average true conservation values and the average predicted values of 20 samples for the different networks. C.Q. represents conserved quantity.
  • Figure 3: Numerical results, adapted from zhu2022neural, for the KM soliton using PINN and S-PINN. Top panels show the spatio-temporal evolution of the amplitude $|\psi(n,t)|$ for the exact solution (a), PINN (b), and S-PINN (c). Bottom panels show the spatial profile at $t = -0.67$ and the temporal evolution at $n = 0$. Solid blue lines indicate the exact solution, while dashed green and red lines correspond to PINN and S-PINN, respectively. Standard PINN fails to capture the time-periodicity and spatio-temporal parity symmetry \ref{['eq:parity_symmetry']}, whereas S-PINN accurately preserves both.
  • Figure 4: PINN prediction of the self-similar dynamics for the Burgers equation adapted from the work of kavousanakis2025symmetrypinns. (a) Convergence of the total loss $\mathcal{E}_{loss}$ during training for the rescaled/co-moving Burgers equation. (d) Snapshots of rescaled PINN predicted solution $w$ at different $\tau$ values. The converged self-similar solution $w$ (at $\tau \approx 3$) and the analytically predicted self-similar solution whitham2011linear are practically coincident. (c) Evolution of the scaling rates $\partial_\tau A/A$ and $\partial_\tau c/A$ over rescaled time $\tau$. For sufficiently long $\tau$ values ($\tau>3$), we can observe the convergence of the rescaled solution to a stationary, self-similar profile.
  • Figure 5: [Adapted from suyang] Comparison of the moment evolutions of $[I_2, V_1, K, J, K+J]$ between SINDy and the ground truth. The training occurs for SINDy only on the selected moments $\mathbf{x} = [I_2, V_1, K, J]$, where a closure does not exist, using a linear library $\mathbf{\Theta}_{\deg=1}(\mathbf{x})$. Interestingly, SINDy captures the correct dynamics for $E=K+J$, although not so the individual ones of $K$ and $J$.
  • ...and 7 more figures

Theorems & Definitions (2)

  • Theorem 1: Universal Approximation of Operators
  • Theorem 2: Existence of a new KdV Lax pair