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Computer Assisted Discovery of Integrability via SILO: Sparse Identification of Lax Operators

Jimmie Adriazola, Wei Zhu, Panayotis Kevrekidis, Alejandro Aceves

TL;DR

This work presents SILO, a sparse symbolic-regression framework that casts the discovery of Lax integrability for Hamiltonian systems as an optimization over candidate Lax pairs $(L,P)$ via the equation $\frac{dL}{dt}=[L,P]$ and the condition $\{L,H\}-[L,P]=0$. By enforcing sparsity through a thresholded $l^0$ regularization and solving a nonconvex loss with cross-entropy followed by local refinement, SILO achieves high-precision integrability detection and recovers interpretable Lax pairs across SHO, HH, KdV, and NLS, including novel Lax-pair structures. The approach also demonstrates robustness to nonintegrable perturbations, revealing integrable parameter regimes and exposing weak Lax-pair forms that still reproduce the target dynamics. While successful in many cases, the method also highlights limitations tied to operator-hypothesis choices and function-space selections, suggesting future work on broader libraries, stronger guarantees, and potential quantum-field-theoretic extensions through similar sparse-regression strategies. Overall, SILO offers a transparent, data-driven tool for automated exploration of integrable Hamiltonian dynamics with potential extensions toward Liouville integrability and related integrable structures.

Abstract

We formulate the discovery of Lax integrability of Hamiltonian dynamical systems as a symbolic regression problem, which, loosely speaking, seeks to maximize the compatibility between a pair of Lax operators and the known Hamiltonian of the dynamical system. Our approach is first tested on the simple harmonic oscillator. We then move on to the Henon-Heiles system, i.e. a two-degree-of-freedom system of nonlinear oscillators. The integrability of the Henon-Heiles system is critically dependent on a set of three parameters within its Hamiltonian, a fact that we leverage to assess the robustness of our approach in detecting the integrability of this system with respect to the parameter dependence of the Hamiltonian. We then adapt our method to canonical examples of Hamiltonian partial differential equations, including the Korteweg-de Vries and cubic nonlinear Schrödinger equations, again testing robustness against nonintegrable perturbations of their respective Hamiltonians. In all examples, our approach reliably confirms or denies the integrability of the equations of interest. Moreover, by appropriately adjusting the loss function and applying thresholded $l^0$ regularization to enforce sparsity in the operator weights, we successfully recover accurate forms of the Lax pairs despite wide initial hypotheses on the operators. Some of the relevant Lax pairs, notably for the Henon-Heiles system and the Korteweg-deVries equation, are distinct from the ones that are typically reported in the literature. The Lax pairs that our methodology discovers warrant further mathematical and computational investigation, and we discuss extensively the opportunities for further improvement of SILO as a viable tool for interpretable exploration of integrable Hamiltonian dynamical systems.

Computer Assisted Discovery of Integrability via SILO: Sparse Identification of Lax Operators

TL;DR

This work presents SILO, a sparse symbolic-regression framework that casts the discovery of Lax integrability for Hamiltonian systems as an optimization over candidate Lax pairs via the equation and the condition . By enforcing sparsity through a thresholded regularization and solving a nonconvex loss with cross-entropy followed by local refinement, SILO achieves high-precision integrability detection and recovers interpretable Lax pairs across SHO, HH, KdV, and NLS, including novel Lax-pair structures. The approach also demonstrates robustness to nonintegrable perturbations, revealing integrable parameter regimes and exposing weak Lax-pair forms that still reproduce the target dynamics. While successful in many cases, the method also highlights limitations tied to operator-hypothesis choices and function-space selections, suggesting future work on broader libraries, stronger guarantees, and potential quantum-field-theoretic extensions through similar sparse-regression strategies. Overall, SILO offers a transparent, data-driven tool for automated exploration of integrable Hamiltonian dynamics with potential extensions toward Liouville integrability and related integrable structures.

Abstract

We formulate the discovery of Lax integrability of Hamiltonian dynamical systems as a symbolic regression problem, which, loosely speaking, seeks to maximize the compatibility between a pair of Lax operators and the known Hamiltonian of the dynamical system. Our approach is first tested on the simple harmonic oscillator. We then move on to the Henon-Heiles system, i.e. a two-degree-of-freedom system of nonlinear oscillators. The integrability of the Henon-Heiles system is critically dependent on a set of three parameters within its Hamiltonian, a fact that we leverage to assess the robustness of our approach in detecting the integrability of this system with respect to the parameter dependence of the Hamiltonian. We then adapt our method to canonical examples of Hamiltonian partial differential equations, including the Korteweg-de Vries and cubic nonlinear Schrödinger equations, again testing robustness against nonintegrable perturbations of their respective Hamiltonians. In all examples, our approach reliably confirms or denies the integrability of the equations of interest. Moreover, by appropriately adjusting the loss function and applying thresholded regularization to enforce sparsity in the operator weights, we successfully recover accurate forms of the Lax pairs despite wide initial hypotheses on the operators. Some of the relevant Lax pairs, notably for the Henon-Heiles system and the Korteweg-deVries equation, are distinct from the ones that are typically reported in the literature. The Lax pairs that our methodology discovers warrant further mathematical and computational investigation, and we discuss extensively the opportunities for further improvement of SILO as a viable tool for interpretable exploration of integrable Hamiltonian dynamical systems.

Paper Structure

This paper contains 7 sections, 4 theorems, 64 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Warm Up Problem: The Simple Harmonic Oscillator
  3. A Two Degree of Freedom Scenario: Henon-Heiles
  4. The Korteweg-deVries Equation
  5. The Cubic Nonlinear Schrödinger Equation
  6. Conclusion and Outlook
  7. Acknowledgments

Key Result

Theorem 4.1

For every $u\in C^1([0,T];C^3(\mathbb{R}))$, there exists a parameter $v\in\mathbb{R}$ such that the pair of operators satisfying Lax's equation $\partial_tL=[L,P],$ understood as acting on the function space $C^3(\mathbb{R}),$ reproduces the KdV equation in the co-traveling reference frame $x\to x-vt$.

Figures (8)

  • Figure 1: Two numerical results from studying Problem \ref{['eq:SHOProb']} for the simple harmonic oscillator \ref{['eq:SHO']}. We show the result of optimizing before and after thresholding. SILO correctly identifies that there should only be 6 parameters in the 24 parameter operator hypothesis. SILO also finds both families of Lax pairs as given by Equations \ref{['eq:FirstSHOLax']} and \ref{['eq:SecondSHOLax']}.
  • Figure 2: The correct identification, to 6 digits of precision, of a Lax pair for the integrable case of the HH system given by Hamiltonian \ref{['eq:HHHam']}. The panel on the left reproduces the expected structure of the known Lax pair. The panel on the right discovers that, up to a sign, the transpose of the known Lax pair is also valid for reproducing the HH system.
  • Figure 3: A broad parameter search (with $B$ fixed to 1) for integrability detection in the HH system given by Hamiltonian \ref{['eq:HHHam']}. The optimization of the loss function, shown on a logarithmic scale, identifies a distinct position at $(A, \varepsilon) = (1, 1/3)$ where integrability is meaningfully detected, differing by several orders of magnitude from background loss values. Cubic spline interpolation of the landscape is used for visualization.
  • Figure 4: A numerical result of solving Problem \ref{['eq:KdVProb']} without sparsification. Visualized here is a cross-validation study displaying the generalized Poisson brackets and commutators evaluated at the optimal point $\eta^*$ and on four samples from the function space $\Omega$ that were unseen during training. For all four cases, the loss is on the order of $10^{-11}$.
  • Figure 5: A perturbation study using the perturbed Hamiltonian density $h_1=\frac{\varepsilon_1}{2}\left(\partial_x^2 u\right)^2$. Shown here is the numerical solution of Problem \ref{['eq:KdVProb']}, without sparsification, with density $h+\varepsilon h_1$. We observe a near-smooth dependence on $\varepsilon$ with a clearly discernible "special" point associated with the detection of integrability at $\varepsilon=0$.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Theorem 4.1: Existence of a new KdV Lax pair
  • proof
  • Theorem 4.2: Existence of a Weak Lax Pair
  • proof
  • Theorem 5.1: Existence of Weak Lax Pairs for the Linear and Nonlinear Schrödinger Equations
  • proof
  • Theorem 5.2: Sparsest Weak Lax Pair for the NLS Equation
  • proof