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Deep Learning based discovery of Integrable Systems

Shailesh Lal, Suvajit Majumder, Evgeny Sobko

TL;DR

This work introduces an AI-assisted pipeline to systematically discover new integrable spin-chain models by first solving for high-precision numerical R-matrices within a constrained class using a neural network ensemble, and then extracting exact analytical Hamiltonians from the numerical seeds via the Reshetikhin condition. The approach combines a neural R-matrix solver with algebraic-geometry-based extraction to produce explicit Hamiltonian families that lie on algebraic varieties, often rational, across 3- and 4-dimensional site spaces. Key contributions include hundreds of new integrable Hamiltonians, detailed examples (including a 25-vertex model with a three-parameter linear variety) and a general, scalable workflow for translating numerical solutions into exact algebraic relations. The method offers a scalable platform for automated discovery of integrable systems, with potential extensions to 2D QFTs, AdS sigma models, and other areas where exact solvability and algebraic structure are central.

Abstract

We introduce a novel machine learning based framework for discovering integrable models. Our approach first employs a synchronized ensemble of neural networks to find high-precision numerical solution to the Yang-Baxter equation within a specified class. Then, using an auxiliary system of algebraic equations, [Q_2, Q_3] = 0, and the numerical value of the Hamiltonian obtained via deep learning as a seed, we reconstruct the entire Hamiltonian family, forming an algebraic variety. We illustrate our presentation with three- and four-dimensional spin chains of difference form with local interactions. Remarkably, all discovered Hamiltonian families form rational varieties.

Deep Learning based discovery of Integrable Systems

TL;DR

This work introduces an AI-assisted pipeline to systematically discover new integrable spin-chain models by first solving for high-precision numerical R-matrices within a constrained class using a neural network ensemble, and then extracting exact analytical Hamiltonians from the numerical seeds via the Reshetikhin condition. The approach combines a neural R-matrix solver with algebraic-geometry-based extraction to produce explicit Hamiltonian families that lie on algebraic varieties, often rational, across 3- and 4-dimensional site spaces. Key contributions include hundreds of new integrable Hamiltonians, detailed examples (including a 25-vertex model with a three-parameter linear variety) and a general, scalable workflow for translating numerical solutions into exact algebraic relations. The method offers a scalable platform for automated discovery of integrable systems, with potential extensions to 2D QFTs, AdS sigma models, and other areas where exact solvability and algebraic structure are central.

Abstract

We introduce a novel machine learning based framework for discovering integrable models. Our approach first employs a synchronized ensemble of neural networks to find high-precision numerical solution to the Yang-Baxter equation within a specified class. Then, using an auxiliary system of algebraic equations, [Q_2, Q_3] = 0, and the numerical value of the Hamiltonian obtained via deep learning as a seed, we reconstruct the entire Hamiltonian family, forming an algebraic variety. We illustrate our presentation with three- and four-dimensional spin chains of difference form with local interactions. Remarkably, all discovered Hamiltonian families form rational varieties.

Paper Structure

This paper contains 18 sections, 45 equations, 3 figures, 1 table.

Table of Contents

  1. Overview
  2. Quantum integrable spin-chains
  3. Families of integrable models as algebraic varieties
  4. Neural Networks as numerical solvers of the Yang-Baxter Equation
  5. Averting Mode Collapse
  6. Neural Network Implementation and Training
  7. Hamiltonian Extraction
  8. Examples of New Models
  9. Example 1
  10. Example 2
  11. Example 3
  12. Example 4
  13. Hundreds of other examples:
  14. Discussion and future directions
  15. R-matrix for the upper-triangular 15-vertex Model
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Compact Scheme of our workflow
  • Figure 2: The evolution of the training and validation losses for a typical round of training the neural network. The x axis is plotted in units of 100 iterations of gradient descent. The y axis is in log scale. The mode collapse loss is turned off at $200\times 100$ iterations, indicated by the gray vertical line.
  • Figure 3: The decomposition of the validation loss into its various constituents in a typical round of training the network. The x axis is plotted in units of 100 iterations of gradient descent. The y axis is in log scale. The mode collapse loss is turned off at $200\times 100$ iterations, indicated by the gray vertical line.