Deep Learning based discovery of Integrable Systems
Abstract : Integrable systems are exactly solvable models that play a central role in QFT,
string theory and statistical physics offering an ideal setting for understanding complex physical
phenomena and developing novel analytical methods. However, the discovery of new integrable sys-
tems remains a major open challenge due to the nonlinearity of the Yang–Baxter equation (YBE)
that defines them, and the vastness of its solution space. Here we present the first AI-based frame-
work that enables the discovery of new quantum integrable systems in exact analytical form. Our
method combines an ensemble of neural networks, trained to identify high-precision numerical so-
lutions to the YBE, with an algebraic extraction procedure based on the Reshetikhin integrability
condition, which reconstructs the corresponding Hamiltonian families analytically. When applied
to spin chains with three- and four-dimensional site spaces, we discover hundreds of previously un-
known integrable Hamiltonians. Remarkably, these Hamiltonians organize into rational algebraic
varieties, and we conjecture that this rationality holds universally — revealing a previously
unexplored connection between quantum integrability and algebraic geometry. By unlocking inte-
grable systems far beyond the reach of traditional methods, this AI-driven approach substantially
expands the landscape of exactly solvable models and opens a scalable path to further discoveries.