D-Lax-Pair-FIND: Data-driven reconstruction of the evolution operators in Lax pairs for discrete integrable equations

The Lax pair stands as a central concept in the theory of integrable systems, whose discovery holds considerable significance for solving and analyzing nonlinear evolution equations. To address the challenge of discovering Lax pairs from observational data, we propose D-Lax-Pair-FIND, a data-driven methodology that synergistically combines deep learning with sparse regression to automatically discover the evolution operators in Lax pairs of discrete integrable equations with known spectral operators. The approach employs fully connected feedforward neural networks to approximate solutions of semi-discrete systems, treating the discrete spatial variable as continuous to enable interpolation at non-integer lattice points. By constructing a predefined library of candidate operators and leveraging sparse regularization techniques, the method automatically identifies the linear evolution operators. The loss function comprises three key components: data-fitting loss, discrete Lax compatibility loss, and sparse regularization loss. Through joint optimization of the neural network parameters and the sparse coefficient vector, the method achieves robust discovery of Lax pairs. Numerical experiments demonstrate that the proposed framework successfully recovers the evolution operators in Lax pairs for several classical discrete integrable systems, including the self-dual network equation, the Ablowitz-Ladik equation, and the Toda lattice-covering both the form with explicit spectral parameters and the operator form. This research provides an effective and general computational framework for automatically uncovering the intrinsic structure of discrete integrable systems from limited data, with potential implications for advancing data-driven studies in integrable systems theory.