Lattice fermion formulation via Physics-Informed Neural Networks: Ginsparg-Wilson relation and Overlap fermions

TL;DR

This paper introduces a physics-informed neural network framework to construct lattice fermions, accurately reproducing the overlap fermion operator and discovering new solutions to the Ginsparg-Wilson relation.

Contribution

It presents a novel machine-learning approach that formulates lattice fermion construction as an optimization problem guided by physical constraints, enabling autonomous discovery of solutions.

Findings

Neural networks reproduce the overlap fermion operator with high accuracy.

The framework can discover solutions to the Ginsparg-Wilson relation autonomously.

It finds a new solution corresponding to a Fujikawa-type generalized GW relation.

Abstract

We propose a novel, machine-learning-based framework for constructing lattice fermions using Physics-Informed Neural Networks (PINNs). Our approach treats the formulation of the Dirac operator as an optimization problem guided by physical requirements, such as symmetries, locality and doubler-decoupling conditions. We first demonstrate that, when trained to satisfy the Ginsparg-Wilson (GW) relation as a soft constraint, a neural network reproduces the overlap fermion operator to high numerical accuracy and learns an effective sign-function mapping without explicitly using a prescribed polynomial or rational approximation. Secondly, we extend the framework from operator construction to machine-assisted algebraic discovery. Within a generalized polynomial ansatz, the network autonomously drives higher-order terms to zero and recovers the standard Ginsparg-Wilson relation. Remarkably, by changing the initial search bias, the same framework also finds a distinct solution corresponding to a Fujikawa-type generalized GW relation.