Enabling local neural operators to perform equation-free system-level analysis

Abstract

Neural operators (NOs) offer a powerful computational framework to learn complex spatiotemporal dynamics as mappings between infinite-dimensional function spaces. Still, NOs have primarily been used as surrogates for brute-force temporal simulations and predictions. Their potential for systematic, system-level numerical analysis, such as fixed-point, stability and bifurcation analysis—crucial for predicting irreversible transitions in real-world phenomena—remains largely unexplored. Here, motivated by the equation-free approach, we develop a framework that integrates (local) NOs with iterative numerical analysis methods in the Krylov subspace. This approach expands their potential beyond simulation to efficiently analyse large-scale dynamical systems and tackle fundamental challenges in computer-assisted modelling and numerical analysis of complex systems. Furthermore, we demonstrate the utility of learning local in space–time NOs, which, combined with multiscale equation-free schemes—such as projective integration, Gap-Tooth and Patch Dynamics—accelerate system-level computations, improve the conditioning of Krylov solvers and reduce memory requirements, enabling efficient multiscale analysis of complex spatiotemporal dynamics. We illustrate our framework via three nonlinear partial differential equations (PDE) benchmarks: the one-dimensional Allen–Cahn equation, which undergoes multiple concatenated pitchfork bifurcations, the Liouville–Bratu–Gelfand PDE, which features a saddle-node tipping point, and the FitzHugh–Nagumo model, consisting of two coupled PDEs that exhibit both Hopf and saddle-node bifurcations. Moving beyond brute-force simulations, local neural operators—combined with equation-free methods and Krylov subspace techniques—enable system-level stability and bifurcation analysis of complex spatiotemporal systems directly from data.