Enabling Local Neural Operators to perform Equation-Free System-Level Analysis

TL;DR

This paper introduces a framework that combines neural operators with numerical methods to enable efficient system-level analysis of large-scale dynamical systems modeled by PDEs, including stability and bifurcation analysis.

Contribution

The paper presents a novel integration of local neural operators with iterative numerical methods for system-level stability and bifurcation analysis of PDE-based models.

Findings

Successfully applied to 1D Allen-Cahn equation with bifurcations

Demonstrated analysis of saddle-node tipping point in Liouville-Bratu-Gelfand PDE

Analyzed Hopf and saddle-node bifurcations in FitzHugh-Nagumo model

Abstract

Neural Operators (NOs) provide a powerful framework for computations involving physical laws that can be modelled by (integro-) partial differential equations (PDEs), directly learning maps between infinite-dimensional function spaces that bypass both the explicit equation identification and their subsequent numerical solving. Still, NOs have so far primarily been employed to explore the dynamical behavior as surrogates of brute-force temporal simulations/predictions. Their potential for systematic rigorous numerical system-level tasks, such as fixed-point, stability, and bifurcation analysis - crucial for predicting irreversible transitions in real-world phenomena - remains largely unexplored. Toward this aim, inspired by the Equation-Free multiscale framework, we propose and implement a framework that integrates (local) NOs with advanced iterative numerical methods in the Krylov subspace, so as to perform efficient system-level stability and bifurcation analysis of large-scale dynamical systems. Beyond fixed point, stability, and bifurcation analysis enabled by local in time NOs, we also demonstrate the usefulness of local in space as well as in space-time ("patch") NOs in accelerating the computer-aided analysis of spatiotemporal dynamics. We illustrate our framework via three nonlinear PDE benchmarks: the 1D 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 (FHN) model, consisting of two coupled PDEs that exhibit both Hopf and saddle-node bifurcations.