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

2025-05-05

AI Summary

This study addresses the challenge of predicting irreversible phase transitions in real-world systems by proposing an equation-free dynamical systems analysis framework based on Local Neural Operators (LNOs). Methodologically, it introduces the first integration of spatiotemporal LNOs into Krylov subspace iterative methods, enabling multiscale fixed-point computation, stability assessment, and bifurcation analysis without requiring explicit governing differential equations. Unlike conventional surrogate models, the framework supports rigorous system-level analysis and directly identifies critical phenomena. Evaluated on three canonical nonlinear PDE benchmarks—Allen–Cahn, Liouville–Bratu–Gelfand, and FitzHugh–Nagumo—the approach successfully detects multiple pitchfork, Hopf, and saddle-node bifurcations. It significantly accelerates computer-aided modeling and analysis of spatiotemporal dynamics while preserving mathematical fidelity and interpretability.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Learning with Manifolds

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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.

Problem

• Enable local Neural Operators for system-level stability analysis
• Integrate NOs with Krylov subspace methods for bifurcation analysis
• Apply local NOs to accelerate spatiotemporal dynamics analysis

Innovation

• Integrates local Neural Operators with Krylov subspace methods
• Enables system-level stability and bifurcation analysis
• Uses space-time patch NOs for spatiotemporal dynamics