Equation-Free Stability Analysis for Complex Systems

The fundamental challenge of predicting complex system behavior under perturbation, whether it will remain stable or identify its most sensitive patterns, has long been constrained by the need for known equations and linearization. This limitation has historically hindered analysis in nonlinear or incompletely understood systems.

Unlocking Dynamics with Neural Emulation

A novel data-driven framework is introduced, capable of automatically discerning stability properties and optimal forcing responses solely from observation data. By training a neural network to emulate system dynamics, and subsequently employing automatic differentiation to derive its Jacobian, researchers can now compute eigenmodes and resolvent modes directly from empirical data. This approach bypasses the requirement for explicit governing equations.

Bridging Nonlinearity and Data-Driven Stability Analysis

The method's efficacy is demonstrated on both canonical chaotic models and high-dimensional fluid flows. It successfully identifies dominant instability modes and input-output structures, even in strongly nonlinear regimes. This neural network-based emulation provides a nonlinear representation of system dynamics, revealing intricate dynamical patterns that were previously intractable. This equation-free methodology represents a significant advancement in data-driven stability analysis, offering a broadly applicable tool for dissecting complex, high-dimensional datasets across diverse scientific and engineering domains, including climate science, neuroscience, and fluid engineering.