A neural operator framework for data-driven discovery of stability and receptivity in physical systems

Understanding how complex systems respond to perturbations, such as whether they will remain stable or what their most sensitive patterns are, is a fundamental challenge across science and engineering. Traditional stability and receptivity (resolvent) analyses are powerful but rely on known equations and linearization, limiting their use in nonlinear or poorly modeled systems. Here, we introduce a data-driven framework that automatically identifies stability properties of both equilibrium points and periodic orbits, as well as optimal forcing responses from observation data alone, without requiring governing equations. By training a neural network as a dynamics emulator and using automatic differentiation to extract its Jacobian, we can perform stability and receptivity analyses directly from data. We demonstrate the method on systems of increasing complexity, from canonical models to high-dimensional fluid flows and a real-world climate dataset, employing model reduction where needed and successfully identifying dominant instability modes and optimal input-output structures, even in strongly nonlinear regimes. By leveraging a neural network emulator, we readily obtain a nonlinear representation of system dynamics while additionally retrieving intricate dynamical patterns that were previously difficult to resolve. This equation-free methodology establishes a broadly applicable tool for analyzing complex, high-dimensional datasets, with potential relevance to grand challenges in fields such as climate science, neuroscience, and fluid engineering.