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

arXiv: https://arxiv.org/abs/2604.19465 · HTML: https://mlsanity.com/paper/2604.19465/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 92 references that are in the map.
- First posted on arXiv 21 Apr 2026.

## Builds on (most cited first, 20 of 20)

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- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Deep Reinforcement Learning in a Handful of Trials using Probabilistic Dynamics Models](https://mlsanity.com/paper/1805.12114/) (2018)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
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- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flow](https://mlsanity.com/paper/1802.10123/) (2018)
- [Deep Dynamical Modeling and Control of Unsteady Fluid Flows](https://mlsanity.com/paper/1805.07472/) (2018)
- [APEBench: A Benchmark for Autoregressive Neural Emulators of PDEs](https://mlsanity.com/paper/2411.00180/) (2024)
- [Towards Stability of Autoregressive Neural Operators](https://mlsanity.com/paper/2306.10619/) (2023)
- [Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition](https://mlsanity.com/paper/1409.5496/) (2014)
- [Data-driven discovery of intrinsic dynamics](https://mlsanity.com/paper/2108.05928/) (2021)
- [Learned Turbulence Modelling with Differentiable Fluid Solvers: Physics-based Loss-functions and Optimisation Horizons](https://mlsanity.com/paper/2202.06988/) (2022)
- [Sparsity-promoting dynamic mode decomposition](https://mlsanity.com/paper/1309.4165/) (2013)
- [Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)](https://mlsanity.com/paper/2106.01510/) (2021)
- [Dynamic Mode Decomposition for Real-Time Background/Foreground Separation in Video](https://mlsanity.com/paper/1404.7592/) (2014)
- [From bypass transition to flow control and data-driven turbulence modeling: An input-output viewpoint](https://mlsanity.com/paper/2003.10104/) (2020)

## Abstract (arXiv)

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 and 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 compute eigenmodes and resolvent modes directly from data. We demonstrate the method on both canonical chaotic models and high-dimensional fluid flows, successfully identifying dominant instability modes and input-output structures even in strongly nonlinear regimes. By leveraging a neural network-based 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 immediate relevance to grand challenges in fields such as climate science, neuroscience, and fluid engineering.