Data-driven algorithm workflow: (1) Collect trajectory snapshots, (2) Train NN emulator, (3) Extract Jacobian via automatic differentiation, (4) Perform modal analysis

Understanding how complex systems evlove and respond to perturbations is fundamental across science and engineering. This work introduces a data-driven framework that automatically identifies stability properties and optimal forcing responses from observation data alone, without requiring governing equations.

Our approach bridges the gap between data-driven and operator-based methods by:

1. Training neural emulators on trajectory data to learn nonlinear dynamics
2. Computing local Jacobians via automatic differentiation at any state
3. Extracting stability modes through eigenvalue decomposition
4. Identifying optimal forcings via resolvent analysis

The key relationship: N ≈ exp(A∆t) where N is the NN-based dicrete-time Jacobian and A is the operator-based continuous-time Jacobian.

# Clone the repository
git clone https://github.com/tum-pbs/NonlinearRA.git
cd NonlinearRA
 
# Create conda environment
conda create -n NonlinearRA python=3.12
conda activate NonlinearRA
 
# Install dependencies
pip install -r requirements.txt

See the README in each case folder for specific usage:

• lorenz/README.md - Lorenz system examples
• channel2D/README.md - 2D channel flow examples

Note: Only Lorenz and 2D channel flow cases are provided here. More complex examples (Complex Ginzburg-Landau, reduced 3D channel flow) are available upon request from the author.

If you use this code in your research, please cite:

@misc{wang2026neuraloperatorframeworkdatadriven,
      title={A neural operator framework for data-driven discovery of stability and receptivity in physical systems}, 
      author={Chengyun Wang and Liwei Chen and Nils Thuerey},
      year={2026},
      eprint={2604.19465},
      archivePrefix={arXiv},
      primaryClass={physics.flu-dyn},
      url={https://arxiv.org/abs/2604.19465}, 
}

• Chengyun Wang: chengyun.wang@tum.de
• Nils Thürey: nils.thuerey@tum.de

This repository is released under the license provided in LICENSE.