PERC: Physics Enforced Reservoir Computing

Apply hard physical constraints directly to Reservoir Computers! This repository contains the supporting code for the L4DC 2025 paper Physics-Enforced Reservoir Computing for Forecasting Spatiotemporal Systems by Dima Tretiak, Anastasia Bizyaeva, J. Nathan Kutz, and Steven L. Brunton.

PERC enforces hard constraints by solving a constrained optimization problem during the Ridge Regression step of RC training. For a given time series $u(t)$, PERC guarantees adhereance to linear invariants (any linear conservation laws) of the form $Cu(t) = d$. The examples are split into categories where $d=0$ (linear homogenous constraints) and $d \neq 0$ (linear inhomogenous constraints). Furthermore, nonlinear constraints of the form $p(u(t)) = d$ can be also promoted with PERC.

Installation

To install perc please clone the repository then install via pip

git clone https://github.com/dtretiak/PhysicsEnforcedReservoirComputing.git
cd PhysicsEnforcedReservoirComputing
pip install .

Examples

All examples from the paper are included as jupyter notebooks ./notebooks/.

• For a basic unconstrained RC example please see basic_RC_example.ipynb
• For linear homogeneous constraints of the form $Cu = 0$, please see kol_flow.ipynb and ks.ipynb
• For linear inhomogenous constraints of the form $Cu = d$, please see lotka_volterra.ipynb and heat_eq.ipynb
• For nonlinear constraints, please see hamiltonian.ipynb

Data Availability

Most of the examples shown in the paper contain simple enough datasets that can be simulated directly prior to training. All of the integrators can be found in perc.integrators. However, the Kolmogorov Flow simulation is performed separately via the integrator found here. Some data is provided below in order to demonstrate the PERC algorithm, but please note the usage guidelines for the Kolmogorov Flow integrator and contact the owner of that repository prior to using the data for other purposes.

In order to download the fluids data, first install the hugging face CLI

pip install -U "huggingface_hub[cli]"

Then download the data set into ./data/

huggingface-cli download dtretiak/PERC_data --include "kol_flow/*" --repo-type dataset --local-dir ./data

Citation

If you find this work useful in your endeavors, please consider citing it:

@InProceedings{tretiak25a_pmlr,
  title = {Physics-Enforced Reservoir Computing for Forecasting Spatiotemporal Systems},
  author = {Tretiak, Dima and Bizyaeva, Anastasia and Kutz, J. Nathan and Brunton, Steven L.},
  booktitle = {Proceedings of the 7th Annual Learning for Dynamics \& Control Conference},
  pages = {350--364},
  year = {2025},
  editor = {Ozay, Necmiye and Balzano, Laura and Panagou, Dimitra and Abate, Alessandro},
  volume = {283},
  series = {Proceedings of Machine Learning Research},
  month = {04--06 Jun},
  publisher = {PMLR},
  pdf = {https://raw.githubusercontent.com/mlresearch/v283/main/assets/tretiak25a/tretiak25a.pdf},
  url = {https://proceedings.mlr.press/v283/tretiak25a.html},
}