Data-driven discovery of bounded polynomial ODEs

Welcome to SILAS! This repository contains code and data used to produce results in the following paper:

Albert Alcalde and Giovanni Fantuzzi, Data-driven discovery of polynomial ODEs with provably bounded solutions. arXiv:2604.26933 (2026).

Most of the code is in MATLAB, with some Python scripts to generate data and post-process results. Details on the repository structure and a short description of each file are given below. To replicate the examples of the paper, you can simply run the scripts ex01_discovery.m, ex02_main.m and ex03_main.m, adjusting parameters as needed to match those given in the paper.

The code is provided "as-is" and without any guarantees. See the license file for more information.

Authors: Albert Alcalde, Giovanni Fantuzzi (FAU Erlangen-Nuernberg)

Dependencies

To run the code, you must first install the following MATLAB packages:

• YALMIP
• MOSEK (or another SDP solver compatible with YALMIP)
• ChebFun

Cite us

If you find this code useful in your work and publish your results, please cite the arXiv preprint above. An sample .bib code follows.

    @article{FantuzziAlcalde2026,
        AUTHOR = {Alcalde, Albert and Fantuzzi, Giovanni},
        TITLE = {Data-driven discovery of polynomial ODEs with provably bounded solutions},
        YEAR = {2026},
        HOWPUBLISHED = {arXiv:2604.26933 [math.DS]},
        URL = {https://arxiv.org/abs/2604.26933}
    }

Repository Structure

silas-main/
├── LICENSE.txt
├── README.md
├── ex01-cubic/
├── ex02-dysts/
├── ex03-pde/
└── utils/

Files

Root

File	Description
LICENSE.txt	Project license.
README.md	Original project documentation.

ex01-cubic

File	Description
ex01_simulate.m	Simulates the cubic example and generates data.
ex01_discovery.m	Runs model discovery on the cubic example.

ex02-dysts

File	Description
ex02_main.m	Main script for learning models from Dysts dynamical-system data.
ex02_preprocess_data.m	Prepares raw Dysts data for training and testing.
ex02_attractor_plots.m	Generates attractor plots for Dysts systems.

ex02-dysts/private

File	Description
exact_AtmosphericRegime.m	Exact reference equations for the AtmosphericRegime system.
exact_Bouali2.m	Exact reference equations for the Bouali2 system.
exact_GuckenheimerHolmes.m	Exact reference equations for the Guckenheimer-Holmes system.
exact_HyperRossler.m	Exact reference equations for the HyperRossler system.
exact_SprottA.m	Exact reference equations for the SprottA system.
exact_Thomas.m	Exact reference equations for the Thomas system.
exact_YuWang.m	Exact reference equations for the YuWang system.
plot_lyap.m	Plots learned Lyapunov functions.

ex02-dysts/python-scripts

File	Description
data_generate.ipynb	Notebook for generating Dysts datasets.
process_results.ipynb	Notebook for processing and summarizing experiment results.

ex03-pde

File	Description
ex03_main.m	Main MATLAB script for the PDE reduced-order modeling example.
ex03_helpers.py	Python helper functions for PDE simulations and preprocessing.
ex03_simulation.py	Generates PDE simulation data.

ex03-pde/private

File	Description
lyap_plotProjDeg2.m	Plots projected degree-2 Lyapunov functions.
plot_attractors.m	Plots attractors for the PDE reduced model.
plot_pde_solution.m	Visualizes PDE solution trajectories.

utils

File	Description
build_initial_lyap_constrained.m	Builds an initial constrained Lyapunov candidate.
build_initial_lyap_penalized.m	Builds an initial penalized Lyapunov candidate.
build_lyap_model.m	Constructs a Lyapunov model.
build_model_lyap.m	Builds the combined learned model and Lyapunov structure.
cheb_evaluate.m	Evaluates Chebyshev polynomial expansions.
cheb_gradient.m	Computes gradients of Chebyshev expansions.
cheb_mommat.m	Builds Chebyshev moment matrices.
cheb_multiply.m	Multiplies Chebyshev polynomial terms.
lyap_initialize.m	Initializes Lyapunov-function learning.
lyap_load.m	Loads saved Lyapunov-function results.
make_parameters.m	Creates parameter settings for experiments.
model_cheb2mon.m	Converts models from Chebyshev to monomial form.
model_evaluate.m	Evaluates learned dynamical models.
model_initialize.m	Initializes model-learning variables.
model_learn.m	Learns a dynamical model from data.
model_load.m	Loads saved learned models.
multi_indices.m	Generates polynomial multi-index sets.
save_results.m	Saves experiment outputs.
select_fekete.m	Selects interpolation/sample points using Fekete-style criteria.
viridis.m	Provides the Viridis colormap for plots.

Funding

Funded bu:

• The European Union’s Horizon Europe MSCA project ModConFlex (grant number 101073558)
• The DFG-ANR project MONET (DFG grant number 568735456)