Modeling and Control of Asymptotically Stable Systems Using a Foward Tracking Neural ODE Approach (FTNODE)

This repository contains the code to reproduce the examples from the paper: Modeling and feedback control of asymptotically stable nonlinear dynamical systems

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Overview

This work presents a methodology for learning dynamical systems with enforced asymptotic stability using Neural ODEs, enabling reliable short-horizon learning and accurate, noise-robust feedback control across systems exhibiting nontrivial bifurcations.

Methodology

Model Architecture

The approach uses a tunable model of the form:

F_θ(x, u) = f_θ(x)(x - g_θ(x, u))

Where:

• f_θ and g_θ are multi-layer perceptrons (MLPs) with Sigmoid-weighted Linear Unit (SiLU) activation functions and sigmoidal output layers
• Asymptotic stability is enforced by requiring that f_θ(x)<0 everywhere on the domain. and g_θ remains bounded under bounded input

Training Objectives

Two training objectives are implemented:

1. Trajectory Matching: Standard NODE training using backpropagation to compare simulated trajectories against observed trajectories
2. Gradient Matching: Compares the learned model against gradient information computed using centered finite differences

Examples

The repository includes implementations for the following systems:

• Mixing Tanks
• Symmetric Hysteresis
• Budworm Population
• Genetic Toggle Switch

Requirements

• PyTorch
• torchode (Parallel ODE Solver for PyTorch)

Installation & Reproducibility

This repository supports both general use as a library and exact reproducibility for the results presented in the associated paper.

General Use

If you want to use the Forward Tracking Neural ODE Approach (ftnode) in your own projects alongside your existing packages, you can install it directly.

# Clone the repository
git clone https://github.com/pnnl/FTNODE.git
cd FTNODE

# Install the package
pip install .

Reproducing Results

If you are looking to reproduce the results from examples/, you may use the frozen requirements.txt file.

# 1. Clone the repository and navigate into it
git clone https://github.com/pnnl/FTNODE.git
cd FTNODE

# 2. Create and activate a fresh virtual environment
python -m venv ftnode_env

# On Windows:
# ftnode_env\Scripts\activate
# On macOS/Linux:
source ftnode_env/bin/activate

# 3. Install the exact dependency tree
pip install -r requirements.txt

# 4. Install the package locally in editable mode
pip install -e .

Usage

Each example contains specific hyperparameters including:

• Number of training epochs
• Batch size
• Initial learning rate
• Model architecture specifications (layer widths, bounds)

Refer to individual example scripts for detailed configuration.

Citation

If you use this code, please cite the associated paper.

License

Copyright Battelle Memorial Institute 2026

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.

2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

Disclaimer

DISCLAIMER This material was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the United States Department of Energy, nor Battelle, nor any of their employees, nor any jurisdiction or organization that has cooperated in the development of these materials, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness or any information, apparatus, product, software, or process disclosed, or represents that its use would not infringe privately owned rights.

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