Neural Network Framekwork for Modelling, Control, and Estimation of Physical Systems

Modeling, control, and estimation of physical systems are central to many engineering disciplines. While data-driven methods like neural networks offer powerful tools, they often struggle to incorporate prior domain knowledge, limiting their interpretability, generalizability, and safety.

To bridge this gap, we present nnodely (where "nn" can be read as "m," forming Modely) — a framework that facilitates the creation and deployment of Model-Structured Neural Networks (MS-NNs).
MS-NNs combine the learning capabilities of neural networks with structural priors grounded in physics, control, and estimation theory, enabling:

• Reduced training data requirements
• Generalization to unseen scenarios
• Real-time deployment in real-world applications

In short:

nnodely is not a replacement for a general purpose deep learning frameworks — it is a structured layer on top of them, purpose-built for physical systems.

📖 Documentation • 🔬 Case Studies • 🚀 Other Applications

Table of Contents

1. Getting Started
• Installation
• Hello, World!
2. Structure of the Repository
3. How to contribute
4. License
5. References

Getting Started

Installation

You can install nnodely from PyPI via:

pip install nnodely

Alternatively, you can build it from source by first cloning the repository and installing the requirements and the nnodely library:

git clone https://github.com/tonegas/nnodely.git
cd nnodely
pip install -r requirements.txt
pip install .

Hello, World!

To check if nnodely is installed correctly try running the following script.

from nnodely import Input, Output, nnodely, Parameter

x = Input("x")
l = x.last()
o = (Parameter('A')*l.sw([-2,-1])+Parameter('B')*l).closedLoop(x)
f = Output("fib",o)
model = nnodely()
model.addModel("Fibonacci",f)
model.addMinimize("target", l.sw([-2,-1])+l, f)
model.neuralizeModel(1)
model.loadData("data",{ "x" : list(range(100)) } )
model.trainModel(prediction_samples = 2, lr = 0.5, num_of_epochs = 500)
model.exportPythonModel(models = "Fibonacci")
print(model({ "x" : [1] },prediction_samples = 20,num_of_samples = 20))

In the example, the neural network is trained to mimic the Fibonacci series. Finally, a native pytorch network is exported in a file.

(back to top)

Structure of the Repository

nnodely/              # root directory
├── nnodely/          # source code
│   ├── basic/        # core low-level classes
│   ├── exporter/     # model export utilities
│   ├── layers/       # supported layers
│   ├── operators/    # core operators
│   ├── support/      # utility functions
│   └── visualizer/   # visualization tools
├── case-studies/     # main case studies
├── docs/             # documentation
├── tests/            # unit and integration tests
├── imgs/             # images used in the documentation
└── mplplots/         # utilities for MatPlotLib

(back to top)

How to Contribute

To contribute to the nnodely framework, you can:

• Open a pull request if you have a new feature or bug fix.
• Open an issue if you have a question or suggestion.

We welcome contributions and collaborations.

(back to top)

License

This project is released under the license License: MIT.

(back to top)

References

[1] Mauro Da Lio, Daniele Bortoluzzi, Gastone Pietro Rosati Papini. (2019). Modelling longitudinal vehicle dynamics with neural networks. Vehicle System Dynamics. https://doi.org/10.1080/00423114.2019.1638947 (look the [code])

[2] Alice Plebe, Mauro Da Lio, Daniele Bortoluzzi. (2019). On Reliable Neural Network Sensorimotor Control in Autonomous Vehicles. IEEE Transaction on Intelligent Transportation System. https://doi.org/10.1109/TITS.2019.2896375

[3] Mauro Da Lio, Riccardo Donà, Gastone Pietro Rosati Papini, Francesco Biral, Henrik Svensson. (2020). A Mental Simulation Approach for Learning Neural-Network Predictive Control (in Self-Driving Cars). IEEE Access. https://doi.org/10.1109/ACCESS.2020.3032780 (look the [code])

[4] Edoardo Pagot, Mattia Piccinini, Enrico Bertolazzi, Francesco Biral. (2023). Fast Planning and Tracking of Complex Autonomous Parking Maneuvers With Optimal Control and Pseudo-Neural Networks. IEEE Access. https://doi.org/10.1109/ACCESS.2023.3330431 (look the [code])

[5] Mattia Piccinini, Sebastiano Taddei, Matteo Larcher, Mattia Piazza, Francesco Biral. (2023). A Physics-Driven Artificial Agent for Online Time-Optimal Vehicle Motion Planning and Control. IEEE Access. https://doi.org/10.1109/ACCESS.2023.3274836 (look [code basic] and [code extended])

[6] Hector Perez-Villeda, Justus Piater, Matteo Saveriano. (2023). Learning and extrapolation of robotic skills using task-parameterized equation learner networks. Robotics and Autonomous Systems. https://doi.org/10.1016/j.robot.2022.104309 (look the [code])

[7] M. Raissi. P. Perdikaris b, G.E. Karniadakis a. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations Journal of Computational Physics. https://doi.org/10.1016/j.jcp.2018.10.045 (look the [example Burger's equation])

[8] Wojciech Marian Czarnecki, Simon Osindero, Max Jaderberg, Grzegorz Świrszcz, Razvan Pascanu. (2017). Sobolev Training for Neural Networks. arXiv. https://doi.org/10.48550/arXiv.1706.04859 (look the [code])

[9] Mattia Piccinini, Matteo Zumerle, Johannes Betz, Gastone Pietro Rosati Papini. (2025). A Road Friction-Aware Anti-Lock Braking System Based on Model-Structured Neural Networks. IEEE Open Journal of Intelligent Transportation Systems. https://doi.org/10.1109/OJITS.2025.3563347 (look at the [code])

[10] Mauro Da Lio, Mattia Piccinini, Francesco Biral. (2023). Robust and Sample-Efficient Estimation of Vehicle Lateral Velocity Using Neural Networks With Explainable Structure Informed by Kinematic Principles. IEEE Transactions on Intelligent Transportation Systems. https://doi.org/10.1109/TITS.2023.3303776

(back to top)