nnodely Applications

Applications of Model-Structured Neural Networks using nnodely. This repository contains a list of applications of the nnodely framework with the related references.

General Applications

Mass-Spring-Damper System

The file presents the modeling of a mass-spring-damper system. The network estimate the next position and the next velocity of the mass.

Pendulum

The file presents the modeling of a pendulum. The network estimate the next position of the pendulum.

Nonlinear Function Fitting

The file presents the modeling of a nonlinear function. The network estimate the value of the function with a family of models.

Vehicle Applications

Longitudinal Vehicle Dynamics

The file presents the modeling of the longitudinal vehicle dynamics described in:

@article{DaLio2020Modelling,
    author = {Mauro Da Lio, Daniele Bortoluzzi and Gastone Pietro Rosati Papini},
    title = {Modelling longitudinal vehicle dynamics with neural networks},
    journal = {Vehicle System Dynamics},
    volume = {58},
    number = {11},
    pages = {1675--1693},
    year = {2020},
    publisher = {Taylor \& Francis},
    doi = {10.1080/00423114.2019.1638947}
}

Lateral Vehicle Dynamics

The file presents the modeling of the lateral vehicle dynamics described in:

@article{DaLio2020Mental,
  author={Da Lio, Mauro and Donà, Riccardo and Papini, Gastone Pietro Rosati and Biral, Francesco and Svensson, Henrik},
  journal={IEEE Access}, 
  title={A Mental Simulation Approach for Learning Neural-Network Predictive Control (in Self-Driving Cars)}, 
  year={2020},
  volume={8},
  number={},
  pages={192041-192064},
  doi={10.1109/ACCESS.2020.3032780}
}

Control Steer Car Parking

The file presents a neural network for the control of the steering angle for parking maneuvers, as described in:

@article{Pagot2023Fast,
  author={Pagot, Edoardo and Piccinini, Mattia and Bertolazzi, Enrico and Biral, Francesco},
  journal={IEEE Access}, 
  title={Fast Planning and Tracking of Complex Autonomous Parking Maneuvers With Optimal Control and Pseudo-Neural Networks}, 
  year={2023},
  volume={11},
  number={},
  pages={124163-124180},
  doi={10.1109/ACCESS.2023.3330431}
}

Control Steer Artificial Race Driver

The file presents a neural network for the control of the steering angle for an artificial race driver, as described in:

@article{piccinini2023physics,
  author={Piccinini, Mattia and Taddei, Sebastiano and Larcher, Matteo and Piazza, Mattia and Biral, Francesco},
  journal={IEEE Access}, 
  title={A Physics-Driven Artificial Agent for Online Time-Optimal Vehicle Motion Planning and Control}, 
  year={2023},
  volume={11},
  number={},
  pages={46344-46372},
  keywords={Motion planning;Biological system modeling;Vehicle dynamics;Tracking;Load modeling;Artificial neural networks;Computational modeling;Neural networks;Autonomous racing;model learning;model predictive control (MPC);motion planning;neural networks;trajectory optimization},
  doi={10.1109/ACCESS.2023.3274836}
}

@INPROCEEDINGS{piccinini2025model,
  author={Piccinini, Mattia and Mungiello, Aniello and Jank, Georg and Papini, Gastone Pietro Rosati and Biral, Francesco and Betz, Johannes},
  booktitle={2025 IEEE 28th International Conference on Intelligent Transportation Systems (ITSC)}, 
  title={Model-Structured Neural Networks to Control the Steering Dynamics of Autonomous Race Cars}, 
  year={2025},
  volume={},
  number={},
  pages={},
  note={accepted}
}

Vehicle Mass Estimation

The file presents a neural network for the estimation of the vehicle mass.

Road Friction Aware ABS

The file presents a neural network for a road friction-aware ABS system, by learning the longitudinal vehicle dynamics in different road conditions. This is described in:

@ARTICLE{Piccinini_2025_RoadFrictionAwareABS,
  author={Piccinini, Mattia and Zumerle, Matteo and Betz, Johannes and Pietro Rosati Papini, Gastone},
  journal={IEEE Open Journal of Intelligent Transportation Systems}, 
  title={A Road Friction-Aware Anti-Lock Braking System Based on Model-Structured Neural Networks}, 
  year={2025},
  volume={6},
  number={},
  pages={522-536},
  doi={10.1109/OJITS.2025.3563347}
}

Other Applications

Equation Learner Network

The file presents a simple example of the Equation Learner Network via nnodely. The core ideas is presented in:

@article{perezvilleda2023learning,
    title = {Learning and extrapolation of robotic skills using task-parameterized equation learner networks},
    journal = {Robotics and Autonomous Systems},
    volume = {160},
    pages = {104309},
    year = {2023},
    issn = {0921-8890},
    doi = {https://doi.org/10.1016/j.robot.2022.104309},
    author = {Hector Perez-Villeda and Justus Piater and Matteo Saveriano},
    keywords = {Learning from demonstration, Learning parameterized skills, Skill generalization and extrapolation, Equation learner neural networks}
}

Sobolev Learning

The file presents a simple implementation and test of a Sobolev learning via nnodely. The core ideas is presented in:

@misc{czarnecki2017sobolev,
      title={Sobolev Training for Neural Networks}, 
      author={Wojciech Marian Czarnecki and Simon Osindero and Max Jaderberg and Grzegorz Świrszcz and Razvan Pascanu},
      year={2017},
      eprint={1706.04859},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/1706.04859}, 
}

Physics-Informed Neural Networks

The file presents a simple implementation for solving the Burger's equation using Physics-Informed Neural Networks via nnodely. The main idea of PINN is presented in:

@article{RAISSI2019686,
    title = {Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations},
    journal = {Journal of Computational Physics},
    volume = {378},
    pages = {686-707},
    year = {2019},
    issn = {0021-9991},
    doi = {https://doi.org/10.1016/j.jcp.2018.10.045},
    author = {M. Raissi and P. Perdikaris and G.E. Karniadakis},
}