# Guaranteed Conformance of Neurosymbolic Models to Natural Constraints

arXiv: https://arxiv.org/abs/2212.01346 · HTML: https://mlsanity.com/paper/2212.01346/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 51 references that are in the map.
- First posted on arXiv 2 Dec 2022.

## Builds on (most cited first, 19 of 19)

- [CARLA: An Open Urban Driving Simulator](https://mlsanity.com/paper/1711.03938/) (2017)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [Neural Lander: Stable Drone Landing Control using Learned Dynamics](https://mlsanity.com/paper/1811.08027/) (2018)
- [Learning to Fly -- a Gym Environment with PyBullet Physics for Reinforcement Learning of Multi-agent Quadcopter Control](https://mlsanity.com/paper/2103.02142/) (2021)
- [Lagrangian Duality for Constrained Deep Learning](https://mlsanity.com/paper/2001.09394/) (2020)
- [Imposing Hard Constraints on Deep Networks: Promises and Limitations](https://mlsanity.com/paper/1706.02025/) (2017)
- [Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modeling](https://mlsanity.com/paper/2109.06407/) (2021)
- [Hamiltonian-based Neural ODE Networks on the SE(3) Manifold For Dynamics Learning and Control](https://mlsanity.com/paper/2106.12782/) (2021)
- [Modeling System Dynamics with Physics-Informed Neural Networks Based on Lagrangian Mechanics](https://mlsanity.com/paper/2005.14617/) (2020)
- [Constrained Deep Networks: Lagrangian Optimization via Log-Barrier Extensions](https://mlsanity.com/paper/1904.04205/) (2019)
- [Structured Mechanical Models for Robot Learning and Control](https://mlsanity.com/paper/2004.10301/) (2020)
- [Training neural networks under physical constraints using a stochastic augmented Lagrangian approach](https://mlsanity.com/paper/2009.07330/) (2020)
- [A Framework for Checkpointing and Recovery of Hierarchical Cyber-Physical Systems](https://mlsanity.com/paper/2205.08650/) (2022)

## Built on by (most cited first, 6 of 6)

- [REGENT: A Retrieval-Augmented Generalist Agent That Can Act In-Context in New Environments](https://mlsanity.com/paper/2412.04759/) (2024)
- [Memory-Consistent Neural Networks for Imitation Learning](https://mlsanity.com/paper/2310.06171/) (2023)
- [Four Principles for Physically Interpretable World Models](https://mlsanity.com/paper/2503.02143/) (2025)
- [Physically Interpretable World Models via Weakly Supervised Representation Learning](https://mlsanity.com/paper/2412.12870/) (2024)
- [Automating Weak Label Generation for Data Programming with Clinicians in the Loop](https://mlsanity.com/paper/2407.07982/) (2024)
- [Curating Naturally Adversarial Datasets for Learning-Enabled Medical Cyber-Physical Systems](https://mlsanity.com/paper/2309.00543/) (2023)

## Abstract (arXiv)

Deep neural networks have emerged as the workhorse for a large section of robotics and control applications, especially as models for dynamical systems. Such data-driven models are in turn used for designing and verifying autonomous systems. They are particularly useful in modeling medical systems where data can be leveraged to individualize treatment. In safety-critical applications, it is important that the data-driven model is conformant to established knowledge from the natural sciences. Such knowledge is often available or can often be distilled into a (possibly black-box) model. For instance, an F1 racing car should conform to Newton's laws (which are encoded within a unicycle model). In this light, we consider the following problem - given a model M and a state transition dataset, we wish to best approximate the system model while being a bounded distance away from M. We propose a method to guarantee this conformance. Our first step is to distill the dataset into a few representative samples called memories, using the idea of a growing neural gas. Next, using these memories we partition the state space into disjoint subsets and compute bounds that should be respected by the neural network in each subset. This serves as a symbolic wrapper for guaranteed conformance. We argue theoretically that this only leads to a bounded increase in approximation error; which can be controlled by increasing the number of memories. We experimentally show that on three case studies (Car Model, Drones, and Artificial Pancreas), our constrained neurosymbolic models conform to specified models (each encoding various constraints) with order-of-magnitude improvements compared to the augmented Lagrangian and vanilla training methods. Our code can be found at: https://github.com/kaustubhsridhar/ConstrainedModels