# Learning Physical Constraints with Neural Projections

arXiv: https://arxiv.org/abs/2006.12745 · HTML: https://mlsanity.com/paper/2006.12745/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jun 2020.

## Built on by (most cited first, 19 of 19)

- [Equivariant Graph Mechanics Networks with Constraints](https://mlsanity.com/paper/2203.06442/) (2022)
- [Learning to Accelerate Partial Differential Equations via Latent Global Evolution](https://mlsanity.com/paper/2206.07681/) (2022)
- [Extending Lagrangian and Hamiltonian Neural Networks with Differentiable Contact Models](https://mlsanity.com/paper/2102.06794/) (2021)
- [Constraint-based graph network simulator](https://mlsanity.com/paper/2112.09161/) (2021)
- [Learning Articulated Rigid Body Dynamics with Lagrangian Graph Neural Network](https://mlsanity.com/paper/2209.11588/) (2022)
- [Adaptable Hamiltonian neural networks](https://mlsanity.com/paper/2102.13235/) (2021)
- [FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantities](https://mlsanity.com/paper/2210.00272/) (2022)
- [ElastoGen: 4D Generative Elastodynamics](https://mlsanity.com/paper/2405.15056/) (2024)
- [Locally-symplectic neural networks for learning volume-preserving dynamics](https://mlsanity.com/paper/2109.09151/) (2021)
- [Differentiable Physics Simulations with Contacts: Do They Have Correct Gradients w.r.t. Position, Velocity and Control?](https://mlsanity.com/paper/2207.05060/) (2022)
- [Improving Gradient Computation for Differentiable Physics Simulation with Contacts](https://mlsanity.com/paper/2305.00092/) (2023)
- [Projected Neural Differential Equations for Learning Constrained Dynamics](https://mlsanity.com/paper/2410.23667/) (2024)
- [Zero-Shot Cyclic Peptide Design via Composable Geometric Constraints](https://mlsanity.com/paper/2507.04225/) (2025)
- [PH-Dreamer: A Physics-Driven World Model via Port-Hamiltonian Generative Dynamics](https://mlsanity.com/paper/2605.18303/) (2026)
- [A Structural Dynamics Graph World Model: Unified Modeling, Constrained Rollout, and Interpretable Calibration](https://mlsanity.com/paper/2608.08689/) (2026)
- [FNODE: Flow-Matching for data-driven simulation of constrained multibody systems](https://mlsanity.com/paper/2509.00183/) (2025)
- [Physically Accurate Differentiable Inverse Rendering for Radio Frequency Digital Twin](https://mlsanity.com/paper/2603.18026/) (2026)
- [Enabling Structure-Only Initialization and Out-of-Distribution Generalization in GNN-based Molecular Dynamics Simulators](https://mlsanity.com/paper/2605.09495/) (2026)
- [Inverse Learning of Latent Risk-Neutral Densities from Irregular Option Quotes](https://mlsanity.com/paper/2607.27188/) (2026)

## Abstract (arXiv)

We propose a new family of neural networks to predict the behaviors of physical systems by learning their underpinning constraints. A neural projection operator lies at the heart of our approach, composed of a lightweight network with an embedded recursive architecture that interactively enforces learned underpinning constraints and predicts the various governed behaviors of different physical systems. Our neural projection operator is motivated by the position-based dynamics model that has been used widely in game and visual effects industries to unify the various fast physics simulators. Our method can automatically and effectively uncover a broad range of constraints from observation point data, such as length, angle, bending, collision, boundary effects, and their arbitrary combinations, without any connectivity priors. We provide a multi-group point representation in conjunction with a configurable network connection mechanism to incorporate prior inputs for processing complex physical systems. We demonstrated the efficacy of our approach by learning a set of challenging physical systems all in a unified and simple fashion including: rigid bodies with complex geometries, ropes with varying length and bending, articulated soft and rigid bodies, and multi-object collisions with complex boundaries.