# Constraint-based graph network simulator

arXiv: https://arxiv.org/abs/2112.09161 · HTML: https://mlsanity.com/paper/2112.09161/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 56 references that are in the map.
- First posted on arXiv 16 Dec 2021.

## Builds on (most cited first, 29 of 29)

- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Graph networks as learnable physics engines for inference and control](https://mlsanity.com/paper/1806.01242/) (2018)
- [Learning Particle Dynamics for Manipulating Rigid Bodies, Deformable Objects, and Fluids](https://mlsanity.com/paper/1810.01566/) (2018)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [Flexible Neural Representation for Physics Prediction](https://mlsanity.com/paper/1806.08047/) (2018)
- [Reviving and Improving Recurrent Back-Propagation](https://mlsanity.com/paper/1803.06396/) (2018)
- [Application of Convolutional Neural Network to Predict Airfoil Lift Coefficient](https://mlsanity.com/paper/1712.10082/) (2017)
- [Learning Physical Constraints with Neural Projections](https://mlsanity.com/paper/2006.12745/) (2020)

## Built on by (most cited first, 27 of 27)

- [GraphCast: Learning skillful medium-range global weather forecasting](https://mlsanity.com/paper/2212.12794/) (2022)
- [Learning rigid dynamics with face interaction graph networks](https://mlsanity.com/paper/2212.03574/) (2022)
- [Learning Physical Dynamics with Subequivariant Graph Neural Networks](https://mlsanity.com/paper/2210.06876/) (2022)
- [Learning Iterative Reasoning through Energy Diffusion](https://mlsanity.com/paper/2406.11179/) (2024)
- [Learning Iterative Reasoning through Energy Minimization](https://mlsanity.com/paper/2206.15448/) (2022)
- [A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applications](https://mlsanity.com/paper/2403.00485/) (2024)
- [Learning Articulated Rigid Body Dynamics with Lagrangian Graph Neural Network](https://mlsanity.com/paper/2209.11588/) (2022)
- [MTP-GO: Graph-Based Probabilistic Multi-Agent Trajectory Prediction with Neural ODEs](https://mlsanity.com/paper/2302.00735/) (2023)
- [Learning Flexible Body Collision Dynamics with Hierarchical Contact Mesh Transformer](https://mlsanity.com/paper/2312.12467/) (2023)
- [Graph Neural Network-based surrogate model for granular flows](https://mlsanity.com/paper/2305.05218/) (2023)
- [A review of graph neural network applications in mechanics-related domains](https://mlsanity.com/paper/2407.11060/) (2024)
- [FlashMD: long-stride, universal prediction of molecular dynamics](https://mlsanity.com/paper/2505.19350/) (2025)
- [A composable machine-learning approach for steady-state simulations on high-resolution grids](https://mlsanity.com/paper/2210.05837/) (2022)
- [AMR-Transformer: Enabling Efficient Long-range Interaction for Complex Neural Fluid Simulation](https://mlsanity.com/paper/2503.10257/) (2025)
- [Equilibrium Aggregation: Encoding Sets via Optimization](https://mlsanity.com/paper/2202.12795/) (2022)
- [Inferring Relational Potentials in Interacting Systems](https://mlsanity.com/paper/2310.14466/) (2023)
- [Equivariance Is Not All You Need: Characterizing the Utility of Equivariant Graph Neural Networks for Particle Physics Tasks](https://mlsanity.com/paper/2311.03094/) (2023)
- [EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions](https://mlsanity.com/paper/2410.03779/) (2024)
- [DEL: Discrete Element Learner for Learning 3D Particle Dynamics with Neural Rendering](https://mlsanity.com/paper/2410.08983/) (2024)
- [Learning Physical Simulation with Message Passing Transformer](https://mlsanity.com/paper/2406.06060/) (2024)
- [Generalizable Reasoning through Compositional Energy Minimization](https://mlsanity.com/paper/2510.20607/) (2025)
- [MBDS: A Multi-Body Dynamics Simulation Dataset for Graph Networks Simulators](https://mlsanity.com/paper/2410.03107/) (2024)
- [Thickness-aware E(3)-Equivariant 3D Mesh Neural Networks](https://mlsanity.com/paper/2505.21572/) (2025)
- [EqCollide: Equivariant and Collision-Aware Deformable Objects Neural Simulator](https://mlsanity.com/paper/2506.05797/) (2025)
- [Exactness Matters for Physical Rule Enforcement](https://mlsanity.com/paper/2605.08285/) (2026)
- [Enabling Structure-Only Initialization and Out-of-Distribution Generalization in GNN-based Molecular Dynamics Simulators](https://mlsanity.com/paper/2605.09495/) (2026)
- [TRACE: A spatiotemporal contact memory graph network simulator for granular dynamics](https://mlsanity.com/paper/2609.02991/) (2026)

## Abstract (arXiv)

In the area of physical simulations, nearly all neural-network-based methods directly predict future states from the input states. However, many traditional simulation engines instead model the constraints of the system and select the state which satisfies them. Here we present a framework for constraint-based learned simulation, where a scalar constraint function is implemented as a graph neural network, and future predictions are computed by solving the optimization problem defined by the learned constraint. Our model achieves comparable or better accuracy to top learned simulators on a variety of challenging physical domains, and offers several unique advantages. We can improve the simulation accuracy on a larger system by applying more solver iterations at test time. We also can incorporate novel hand-designed constraints at test time and simulate new dynamics which were not present in the training data. Our constraint-based framework shows how key techniques from traditional simulation and numerical methods can be leveraged as inductive biases in machine learning simulators.