# Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamics

arXiv: https://arxiv.org/abs/2210.06036 · HTML: https://mlsanity.com/paper/2210.06036/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 56 references that are in the map.
- First posted on arXiv 12 Oct 2022.

## Builds on (most cited first, 30 of 34)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Deep High-Resolution Representation Learning for Human Pose Estimation](https://mlsanity.com/paper/1902.09212/) (2019)
- [VoxelNet: End-to-End Learning for Point Cloud Based 3D Object Detection](https://mlsanity.com/paper/1711.06396/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [A Point Set Generation Network for 3D Object Reconstruction from a Single Image](https://mlsanity.com/paper/1612.00603/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [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)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [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)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [DiffTaichi: Differentiable Programming for Physical Simulation](https://mlsanity.com/paper/1910.00935/) (2019)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [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)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Deep Sliding Shapes for Amodal 3D Object Detection in RGB-D Images](https://mlsanity.com/paper/1511.02300/) (2015)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [Lie Point Symmetry Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2202.07643/) (2022)
- [Deep Dynamical Modeling and Control of Unsteady Fluid Flows](https://mlsanity.com/paper/1805.07472/) (2018)
- [SPNets: Differentiable Fluid Dynamics for Deep Neural Networks](https://mlsanity.com/paper/1806.06094/) (2018)

## Built on by (most cited first, 30 of 30)

- [Scalable Transformer for PDE Surrogate Modeling](https://mlsanity.com/paper/2305.17560/) (2023)
- [Neural Material Adaptor for Visual Grounding of Intrinsic Dynamics](https://mlsanity.com/paper/2410.08257/) (2024)
- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [LagrangeBench: A Lagrangian Fluid Mechanics Benchmarking Suite](https://mlsanity.com/paper/2309.16342/) (2023)
- [Inferring Hybrid Neural Fluid Fields from Videos](https://mlsanity.com/paper/2312.06561/) (2023)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [Latent Intuitive Physics: Learning to Transfer Hidden Physics from A 3D Video](https://mlsanity.com/paper/2406.12769/) (2024)
- [Invariant preservation in machine learned PDE solvers via error correction](https://mlsanity.com/paper/2303.16110/) (2023)
- [PICT -- A Differentiable, GPU-Accelerated Multi-Block PISO Solver for Simulation-Coupled Learning Tasks in Fluid Dynamics](https://mlsanity.com/paper/2505.16992/) (2025)
- [Latent Neural PDE Solver: a reduced-order modelling framework for partial differential equations](https://mlsanity.com/paper/2402.17853/) (2024)
- [A Neural Material Point Method for Particle-based Emulation](https://mlsanity.com/paper/2408.15753/) (2024)
- [Overtone: Cyclic Patch Modulation for Clean, Efficient, and Flexible Physics Emulators](https://mlsanity.com/paper/2507.09264/) (2025)
- [Towards Universal Mesh Movement Networks](https://mlsanity.com/paper/2407.00382/) (2024)
- [EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions](https://mlsanity.com/paper/2410.03779/) (2024)
- [Enforcing governing equation constraints in neural PDE solvers via training-free projections](https://mlsanity.com/paper/2511.17258/) (2025)
- [Stabilizing Backpropagation Through Time to Learn Complex Physics](https://mlsanity.com/paper/2405.02041/) (2024)
- [FluidFormer: Transformer with Continuous Convolution for Particle-based Fluid Simulation](https://mlsanity.com/paper/2508.01537/) (2025)
- [MaNGO - Adaptable Graph Network Simulators via Meta-Learning](https://mlsanity.com/paper/2510.05874/) (2025)
- [Learning the dynamics of a one-dimensional plasma model with graph neural networks](https://mlsanity.com/paper/2310.17646/) (2023)
- [Physics-embedded Fourier Neural Network for Partial Differential Equations](https://mlsanity.com/paper/2407.11158/) (2024)
- [Variable Selection for Comparing High-dimensional Time-Series Data](https://mlsanity.com/paper/2412.06870/) (2024)
- [A Pioneering Neural Network Method for Efficient and Robust Fluid Simulation](https://mlsanity.com/paper/2412.10748/) (2024)
- [Reversible GNS for Dissipative Fluids with Consistent Bidirectional Dynamics](https://mlsanity.com/paper/2509.22207/) (2025)
- [CORGI: GNNs with Convolutional Residual Global Interactions for Lagrangian Simulation](https://mlsanity.com/paper/2511.22938/) (2025)
- [Generalized Neural Operator for Parametric and Boundary-Value Problems](https://mlsanity.com/paper/2607.21932/) (2026)
- [Mixing Artificial and Natural Intelligence: From Statistical Mechanics to AI and Back to Turbulence](https://mlsanity.com/paper/2403.17993/) (2024)
- [An Invariant Compiler for Neural ODEs in AI-Accelerated Scientific Simulation](https://mlsanity.com/paper/2603.23861/) (2026)
- [Momentum-Conserving Graph Neural Networks for Deformable Objects](https://mlsanity.com/paper/2604.26097/) (2026)
- [Hybrid Lagrangian-Eulerian Model for Lagrangian Fluid Simulation](https://mlsanity.com/paper/2608.01164/) (2026)
- [TRACE: A spatiotemporal contact memory graph network simulator for granular dynamics](https://mlsanity.com/paper/2609.02991/) (2026)

## Abstract (arXiv)

We present a novel method for guaranteeing linear momentum in learned physics simulations. Unlike existing methods, we enforce conservation of momentum with a hard constraint, which we realize via antisymmetrical continuous convolutional layers. We combine these strict constraints with a hierarchical network architecture, a carefully constructed resampling scheme, and a training approach for temporal coherence. In combination, the proposed method allows us to increase the physical accuracy of the learned simulator substantially. In addition, the induced physical bias leads to significantly better generalization performance and makes our method more reliable in unseen test cases. We evaluate our method on a range of different, challenging fluid scenarios. Among others, we demonstrate that our approach generalizes to new scenarios with up to one million particles. Our results show that the proposed algorithm can learn complex dynamics while outperforming existing approaches in generalization and training performance. An implementation of our approach is available at https://github.com/tum-pbs/DMCF.