A PyTorch Implementation of MLS-MPM (Moving Least Squares Material Point Method)

This repository provides a PyTorch implementation of the MLS-MPM (Moving Least Squares Material Point Method). The algorithm is implemented using a few lines of tensor operations in PyTorch, making it naturally differentiable and optimized for GPU acceleration. The code is vectorized without any explicit loops, which makes it efficient for large-scale simulations.

Gradient Checkpointing is highly recommended when integrating the MLS-MPM into a trainable deep learning framework. See OmniPhysGS (ICLR 2025) for an example.

Installation

From PyPI

pip install mpm-pytorch

Or from source

git clone https://github.com/wgsxm/MPM-PyTorch.git
cd MPM-PyTorch
pip install .

Quick Start

Run the following code to try a simple example of MLS-MPM simulation. The code simulates a 3D elastic jelly falling onto a rigid floor. By default, the code will produce a video of the simulation in the output directory.

python simulate.py --config examples/jelly.yaml

Usage

Refer to simulate.py for more details.

from mpm_pytorch import MPMSolver, set_boundary_conditions
from mpm_pytorch.constitutive_models import *
particles = ... # Particle positions to simulate
# Create a MPM solver with default parameters
mpm_solver = MPMSolver(particles)
# Set boundary conditions (optional)
boundary_conditions = ... # Refer to example configs
set_boundary_conditions(mpm_solver, boundary_conditions)
# Create constitutive models
elasticity = CorotatedElasticity(E=2e6)
plasicity = IdentityPlasticity()
# Init particle state
x = particles
v = torch.zeros_like(x)
C = torch.zeros((x.shape[0], 3, 3), device=x.device)
F = torch.eye(3, device=x.device).unsqueeze(0).repeat(x.shape[0], 1, 1)
# Start simulation for T steps
for i in range(T):
    # Update stress
    stress = elasticity(F)
    # Particle to grid, grid update, grid to particle
    x, v, C, F = mpm_solver(x, v, C, F, stress)
    # Plasticity correction
    F = plasticity(F)

Fast Batched SVD

Batched SVD is a common operation in constitutive models. Original PyTorch SVD is not optimized for batched computation. We adopt a warp-lang implementation of differentiable batched SVD in mpm_pytorch.constitutive_models.warp_svd from NCLaw. It can run on CPU or GPU with CUDA.

The result of the decomposed matrices is not guaranteed to be the same as the original PyTorch SVD. You can also use the original PyTorch SVD or other batched SVD implementations if there is any environment conflict (the version of warp-lang requires numpy<2).

We provide a script to benchmark the batched SVD implementation.

python benchmark_svd.py

If you encounter error as ModuleNotFoundError: No module named 'imp' when using higher version of Python, simply change the import statement from import imp to import importlib as imp.

Citation

If you find our work helpful, please consider citing:

@inproceedings{
  lin2025omniphysgs,
  title={OmniPhys{GS}: 3D Constitutive Gaussians for General Physics-Based Dynamics Generation},
  author={Yuchen Lin and Chenguo Lin and Jianjin Xu and Yadong MU},
  booktitle={The Thirteenth International Conference on Learning Representations},
  year={2025},
}

@article{hu2018moving,
  title={A moving least squares material point method with displacement discontinuity and two-way rigid body coupling},
  author={Hu, Yuanming and Fang, Yu and Ge, Ziheng and Qu, Ziyin and Zhu, Yixin and Pradhana, Andre and Jiang, Chenfanfu},
  journal={ACM Transactions on Graphics (TOG)},
  year={2018},
}

@article{stomakhin2013material,
  title={A material point method for snow simulation},
  author={Stomakhin, Alexey and Schroeder, Craig and Chai, Lawrence and Teran, Joseph and Selle, Andrew},
  journal={ACM Transactions on Graphics (TOG)},
  year={2013},
}