A Neural Material Point Method for Particle-based Emulation

TL;DR

NeuralMPM introduces a grid-based neural emulator for particle-based simulations inspired by the Material Point Method. By voxelizing particle states to a fixed grid, processing with a grid-to-grid neural network (e.g., U-Net) to predict multi-step grid velocities, and interpolating back to particles, it achieves differentiable, long-horizon rollouts with significantly reduced training time compared to prior approaches. Across diverse datasets, NeuralMPM matches or surpasses existing methods in long-term accuracy (EMD and MSE) while enabling efficient generalization to larger domains and enabling inverse design through differentiability. The work demonstrates practical potential for forward simulation, design optimization, and data-driven modeling in fluid-solid interactions, with clear avenues for 3D extension and probabilistic enhancements.

Abstract

Mesh-free Lagrangian methods are widely used for simulating fluids, solids, and their complex interactions due to their ability to handle large deformations and topological changes. These physics simulators, however, require substantial computational resources for accurate simulations. To address these issues, deep learning emulators promise faster and scalable simulations, yet they often remain expensive and difficult to train, limiting their practical use. Inspired by the Material Point Method (MPM), we present NeuralMPM, a neural emulation framework for particle-based simulations. NeuralMPM interpolates Lagrangian particles onto a fixed-size grid, computes updates on grid nodes using image-to-image neural networks, and interpolates back to the particles. Similarly to MPM, NeuralMPM benefits from the regular voxelized representation to simplify the computation of the state dynamics, while avoiding the drawbacks of mesh-based Eulerian methods. We demonstrate the advantages of NeuralMPM on several datasets, including fluid dynamics and fluid-solid interactions. Compared to existing methods, NeuralMPM reduces training times from days to hours, while achieving comparable or superior long-term accuracy, making it a promising approach for practical forward and inverse problems. A project page is available at https://neuralmpm.isach.be

Figures (24)

• Figure 1: NeuralMPM works in 4 steps. (1) The positions $P^t$ and velocities $V^t$ of the particles are used to compute the velocity $V^t_g$ and density $D^t_g$ of each grid node through voxelization. (2) From this grid, the processor neural network predicts the grid velocities at the next $m$ timesteps. The next $m$ positions are computed iteratively by (3) performing bilinear interpolation of the predicted velocities onto the previous positions and (4) updating the positions using the predicted velocities.
• Figure 2: Example snapshots. We train and evaluate NeuralMPM on WaterRamps, SandRamps and Goop, each consisting of a single material, on MultiMaterial that mixes water, sand and goop, and on Dam Break 2D, a rectangular-shaped SPH dataset. NeuralMPM is able to learn various kinds of materials, their interactions, and their interactions with solid obstacles. Despite being inspired by MPM, it is not limited to data showing MPM-like behaviour.
• Figure 3: Ablation results. Mean squared error of full rollouts on unseen test data for Goop. The default parameters are in blue. The dotted orange line ($2.4 \times 10^{-3}$) indicates the MSE we obtained for GNS after $240$ hours ($20$M training steps). The dotted red line is the MSE for DMCF after the same amount of time ($5.25 \times 10^{-3}$). NeuralMPM is robust to hyperparameter changes, with the biggest effects coming from the number of timesteps bundled together ($m$) and grid noise. For a rollout of length $T$, the model is called $T/m$ times, meaning lower values of $m$ require maintaining stability for longer. Autoregressive training coupled with time bundling suffices to stabilize the model, eliminating the need for noise injection. Although GNS reportedly slightly outperforms NeuralMPM, these results could not be reproduced in our experiments.
• Figure 4: Example VariableGravity trajectory against baselines. Each method is unrolled starting from the initial conditions of a random test trajectory not seen during training.
• Figure 5: Training convergence. (Left) NeuralMPM trains and converges much faster than GNS and DMCF. Note the log scale on both axes. (Right) Snapshots of models trained for increasing durations then unrolled until the same timestep on a held-out simulation. For a fair comparison, out-of-bounds particles in GNS and DMCF were clamped.