A Neural Material Point Method for Particle-based Emulation
Authors: Omer Rochman-Sharabi
Sacha Lewin
Gilles Louppe
Venue: Accepted by TMLR ()

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 framework for particle-based emulation. 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 6 datasets, including fluid dynamics and fluid-solid interactions simulated with MPM and Smoothed Particles Hydrodynamics (SPH). Compared to GNS and DMCF, NeuralMPM reduces training time from 10 days to 15 hours, memory consumption by 10x-100x, and increases inference speed by 5x-10x, 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/.

Published in Transactions on Machine Learning Research (02/2025)
A Neural Material Point Method for Particle-based Emulation
Omer Rochman-Sharabi University of Liège
o. rorchmaan Suliege. be
Sacha Lewin University of Liège
vacho.leutt Sulicege.be
Gilles Louppe University of Liège
9. Bulieeeeeuige..... be
Reviewed on OpenReview: https: // openreview. net/ forum? 2SK81A2huQ
Abstract
Mesh-free Lagrangian methods are widely used for simulating fluids, solids, and their com- plex 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 lcarning emulators promise fastor and scalable simulations, yet they often remain expensive and difficult to Irain, limiting their practical use. Inspired by the Material Point Method (MPM), we present NeuralMPM, neural emulation framework for particle-based simulations. NeuralMPM interpolates Lagrangian particles onto a fixed-size grid, computes updates 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 dynam- ics, while avoiding the drawbacks of mosh-based Eulerian methods. We demonstrate the advantages of NeuralMPM on 6 datasets, including fluid dynamics and fluid-solid interac- tions simulated with MPM and Smoothed Particles Hydrodynamies (SPH). Compared to GNS and DMCF, NeuralMPM reduces training time from 10 days to 15 hours, memory consumption by 10x-100x: and increases inference speed by 5x- 10=10x, while achieving compa- rable or superior long-term accuracy, making it a promising approach for practical forward and inverse problems. A project page is available at ftap:://nneralmppan isach. be
1 Introduction
The Navier-Stoker equations describe the time evolution of fluids and their interactions with solid materials. As analytical solutions rarely exist, numerical methods are required to approximate the solutions. On the one hand, Eulerian methods discrctize the fluid domain on a fixed grid, simplifying the computation of the dynamies, but requiring high-resolution meshes to solve small-scale details in the flow. Lagrangian methods, the other hand, represent the fluid as virtual moving particles defining the system's state, hence maintaining high level of detail in regions of high density. While effective at handling deformations and topological changes [Monaghan] 2012). Lagrangian methods struggle with collisions and interactions with rigid objects (Vacondio et. al. 2021 Lind el. al. 2020).
Regardless of the discrctization strategy, largo-scale high-resolution numerical simulations are computation- ally expensive, limiting their practical use in downstream tasks such as forecasting, inverse problems, computational design. To address these issues, deep learning emulators have shown promise in accelerating

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simulations by learning an emulator model that can predict the system's state at a fraction of the cost. Next to their speed, neural emulators also have the strategic ad advantage of being differentiable, enabling their use in
inverse problems and optimization tasks (Allen et al. 2022 Zhao et al. 2022 Forte et al. 2022). Moreover,
they have the potential to be learned directly from real data, bypassing the costly and resource-intensiv
process of building a simulator (Lam el al. 2023: Pfaff et al. 2021: Lemos et al. 2023 Jumper el al. 2021 He
ct al.
2019). In this direction, particle-based neural emulators (Sanchez-Gonzal
et al. 2020 Ummenhofer
jet al., environments. These emulators, however, suffer from the same issues as traditional Lagrangian methods, with collisions and interactions with rigid objects being particularly challenging. These emulators can also require long training and inference times, limiting their practical use.
2020 Prantl al.) 2022) have seen success in accurately simulating fluids and generalizing to unseen
Taking inspiration from the hybrid Material Point Method (MPM) (Sulsky et al. 1993 Nguyen et al. 2023) that combines the strengths of both Fulerian and Lagrangian mothods, we introduce NeuralMPM, a neural emulation framework for particle-basod simulations.
As in MPM, NeuralMPM uses Lagrangian particles to represent the system's state but models system dynamics on voxelized grids. This approach benefits from a regular grid structure for simplified state dynamics computation while avoiding the drawbacks of mesh- based Eulerian methods.
It interpolates particles onto a fixed-size grid, bypassing the need for expensive neighbor searches at each timestep and replacing them with two efficient noxclization-based interpolation steps Xu
et al. 2021
By defining dynamics on a grid, NeuralMPM leverages well-established grid-to-grid neural archi- tectures. This introduces an inductive bias that enables easier processing of global and local point cloud structures, freeing capacity to learn the system's 's dynamics.
Compared to previous data-driven approaches "Sanche-Gonzzleer et al. 2020 Ummenhofer et al., 2020; Prantl et al. 22022), these improvements reduce the training time from days to hours, while achieving higher or comparable accuracy.
2 Computational Fluid Dynamics
Computational fluid dynamics simulations can be classified into two broad categories, Eulerian and La- grangian, depending on the discretization of the fluid (Rakhsha et al. 2021). In Eulerian simulations, the domain is discretized with a mesh, with state variables uf (such as mass or momentum) maintained at each mesh point í. Well-known examples of Eulerian simulations are the finito difference mothod, where the domain divided into a uniform regular grid (also called an Eulerian grid), and the finite element method, where the domain is divided into regions, or elements, that may have different shapes and density, allowing to increase the resolution in only some areas of the domain (Iscrles] 2008 Morton & Mayers) 2005 Another widely used family of Eulerian methods is the spectral and pseudo-spectral family of methods, where the equation is solved in Fourier space or alternating between Fourier and real space, respectively, leveraging the fact that differentiation in real space is multiplication in Fourier space Pope (2000). Lagrangian simulations, on the other hand, discretize the fluid as a set of virtual moving particles {p!, w each described by its position p and state variables of that include the particle velocity of. To simulate the fluid, the particles move according to the dynamics of the system, producing a new set of particles (pi+1, uf uf ) at cach timestep. Simulations in Lagrangian coordinates are particularly useful when the fluid is highly deformable, as the particles move freely and adapt to the fluid's shape. Among Lagrangian methods, Smoothed Particle Hydrodynamics (SPH) (Gingold & Monaghan 1977 Price 2012) is one of the most popular, where the fluid is represented by a set of particles that interact with each other through a kernel function that smooths the interactions.
Hybrid Eulerian-Lagrangian methods combine the strongths of both frameworks. Like Lagrangian mothods, they carry the system state information via particles, thereby automatically adjusting the resolution to the local density of the system. By using a regular grid, however, they simplify gradient computation, make ent
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contact detection easier, and prevent cracks from propagating only along the mesh. Among hybrid methods, the Material Point Method has gained popularity for its ability to handle large deformations and topological changes, MPM combines a regular Eulerian grid with moving Lagrangian particles. It does so in four main steps: (1) the quantitics carried by the particles are interpolated onto a regular grid Go = p2g({p): uf)) using a particle-to-grid (p2g) function, (2) the equations of motion are solved on the grid, where derivatives and other quantities are casier to compute, resulting in new grid state G++1 = f(G)), (3) the resulting dynamics are interpolated back onto the particles as {u, (u - g2p(G*+1 (pf)), using
(g2p) function, (4) the positions of the particles are updated by computing carticle-wise velocities and using an appropriate integrator, such as Euler, i.e., pf+ pt + Atr The grid values are then reset. for the next step. MPM has been used in soft tissue simulations (Lonescu et al. 2005). in molecular dynamics (Lu let al., 2006), in astrophysics (Li & Liu, 2002), in lluid-mebbrane interaction (Yokk II et al., 2000), and in simulating cracks (Daphalapurkar et al. 2007] and landslides (Llano Serna et al. 2015). MPM is also widely
used in the animation industry, perhaps most notably in Disncy's 2013 film Frozen (Stomakhin et al. 2013
where it was used to simulate snow.
Notwithstanding the success of numerical simulators, they remain expensive, slow, and, most of the time, cin-differnntiabl. In recent years, differentiable neural emulators have shown great promise in accelcrating fluid simulations, most notably in a series of works to emulate SPH simulations in a fully data-driven manner. Graph network-based simulators (GNS) and a graph built from the local neighborhood of the particles to predict the acceleration of the system. The approach requires building a graph out of the point cloud at every timestep to obtain structural information about the cloud, which is an expensive operation. In addition, the GNN needs to extract global information from its nodes, which is only possible with high number of message-passingg steps, resulting in a large computational graph and long training and inference times. This large computational graph, along with repeated construction, makes fully autoregressive training over long rollouts impractical, as the gradients need to backpropagate all the way back to the first step. Cheaper strategies exist, like the push-forward trick (Brandstetter et al. 2022b), but they have been shown to be inferior to fully backpropagating through trajectories (List et al.j 2024; Sharabi & Louppe 2023). As autoregressive training not available, the stability of the learned dynamics can be compromised, making the model prone to diverging or oscillating. Noise injection training strategies can be used to incrcase the stability of the rollouts, but the magnitude of the noise becomes critical parameter. Related approaches to GNS include Hann et al. (2022), who introduce improvements to GNS to make them subequivariant to certain transformations and show increased accuracy on sinulations involving solid objects, and Viswanath et al. (2024), who apply GNS on a reduced representation of the particle system obtained by farthest point sampling.
Sanncee--Gnzzlle et al. 2020) use a graph neural network (GNN)
An alternative approach is the continuous convolution (CConv) (Ummenhofer et al. 2020 Winchenbach & Thuerey| 2024), an extension of convolutional networks to point clouds. In this method, a convolutional kernel is applied to each particle by interpolating the values of the kernel at the positions of its neighbors, which are found via spatial hashing on GPU, a cheaper alternative to tree-based searches that allows for autoregressive training. In (Prantl et al. 2022), Deep Momentum Conserving Fluids (DMCF) build upon CConv to design a moomentum-conserviie the authors introduce different branches, with different receptive fields, into their network. The number of branches, and their hyperparameters, nced to be tuned to capture global dependencics, Icading to long training times even with optimized CUDA kernels. Finally, Zhang et al. (2020) propose an approach that uses nearest neighbors to construct the local features of each particle. Those local features are then averaged onto a regular grid. Like GNS, this method suffers from the need to repeat the neighbor search at every simulation timestep. Ultimately, the performance of point cloud-based simulators is tightly linked to the method used to process the spatial structure of the cloud. Brute force neighbour search is O(N)), K-d trees are O(N log N), and voxelization and hashing are ((N) (Xu et al. 2021 Hastings & Mesit 2005 J. Recently, Alkin et al. 22024), attempt to bridge the gap between Lagrangian and Eulerian representations by avoiding explicit grid- or particle- based latent structures, but necessitating extra networks to account for particle movement or empty volumes.
archilecture. Nevertheless, to account for long-range interactions,
An approach different from data-driven modeling is hybrid models, where parts of a classical solver are replaced with learned components. For instance, Yin et al. (2021) employ neural network to learn unknown
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Published in Transactions on Machine Learning Research (02/202)
ptt) ppp pttlal
Update velocities
Upeate
positions (4)
(3)
111
Ail
8" (P', V*)
1/112
(1) Voxelization
Update velocities
Update positions
g7142
Processing
pron-t
d-m
Updatec
Upditte
velocitics
positions
DJ
( Jimi
piin
Figure 1: NeuralMPM works in steps. (1) The positions p and velocities V of the particles are used to compute the velocity Vi and density D 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 the predicted velocities onto the previous positions and (4) updating the positions using the predicted velocities.
physics, which is then integrated into a simulator. Similarly, Li et al. (2024a) use a neural network to bypass computational bottlenecks in MPM simulators, Mhile[Ma al. (2023) learn general constitutive laws, allowing for onc-shot trajectory learning. These approaches achieve impressive results by loveraging extensive physics knowledge, but this reliance also limits their applicability. Hybrid models may inherit both the strengths and weaknesses of classical and ML methods. Our approach falls within the data-driven domain, as the prior constraints on the learnable dynamics induced by the MPM-like algorithm are minimal and can be overcome with data (Table 1
3 NeuralMPM
We consider a Lagrangian system evolving in time and defined by the positions pf and velocities of of a set of particles i = 1, ...,.. N. We denote with pt and yt the set positions and velocities all particles at time and with St = (P', V*) the full state of the system. In a more complex setting, the state of the system can include other local properties such as pressure or elastic stiffness of materials, and global properties, such external force. In this work, for simplicity, we let the network learn the relevant simulation parameters implicitly. The evolution of the particles is described by function f mapping the current state of the system to its next state St+11 = f(S).. Given a starting system S = (Pa, v²), its full trajectory, or rollout, is denoted by SI-,, Our goal is to build emulator fa(-) capable of predicting a full rollout f):T(5)) of T timesteps from the initial state 5 Following MPM, NeuralMPM operates in four steps, described below, as illustrated in Figure I and in Alg [
Step 1: Voxelization. Using the particle positions Pf, the velocities V are interpolated onto regular fixed-size grid. This interpolation is performed through vorelizalion, which divides the domain into regular volumes (voxels). Each grid node is identified as the center of a voxel (e.g., square in 2D) in the domain, and the velocities of the particles in the voxel are averaged to give the node's velocity. Similarly, the density is computed as the normalized number of particles in the voxel. This results the grid tensor Gt that contains the grid velocities V and density Dr.
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Step 2: Processing. Taking advantage of the regular grid representation of the cloud, the grid velocities [V.)+= of the next m timesteps are predicted using neural network. We chose a U-Net (Ronneberger jet al. ++ 2015 as it is a well-established image-to-image model, known to perform well in various tasks, including physical applications. The combination of kernels applicd with different receptive fields (from smaller larger) can allow the U-Net to elficiently extract both local and global information. Nonetheless, any grid-to-grid architecture could be used. We experiment with the FNO (Li et al.) (021)) architecture Appendix D and find underperform, leading us keep the U-Net. A fully convolutional U-Net and FNO have the additional advantage of being able to generalize to different domain shapes, desirable property (Section 4.3
Step 3: Update of particle velocities. The predicted velocities F at the next timestep are then interpolated back to the particle level onto the positions p using bilincar interpolation. The velocity of cach particle is computed as a weighted average of the four surrounding grid velocities, based on its Euclidean distance to each of them.
Step 4: Update of particle positions. Finally, the positions of the particles are updated with Euler integration using the next velocities and known current positions of the particles, that is pt+1 +'+ Steps 3 and 4 are performed m times to compute the next m positions from the set of grid velocities computed al step 2.
Additional features of the individual particles can be included the grid tensor G by interpolating them in the same way as the velocitics. Local, such as boundary conditions, or global, such as gravity or external forces, features represented as grid channels. For simulations with multiple types of particles, the features of each material are interpolated independently and stacked channels in G..
NeuralMPM is trained end-to-end on set of trajectories 50:T to minimize the mean squared error ||P(-1 p (SS)) between the ground-truth and predicted next positions of the partieles. At inference time, the model is exposed to much longer sequences, which requires carefully stabilizing the rollout procedure to prevent the accumulation of large errors over time. To address this, we first make use of autoregressive training (Prantl et al. 2022; Ummenhofer et al. 2020), where the model unrolled K times on its own predictions, producing a sequence of S* = fo(S* (6(S*-1) for = I,.., K and initial input 5 = S before backpropagating the error through the entire rollout. Unlike more costly methods that require alternative stabilization strategies, such as noise injection ion (Sanchez-GGonzalez et al. 2020), NeuralMPN'ss efficiency makes autoregressive training possible. Nevertheless, to further stabilize the training, couple autoregressive training with time bundling (Brandstetter et al. 2022b),, resulting in training strategy where the model predicts m steps f1:m at once from a single initial state, inside an outer autoregressive loop of K steps of length m. We show in Section that this training strategy leads to more accurate rollouts.
Algorithm 1 NeuralMPM
Require: p0_ V0 grid g, i, neural backbone h, functions p2g and g2p pe pt, ft +- Vt
while < T do
3: Voxelization: Interpolate density and velocities onto the grid using G = =D', Vi) = p2g Pr, V2)
4: Processing: [ = h(G")
for in range(1, do
Update velocities: Piti ggggpp/ (Veti, ti puti-1)
7: Update positions: peti = piti-l Aiiti
8: end for
9:
t t
10:
end while
return Full trajectory (P, VHT
11:
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4 Experiments
We conduct a series of experiments to demonstrate the accuracy, speed, and generalization capabilities of NeuralMPM. Specifically, we examine its robustness to hyperparameter and architectural choices through an ablation study 4.1 We compare NeuralMPMM to GNS and DMCF in lerms of accuracy, training time, convergence, and inference speed (1.2). We also evaluate the generalization capabilities of NeuralMPM (4.3)
and illustrate how its differentiability can be leveraged to solve an inverse design problem (1.4). Through these experiments, we demonstrate that NeuraMMPM is flexible, accurate, and fast method for emmlating complex particle-base based simulations. The baselines established by Winchenbach & Thuerey (2024) and hybrid simulators (Li et al., 2024a Ma et al., 2023) have promising results. However, we do not compare against them as they either use different benchmarks or are specifically tailored for certain physical domains, requiring material-speciffc knowledge. In contrast, NeuralMPM, like GNS and DMCF, requires only particle positions without being restricted to any particular domain. Rollouts for all experiments, models, and datascts are available al drive and in the supplementary material.
Data. We consider 6 datasets with variable sequence lengths, numbers of particles, and materials. The first three datasets, WATERRAMPS, SANDRAMPS, and GooP, contain a single material, water, sand, and goop, respectively, with different material properties. The first two datasets contain random ramp obstacles to challenge the model's generalization capacity. The fourth dataset, MULTIMATERIAL, mixes the three matcrials together in the same simulations. These four datascts are taken from Sanchez- Gonzalez ct al. (2020) and were simulated using the Taichi-MPM simulator (Hu et al. 2018b). They each contain 1000 trajectories for training and 30 (GooP) or 100 (WATERRAMPS, SANDRAMPS, MULIIMATEEIAA) for validation and testing. The fifth dataset, DAM BREAK 2D, was generated using SPH and contains 50 trajectories for learning, and 25 for validation and testing. The last dataset, VARLABLEGRAVITY, was also generated using Taichi-MPM. It consists of simulations with variable gravity of a water-like material, and contains 1000 trajectories for training and 100 for validation and testing.
Protocol. NeuralMPM is trained on trajectories with varying initial conditions and number of particles.
The training batches are sampled randomly in time and across sequences. We use Adam (Kingma & Ba 2014] with the following learning rate schedule: linear warm-up over 100 steps from 10-5 to 10-3, 900 steps at 10-3, then a cosine annealing [Loshchilov & Hutter 2017) for 100, 000 iterations. We use batch size of 128, K- 4 autoregressive steps per iteration, bundle m 8 timesteps per model call (resulting in 24 predicted states), and a grid size of 64 x 64. For most of our experiments, we use a U-Net (Ronneberger et al., 2013) with three downsampling blocks with a factor of 2, 64 hidden channels, a kernel size of 3, and MLPs with three hidden layers of size 64 for pixel-wise encoding and decoding into a latent space. For a fair comparison, we ran training and inference for NeuralMPM, DMCF, and GNS on the exact same hardware. GNS and DMCF were trained until convergence (a maximum of 120 and 240 hours, respectively), while NeuralMPM required 20 hours or less to converge. For WATERRAMPS, SANDRAMPS, Goop, and MULTMATERIAL we use the same parameters as those reported by authors. We hyperparameter search DMCF for DAM BREAK 2D and both GNS and DMCF for VARIABLEGRAVITY and report the best performance obtained for a budget of 60 GPU-days per dataset. Further details on training can be found in Appendix B
4.1 Ablation study
To study the robustness of NeuralMPM to hyperparameter and architectural choices, we start with the default architecture and hypepparamoters and ablate its components individually to examine their impact on performance. We vary the number K of autoregressive steps with and without grid and particle noise in the input state the number of bundled timesteps m predicted by single model call, and the depth and number of hidden channels of the network. We also investigate adding noise to stabilize rollouts, either directly to the particles' positions or to the grid-level representation after voxelization.
Figure g summarizes the ablation results. A larger number K of autoregressive steps yields more accurate rollouts without the need to add noise. Indeod, injecting noise does not improve accuracy and is even detrimental for K- 4. Individually tuning the noise levels for grids and particles can modestly lower error rates, but is either very sensitive or negligible. The model performs better when bundling more timesteps,
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WATERRAMPS
SANDRAMPS
GOOP
MULTIMATERIAL
DAM BREAK 2D
Initial
Predictions
Ground Truth
Figure 2: Example snapshots. We train and evaluate NeuralMPM on WATERRAMPS, SANDRAMPS and GooP, each consisting of single material, on MUL.IMATEEIAL that mixes water, sand and goop, and on DAM BREAK 2D, a roctangular-shped SPH datasct. 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.
K without noise
K with noise
Time bundle m
Grid size
MSE (10"))
15
54 Width
125
Grid noise
Particle noise
Depth
MSE
0.055
0.001
0.001
0.0006
0,0000
64
Figure 3: Ablation results. Mean squared error of full rollouts unseen test data for GoOP. The default parameters are in blue. The dotted orange line (2.4 x 10 10-3) indicates the MSE obtained for GNS after 240 hours (20M training steps). The dotted red line is the MSE for DMCF after the same amount of time (5.25 x 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 length T, the model called //m times, meaning lower values of m require maintaining stability for longer. Autoregressive training coupled with time bundling suffices stabilize the model, eliminating the need for noise injection. Although GNS reportedly slightly outperforms NeuralMPM, these results could not be reproduced in experiments.

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Time
Ground Truth
Ours
GNS
DMCF
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.
enabling faster rollouts as single forward pass predicts more steps. We found m = 8 to be optimal with the other default hyperparameters, outperforming larger bundling. This is because more network capacity is needed to extract information for the next. 16 or 32 timesteps from a single state. Instead, we opted for a shallower and narrower network to balance speed and memory footprint with performance gains. In terms of network architecture, we chose a U-Net. We experiment with an FNO (Li ct al.) 2021) in Appendix D and find it to underperform, leading us to keep the U-Net architecture. We find the U-Net's width and depth to have minor impact performance, confirming that a larger network is not needed. The grid size, however, is critical. A low resolution loses fine details, while a high resolution turns meaningful structures, such liquid blobs or walls, into isolated voxels.
4.2 Comparison with previous work
We compare NeuralMPM against GNS and DMCF. We use the official implementations and training in- structions to assess training times, inference times, as well as accuracy. We compare against both GNS and DMCF on WATERRAMPS, SANDRAMPS, GooP, DAM BREAK 2D, and VARIABEI.GRAVITY. We also compare against GNS MULTIMATERIAL, but not against DMCF since it does not support multiple materials.
Accuracy. We report quantitative results comparing the long-term accuracy in Table ! and show trajecto- ries of NeuralMPM in Figure well comparisons against baselines WATERRAMPS in Figure On the mono-material datasets WATERRAMPS. SANDRAMPS, and GooP, NeuralMPM performs competitively with GNS and better than DMCF in terms of mean squared error (MSE). For MUL.IMATTTIAI NeuralMPM reduces the MSE by almost half, which we attribute to it being hybrid method, known to better handle in- teractions, mixing, and collisions between different materials. In DAM BREAK 2D, NeuralMPM outperforms both baselines, despite the data being simulated using SPH. Finally, NeuralMPM surpasses the performance of DMCF in
even though the latter accounts for gravity explicitly. In terms of Earth Mover's Distance (EMD), NeuralMPM outperforms both baselines across all benchmarks, suggesting that NeuralMPM is better capturing the spatial distribution of the particles.
Training. In Figure we report the evolution of the mean squared error of full emulated rollouts on the held-out test set during training, for each method, along with predicted snapshots at increasing training

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Data (Simulator)
N
T
MPM
GNS
DMCF
MSE4
EMD4
MSE1
EMD4
MSEI
EMD4
WATERRAMPS (MPM)
2.3k
600
13.92
68
11.75
20.45
105
SANDRAMPS (MPM)
3.3k
400
3.12
61
3.11
84
6.22
91
GooP (MPM)
1.9k
400
2.18
55
2.4
73
5.25
85
MULTIMATERIAL (MPM)
2k
1000
9.6
66
14.79
105
DAM BREAK 2D
(SPH)
5k
401
29.07
348
87.04
384
74.77
381
VARIABLEGRAVIIY (MPM)
600
1000
14.48
92
134
350
28.77
97
Table 1: Full rollout MSE EMD (both x10-3) for NeuralMPM and the baselines on each dataset, with the maximum number of particles N and sequence length T. Each method was trained until full convergence (NeuralMPM: 15h, GNS: 240h, DMCF: 120h), and the best model was used.
Ground
Hour
5 Hours
15 Hours
Best
Truth
Ne.ralMPM
Oun
100
DMCF
GNS
80
60
MSE (IG-1)
GNS
30
DMCF
15
13
11
4.45
60 60
150 240
Time
Figure 5: Training convergence. (Left) NeuralMPM trains and converges much faster than GNS and DMCF, Note the log scale both axes. (Right) Snapshots of models trained for increasing durations then unrolled until the same timestep on held-out simulation. For fair comparison, aut-of-bbunde particles in GNS and DMCF were clamped.
FPS (Frames per sccond)
1000
500
250
GPU Memory Usage
20GB
10GB
100
1GB
Taichi-MPP MPM
NeuralMPM
DMCF
100MB
GNS
10"
10"
10
10'
10"
10'
10'
10'
10"
Number of particles
Number of particles
Figure 6: Time and memory performance. Average FPS (left) and GPU VRAM usage (right) for increasing numbers of particles for traditional solver (Taichi-MPM Hu et all 2018a)), NeuralMPM, and the two baselines. The two baselines quickly require very large amounts of memory and become very slow. Although Taichi-MPM is more memory efficient for high numbers of particles, NeuralMPM remains much faster, emulating 30 million particles at 25FPS. For the low particle count regime (< 10K) we used the NeuralMPM and baselines WaterRamps models. For the high particle count regime we used untrained models and measured the throughput. The figures measures just FPS, and not the real simulation time. Taichi-MPM needs much smaller step size than the three neural emulators (0.2ms 2.5ms), and is therefore likely slower than all of them

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durations. NeuralMPM converges significantly faster than both baselines while reaching lower error rates. Furthermore, the convergence of the training procedure and quality of the architecture can be assessed much earlier during training, effectively saving compute and enabling the development of more refined final models. Morcover, NeuralMPM also more memory-officient, which enables the use of higher batch sizes of 128, as opposed to only 2 in GNS and DMCF.
Inference time and memory. In Figure we display the time and memory performance of NeuralMPM, the two baselines GNS and DMCF, and the reference solver Taichi-MPM. In terms of speed, NeuralMPM strongly outperforms all three methods, partly thanks to time bundling, which considerably reduces the nunber of model calls required for a given number of frames to emulate. In terms of memory, although NeuralMPM remains inferior to Taichi-MPM, which is highly optimized, it can emulate tens of millions of particles on a single GPU, while GNS and DMCF struggle to reach half a million.
4.3
Generalization
WROERDROP-XL
Larger rectangular domains
+
Initial
Initial
Ground Truth Predictions
Predictions t1 Predictions ta
Figure 7: Generalization. (Teft) NeuralMPM generalizes to domains with more particles (~ 4x here) with minimal inference time overhead due to the processing the voxelized reprosentation. (Right) NeuralMPM model trained on a square domain can naturally generalize to larger rectangular domains (twice as wide here) when using fully convolutional U-Net.
One notable advantage of NeuralMPM is that the procossor is invariant the number of particles, the transition model only processes the voxelized representation, while both p2g and g2p scale linearly. To demonstrate this, we train a model on WATERRAMPS, which contains about 2.3k particles and 600 timesteps, and evaluate it on WATERDROP-XL, which features about four times more particles, 1000 timesteps, and no obstacles. An example snapshot displayed Figure 7 The larger number of particles only affects interpolation steps between the grid and particles, resulting making the model nearly as fast despite 4 times more particles. We also validate generalization quantitatively by comparing the error rales WATERDROP-XL of a model trained directly on it and the model trained solely WATERRAMPS. With the same training budget, the latter achieves lower MSE at 20.92 10 against 28.09 x 10-3, More trajectories are displayed in Figure 23
negligible impact on total inference time,
If a domain-agnostic processor architecture is used, such as a fully convolutional U-Net or an FNO, then NeuralMPM can generalize to different domain shapes without retraining, as shown in Fig We demonstrate this ability by considering model solely trained on WATERRAMPS, a square domain of size 0.84 x 0.84 mapped to 64 64 grids. Without retraining, we perform inference with this model on larger unseen environments of size 1.68 x 0.84, and change the grid size to 128 64. The unseen environments were built
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by merging and modifying initial conditions of held-out test trajectories from WATERRAMPS. NeuralMPM emulates particles in this larger and rectangular domain despite being trained smaller square domain with a smaller grid, showing that U-Net can generalize to other domains. No ground truth is displayed as Sanchez-Gonzalez Gonzalez et al. (0220) provide no information about the data generation. More trajectorics are shown in Figure 24
4.4 Inverse design problem
Finally, we demonstrate the application of NeuralMPM for inverse problems toy inverse design task that consists in optimizing the direction of a ramp to make the particles reach a target location, similar to (Allen et al. 2022). We place a blob of water at different starting locations, and we then place a ramp at some location, with a random initial angle a. The goal to spin the ramp by tuning a in order to make the water end up at a desired location. The main challenges of this task are the long-range time horizon of the goal and the presence of nonlinear physical dynamics. We proceed by selecting the point where we want the water to end up and compute the average distance between the point and particles at the last simulation frame. We then minimize the distance via gradient descent, leveraging the differentiability of NeuralMPM to solve this inverse design problem. We show an example optimization in Figure S and additional examples in Appendix D
Initial
Unoptimized
Optimized
Figure 8: Inverse design problem. We exploit NeuralMPM's diffcrentiability to optimize the angle a of a ramp, anchored at the red dot, in order to get the water close to the red square region.
5 Conclusion
Summary. We presented NeuralMPM, a neural emulation framework for nartcle-based simulations in- spired by the hybrid Fulerian-Taggraniann Material Point Method. We have shown its effectiveness sim- ulating a variety of materials and interactions, its ability to generalize to larger systems and its use in inverse problems. Crucially, NeuralMPM trains in 6% of the time it takes to train GNS and DMCF to comparable accuracy, and is 5x-10x faster at inference time. By interpolating particles onto a fixed-size grid, global information is distilled into voxelized representation that is casier to learn and process with powerful image-to-image models. The use of voxelization allows NeuralMPM to bypass expensive graph constructions, and the interpolation leads to easier generalization to a larger number of particles and constant runtime. The lack of expensive graph construction and message passing also allows for more autoregressive steps and parallel rollouts.
Limitations. Like other approaches, NeuralMPM limited by the computation used to process the structure of the point cloud. In our case, voxelization means we cannot deal with particles that lie outside of the domain and are limited to regular grids. Additionally, the size of the voxels is directly related to the number of particles within given volume. If the voxels are too large, the model will fail to capture finer details. Conversely, if they are too small, the model may struggle due to insufficient local structure. Similarly, performance can degrade in very sparse domains. Compressible fluids might also present challenges, though this requires further verification.
Future work. Our work is only a first step towards hybrid Sulerian-Lagrangian neural emulators, leav- ing many avenues for future rescarch. Extending NeuralMPM 3D systems is a natural continuation of this work. Future studies could also explore alternative particle-to-grid and grid-to-particle functions, like the non-uniform Fourier transform (Fessler & Sutton 2003), more sophisticated interpolation methods
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from classical MPM literature Nguyen al, 2023 A less traditional direction is to make NeuralMPM probabilistic and encode richer distributional information about the particles in the grid nodes, instead of maintaining only a mean value. This could potentially improve NeuraLMPNIs ability to resolve subgrid phenomena. Finally, advances in Lagrangian Particle Tracking (Schröder & Schanz 2023) will eventually make possible to create datasets from real-world data, enabling the training of Ne directly from data without the need for the costly design process of numerical simulator.
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A Supplementary Material
In addition to the supplementary material the project page (lttps://neurallmpm.. isach. we have uploaded videos, minimal code for reproduction, and baseline comparisons to online cloud storage that can be found at
B Training details
Hardware. We run all our experiments using the same hardwarc: CPUs, 24GB of RAM, and NVIDIA RTX A5000 GPU with 24GB of VRAM. For reproducing the results of DMCF, we kept the A5000 GPU but it required up to 96GB of RAM for training.
Data Preprocessing. Similar to Prantl et al. 2022 we slightly alter the original MPM datasets to add boundary particles, as the original data from Sanchez- Gonzalez et al. (2020) does not have them. We define the velocity at a timestep to be V, V2 Vt-1- We therefore skip the first step during training for which no velocity is available.
Implementation. Our implementation, training scripts, experiment configurations, and instructions for reproducing results are publicly available at [URL]. We implement NeuralMPM using PyTorch (Paskke) ct al., 2019), and use PyTorch Geometric (Fey Lenssen 2019) for implementing efficient particle-to-gria functions, more specifically from the Scatter and Cluster modules. For memory efficiency, we do not all (up to) 1,000 training trajectories in memory, and rather use buffer of about 16 trajectories over which several epochs are performed before loading new buffer of random trajectories.
Baselines. We use the official implementations and training instructions of GNS (Sanchez (Sanchez-Gonzalee Gonzalez al, 2020) and DMCF (Prantl et al. 2022 to reproduce their results and conduct new experiments. More specif- ically, we train GNS as instructed for 20M steps on all four datasets, using their provided configuration. For DMCF, we follow their default. configurations and train for 400K iterations for each dataset. Tn datasets that were not used by the original authors, VARIABLEGRAVITY and DAM BREAK 2D, we performed hyper- parameter scarch. GNS and DMCF both were trained for total budget of GPU-days per dataset, and the best performance reported.
Normalization. We normalize the input of the model over each channel individually. We investigated computing the statistics across buffer, resembling (Toffe Szegedy, 2015), and precomputing them the whole training set and found difference in performance. During inference, use the precomputed statistics.
Code.
The code, together with additional videos, is
available at the project's website [URL].
C Additional Discussion
Boundary conditions. Boundary conditions are represented as stationary particles. Like moving parti- cles, their density and zero velocity are interpolated into channels, but their positions enforced to remain fixed during rollout. This approach allows for flexible boundary shapes, such the ramps in WATERRAMPS.
Voxclization and interpolation. The voxelization procedure used is implemented using CUDA by Py- TorchCluster Fey & Lenssen 20199) p2g is implemented by us based the voxelized representation, and g2p uses Py bilinear interpolation
(rrid_sampl) Paszke et al. (2019) based on the voxelized rep- resentation of the data. The cost of those operations is O(N), e.g., a naive implementation of voxelization assigns each point p (x, y, 2) to a voxel index computed as:
V
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where is the voxel size and [-] denotes the floor function. Since this computation is performed in con- stant time for each of the N points, the overall complexity is O(N). Alternative interpolation methods could be considered, although quick experimentation with bicubic interpolation for grid-to-particles or dif- ferent weighting schemes for voxclization did not yicld much difference in our pipeline. Some methods, like GIOROM Viswanath et al. (2024) and GINO Li et al. (2024b), learn the interpolation through the use of graph-based neural networks, but this strongly increases computational costs due to the construction of a graph and the forward pass in the neural network.
Limitations of voxelization. One the main issues of voxclization onto a fixed regular grid means the domain is limited and it must be rectangular, like for the material point method. While a rectangular domain can be defined with local boundary conditions over the irregular domain of interest, this will be a disadvantage for sparse systems, for which Lagrangian methods might be better fit. Likewise, out-of-bounds particles be handled either with larger domain, which might be more computationally expensive, by clamping the positions, simply dropping the out-of-bounds particles. These particular conditions are known disadvantages of MPM, and of some implementations of SPH, and are therefore inherited by NeuralMPM.
Rollout stability. Long-term stability during rollout of neural emulators is an open problem. As such, we decouple this problem from the design of the method or the architecture. There are commonly used tricks in the literature, such as noise injection, backpropagating through the autoregressive rollout, temporal bundling, having two models work in tandem (one to produce large timesteps and one to interpolate between them), using explicit constraints White al. (2024), or modifying the gradients [Schnell & Thuerey (2024). List et al. 2025 perform comparative study of some of these methods.
D
Supplementary results
Additional results DamBreak2D We have also compared the accuracy and inference speed of NeuralMPM against a different implementation of GNS, and one of SEGNN, both provided by (Toshev
ct al. 2024 in Tables 2 and 3 As in Table NeuralMPM outperforms both by a margin basclines.
MSE4
FMD4
Ours
20.76
2.88
GNS
114.40
224.1
SEGNN
124.39
268.4
Table 2: Full-trajectory MSE (x10-3) and Sinkhorn distance (EMD) (x10-4) for NeuralMPM, GNS, and SEGNN BBrandstetter al. (022a) on the DAM BREAK 2D dataset from LagrangeBench. The two latter models are baselines provided by LagrangeBench.
Single call (T'=) [/==1] 1)
Rollout (T' 401)
Ours
7.41
193.50
GNS
20.46
8,170.47
SEGNN
46.04
18,194.10
Table 3: Inference time (in ms) of NeuralMPM, GNS, and SEGNN Brandstetter et al. (022a)) on the DAM BREAK 2D dataset from LagrangeBench. Times were averaged over all test trajectories. NeuralMPM predicts 16 frames in single model call and still outperforms the baselines per call, which further widens the gap for the total rollout time.
Evaluation. In Table 4 we report the numerical MSE rollout values that were reported in the bar plots depicted in Figure 3 for GOOP. Also, Figures 12 and T0 displays the error when rolling out model for each
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NeuraMPMM
MultiMaterial
VariableGraviity
WaterRamps
DamBreak2D
SandRamps
Goop
35
40
350
14
175
30
33
300
10-
150
250
10
125
20
100
25
200
MSE
20
150
15
41
10
100
10
3
50
0-
0-
300 Timestop
500
0
200 400
200
400
0
200
200
Timestep
Timestop
Timestop
Timestop
Timestop
Figure 9: MSE propagation during rollout. We show the 25th, 50th and 75th percentile of the MSE, computed over particles and simulations, at each timestep during the rollout for each model and dataset. The accuracy decreases errors accumulate. While training in 5% of the time, NeuralMPM outperform the baselines on all datasets, except WaterRamps, where it is slightly worse than GNS.
dataset, both terms of MSE and EMD. For both metrics, the error starts low and slowly accumulates over time. For the EMD, the Sinkhorn algorithm provided by (Cuturi et al. 2022
Parameter
Value
MSE (x10-3)
Parameter
Value
MSE (x10-$
3.2
0
3.2
3.3
Grid noise
0.001
2.4
K (No noise)
2.4
0.005
6.9
2.2
2.2
3.5
0.0003
2.4
Particle noise
2.5
0.0006
2.4
K (With noise)
3
2.4
0.001
2.1
3.0
2
3.3
6.6
3
3.0
U-Net Depth
4.5
2.4
3.5
2.3
Time bundling m
2.1
32
2.6
16
2.9
U-Net Width
64
2.3
32
3.5
128
2.2
32
5.5
Grid size
64
2.4
128
7.1
Table 4: Ablation results for GooP.
Grid-to-grid network. Although we have used a U-Net architecture for the grid-to-grid processor, Neu- ralMPM can be used with any grid-to-grid processor and is not limited to that network. For example, in Figure 13 and Table we present qualitative and quantitative ablation results, respectively, for NeuralMPM using an FNO network {Li al. 2021) the grid-to-grid processor. Results show that the FNO processor is slightly worse than the U-Net processor.
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NeuraIMPM
DMCF
GNS
MultiMaterial
VariableGravity
WaterRamps
DamBreak2D
SandRamps
Goop
100
120
400
140
120
500
100
80-
100
300
100
60-
80
300
200
60
40-
200
40
100
20-
100
20
0
0-
01
0
300
1000
500
500
Timestep 100
200 Timestep
200
Timestep
Timestep
Timestep
Timestop
Figure 10: EMD propagation during rollout. We show the 25th, 50th and 75th percentile of the EMD, computed over particles and simulations, at each timestep during the rollout for each model dataset daset. The accuracy decreases as errors accumulate. While training in 5% of the time, NeuralMPM outperform the baselines on datasets.
Nam
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0
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Figure 11: MSE propagation during rollout for the generalization task WaterDrop-XL. We show the 25th, 50th and 75th percentile of the MSE, computed over particles and simulations, at each timestep during the rollout for cach model and daset. The accuracy decreases as errors accumulate. The models used were trained on WaterRamps and tested on WaterDrop-XL to evaluate their generalization. NeuralMPM performs better despite having a slightly worse MSE on WaterRamps than GNS.
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NarrlMPM
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50
200
J00
000
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Timestep
Figure 12: EMD propagation during rollout for the generalization task WaterDrop-XL, We show the 25th, 50th and 75th percentile of the MSE, computed over particles and simulations, at cach timestep during the rollout for each model and daset. The accuracy decreases as errors accumulate. The models used were trained on WaterRamps and tested on WaterDrop-XL to evaluate their generalization. NeuralMPM performs better despite having a slightly worse MSE on WaterRamps than GNS.
Without noise
With noise
15
MSE (10
10
Goop
RandRamps
WaterRamps
Goop
SandRamps
WaterRamps
Figure 13: FNO processor. NeuralMPM with an FNO processor and default architecture. Rollout MSE
(x10-3) 3) for different datasets.
Data
FNO with noise
FNO without noise
WATERRAMPS SANDRAMPS
16.8
16.3
5.5
3.5
GOOp
4.3
3.8
Table 5: Rollout MSE (x10-3) for NeuralMPM with an FNO processor and default architecture, with and without noise.
Additional inverse problem examples.
We show two additional optimization examples in Figure 14
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Unoptimized
Optimized
Figure 14: Inverse design problem. Additional optimization examples. We exploit NeuralMPM's differ- entiability optimize the angle of ramp, anchored at the red dot, in order to get the water close to the red square region.
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Gallery of predicted trajectories
We present additional rollout comparisons in Figures 15 and 16 Further, in addition to the trajectories in Figures
we show additional trajectories emnlaatd with NeuralMPM for all datasets in Fig- and 24 We also release videos in the supplementary material, which we recommend watching to better see the details and limitations of NeuralMPM. This includes 10 videos per dataset of emulated trajectorics on held-out test simulations.
and
ures
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Time
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Figure 15: Example MultiMaterial trajectory against basclines. Each method is unrolled start-
ing from the initial conditions of random test trajectory not seen during training.
Time
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Figure 16: Example WaterRamps trajectory against baselines. Each method is unrolled starting from the initial conditions of random test trajectory not seen during training.
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t=0
= 119
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= 478
I = 598
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Figure 17: Additional WaterRamps predicted trajectories. Evenly spaced in time snapshots of pre- dieted unrolled trajectories against ground truth. All trajectories are from the held-out test set.
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= 79
t t= 159
= 238
= 318
t = 398
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Prediction
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Prediction
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Prediction
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Figure 18: Additional SamdRamps predicted trajectories. Evenly spaced in time snapshots of predicted unroure unrolled trajectories against ground truth. All trajectories are from the held-out test set.
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32
/ = 65
= 97
= 130
/ = 398
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Prediction
Ground Truth
Prediction
Ground Truth
Prediction
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Prediction
Figure 19: Additional Goop predicted trajectorics. Snapshots of predicted unrolled trajectories against
ground Iruth. All trajectories are from the held-out test set. Due to GOOP quickly reaching equilibrium, more snapshots are taken in the first half of the trajectory.
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t 239
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Figure 20: Additional MultiMaterial predicted trajectories. Evenly spaced in time snapshots of predicted unrolled trajectories against ground truth. All trajectories are from the held-out test set. The first. trajectory illustrates a rare failure where the shape of sand particles is not retained, even though all particles are supposed to maintain the same velocity while airborne, as they are thrown against the wall.
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Figure Additional Dam Break 2D predicted trajectories. Snapshots of predicted trajectories against ground truth. All trajectories come from the held-out test set. To better show the differences of these longer scquences, select the following timesteps not in time: t E {0, 125, 400). In the last trajectory, NeuralMPM struggles to follow the gravily direction and breaks down over time.

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Published in Transactions on Machine Learning Research (02/2025)
t t=
100
t -= 150
t = 250
500
/ = 999
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Figure 22: Additional VariablyGravin predicted trajectories. Snapshots of predicted trajectories
against ground truth. All trajectories come from the held-out test set.
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t t= 125
t t= 200
400
t = 997
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Prediction
Figure 23: Generalization to particles Droe Snapshots of predicted trajectories emulated using a model trained solely on WATERRAMPS, against ground truth. All trajectories come from the held-out test set of WATERDROP-XT. To better show the differences these longer sequences, we select the following timesteps not even in timc: E (0, 75, 220,200,4400 200,4000 999). We can observe that the generalizing model struggles to retain the shape of water while it's falling. 29

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Initial Conditions
Snapshot 1
Snapshot
+
Figure 24: Generalization to larger and non-square domains. We train a model on the square domains in WATERRAMPS using 64 64 inpul grids to the U-Net, and then perform inference on manually generated non-square environments that are twice wide and use 128 64 input grid the same U-Net. NeuralMPM flawlessly generalizes and emulates particles in these new environments. Note: no ground truth is available because the authors of GNS did not provide the physical parameters for simulating WATERRAMPS using Taichi. Chosen time steps are 0, 150, 575. We recommend watching the videos in the supplementary material for more detailed evaluation.
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Reviews:
[{"review_id": "dptm1TGc1a", "reviewer": "Reviewer_ramk", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "#### **Strengths**\n1. **Hybrid approach:** The method takes inspiration from MPM to integrate both Lagrangian (particle-based) and Eulerian (grid-based) approaches, which can mitigate the shortcomings of each method. (however, Eulerian methods are not evaluated empirically in the considered experiments).\n2. **Computational efficiency:** The method significantly reduces computational cost, leading to reduced memory requirements as well as training and inference times compared to the considered baselines. \n3. **Performance:** In the considered experiments, NeuralMPM shows stable long-term rollouts, matching or surpassing the accuracy of the considered baselines. Moreover, it can generalize to unseen domains, number of particles, or grid sizes. Finally, it is shown that the differentiability of the emulator can be used to solve inverse design tasks.\n\n---\n\n#### **Weaknesses**\n1. **Baselines:** It seems that several closely related baselines are currently missing (both in the related works as well as empirical evaluation):\n    - https://arxiv.org/pdf/2407.03925: They leverage a learned graph neural operator instead of the interpolation and also predict the dynamics on a regular grid (they explore both a transformer and an FNO architecture). \n    - https://arxiv.org/pdf/2402.12365: They also used a learned neural network (based on message-passing and attention layers) instead of the interpolation. Then, they use a transformer architecture to predict the dynamics on a sparse point cloud. \n    - Further baselines mentioned in the two works above.\n2. **Limitations of voxelization:** The computational efficiency depends on the domain shape and dimension, the distribution of the particles, and the chosen grid resolution.\n    - In particular, the grid domain needs to be rectangular and a too coarse resolution might not be able to capture fine details of the system. While Fourier Neural Operator (FNO) are explored in the appendix, their ability to operate on varying grid-sizes is not leveraged.\n    - The method and its efficiency are not tested in 3D.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces NeuralMPM, a neural emulator for particle-based simulations inspired by the Material Point Method (MPM), which is a hybrid Eulerian-Lagrangian method. In particular, the state of the system is represented using Lagrangian dynamics, while its dynamics are modeled on a regular grid using grid-based neural networks (such as UNets). To map between the different representations, voxelization is used to compute the average velocity and density of the particles in each voxel, and bilinear interpolation is used to infer the velocities for each particle based on the predicted grid velocities. Finally, the particle positions are updated using Euler integration. The method is evaluated on six datasets and compared against graph network-based simulators (GNS) and Deep Momentum Conserving Fluids (DMCF). The grid representation allows to greatly reduce computational costs, leading to significant memory savings and speedups (in particular for larger number of particles) while achieving comparable or superior performance for long-term rollouts.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Include a discussion and empirical evaluation of the baselines mentioned in \"Strengths And Weaknesses\" above.\n2. Since the inference cost is only meaningful in combination with a performance measure, the achieved accuracy for each data point of Figure 6 should be added.\n3. The performance on the generalization and inverse design tasks should also be compared against the baselines.\n\n**Minor:**\n1. Explain how local features, such as boundary conditions (which are only defined on subdomains), are mapped to the grid representation.\n2. The computational cost of neighborhood search could be discussed in more detail, in particular since (bilinear) interpolation (as used in NeuralMPM) internally still relies on a neighborhood search.\n3. The related works could be expanded to include (pseudo-)spectral methods often used for Eulerian formulations.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "NaicJQSTN6", "reviewer": "Reviewer_xTrc", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths:**\n\n*   **Novel Approach:** NeuralMPM presents a novel approach by combining the Material Point Method (MPM) with neural networks for particle-based simulation. This hybrid approach allows the model to leverage the strengths of both Eulerian and Lagrangian methods, which is a significant advantage.\n*   **Computational Efficiency**: NeuralMPM demonstrates significant improvements in computational efficiency compared to previous methods:\n    *   **Reduced Training Time**: Training time is reduced from days to hours, which is a substantial improvement over GNS and DMCF. The paper states that NeuralMPM trains in 6% of the time it takes to train GNS and DMCF to comparable accuracy.\n    *   **Faster Inference Speed**: The model achieves a 5x-10x increase in inference speed, making it more practical for real-world applications. This is partly attributed to time bundling, which reduces the number of model calls needed.\n    *   **Memory Efficiency**: NeuralMPM requires significantly less memory than GNS and DMCF (10x-100x), allowing for larger batch sizes and more complex simulations on a single GPU.\n*   **Effective Use of Voxelization:** The use of voxelization to interpolate particle data onto a fixed-size grid is a key strength. This avoids costly neighbour searches and allows the model to use well-established grid-to-grid neural network architectures such as a U-Net. This simplifies the processing of spatial information and results in a computationally efficient system.\n*   **Generalization Capabilities**: NeuralMPM demonstrates strong generalization capabilities:\n    *   **Invariance to Particle Number**: The model is invariant to the number of particles, meaning it can be used to simulate systems with a wide range of particle counts without needing retraining.\n    *   **Generalization to Different Domain Shapes**: With a fully convolutional U-Net, the model can generalize to different domain shapes without retraining. For example, a model trained on a square domain can generalise to larger rectangular domains.\n*   **High Accuracy:** The model achieves comparable or superior long-term accuracy compared to GNS and DMCF on a range of datasets, including fluid dynamics and fluid-solid interactions. It also outperforms both baselines in terms of Earth Mover's Distance (EMD), suggesting that NeuralMPM is better at capturing the spatial distribution of particles.\n*   **Differentiability**: The model's differentiability enables its use in inverse problems and optimisation tasks, as demonstrated by the ramp optimisation example.\n*   **Robustness**: The model is robust to hyperparameter changes, with the most significant impact coming from the number of timesteps bundled together (m) and grid noise. Autoregressive training with time bundling is sufficient to stabilise the model, eliminating the need for noise injection.\n*   **Open-Source Implementation**: The authors have made their implementation, training scripts, experiment configurations and instructions publicly available, promoting reproducibility and further research.\n\n**Weaknesses:**\n\n*   **Limitations of Voxelization:**  The use of voxelization, while beneficial, also introduces limitations:\n    *   **Particles Outside the Domain**: The model cannot handle particles that lie outside of the defined domain.\n    *  **Dependence on Regular Grids**: It is limited to regular grids, potentially restricting its flexibility in handling complex geometries.\n    *   **Voxel Size Sensitivity**: The model's performance depends on the voxel size. If voxels are too large, fine details may be lost, and if they are too small, the model may struggle due to a lack of local structure.\n    *   **Sparse Domains**: Performance can degrade in very sparse domains, which could limit applicability to some simulation scenarios.\n*   **Potential Issues with Compressible Fluids:** The model may have challenges with compressible fluids, although this requires further investigation.\n*   **Limited Comparison with Hybrid Models**: While the paper compares against GNS and DMCF, it does not compare against hybrid simulators because they are specifically tailored for certain physical domains, requiring material-specific knowledge. This may limit the scope of the analysis.\n*   **Accumulation of Errors**: While the model demonstrates good accuracy, errors tend to accumulate over time during long rollouts, as is common in these types of models.\n\n**Areas for Further Attention:**\n\n*   **Exploring Alternative Interpolation Methods:** The authors could explore alternative particle-to-grid and grid-to-particle functions, potentially improving the accuracy and flexibility of the model.\n*   **Addressing Limitations of Voxelization**: Further work could focus on addressing the limitations imposed by voxelization, such as the inability to deal with particles outside the domain or to use non-regular grids.\n*   **Investigating Performance on Compressible Fluids**: More research is needed to assess the model's performance on compressible fluids and explore potential modifications to address any issues.\n*   **Extending to 3D Systems:** Extending NeuralMPM to 3D systems is a natural direction for future research.\n*   **Making NeuralMPM Probabilistic**: Encoding richer distributional information about the particles in the grid nodes instead of maintaining only a mean value could potentially improve the model's ability to resolve subgrid phenomena.\n\nIn summary, NeuralMPM is a promising approach for particle-based simulations due to its computational efficiency, strong generalization capabilities, and differentiability. The use of voxelization and a hybrid MPM-neural network framework are key strengths. However, the limitations associated with voxelization and the potential challenges with compressible fluids need further attention. Addressing these weaknesses and expanding the model's capabilities in future work could significantly enhance its impact.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces NeuralMPM, a novel neural emulation framework for particle-based simulations, drawing inspiration from the Material Point Method (MPM). The following are the key contributions of this work:\n\n*   **Hybrid Approach**: NeuralMPM combines the strengths of both Eulerian and Lagrangian methods. It uses Lagrangian particles to represent the system's state but models system dynamics on a fixed-size grid, similar to MPM, taking advantage of a regular grid structure for simplified computation. This avoids the drawbacks of mesh-based Eulerian methods.\n*   **Efficient Interpolation**: It interpolates particle data onto a fixed-size grid using voxelisation, avoiding expensive neighbour searches at each time step. This interpolation is done using efficient voxelisation-based interpolation steps.\n*   **Grid-Based Dynamics**: By defining the dynamics on a grid, NeuralMPM uses well-established grid-to-grid neural network architectures, which introduce an inductive bias and facilitate the processing of global and local point cloud structures. This frees up the capacity for learning system dynamics.\n*  **Training Efficiency:** Compared to previous data-driven methods like GNS and DMCF, NeuralMPM significantly reduces training time from days to hours. The paper states that NeuralMPM can train GNS and DMCF 6% of the time to achieve comparable accuracy. The model also achieves higher or comparable accuracy.\n*   **Inference Speed**: NeuralMPM demonstrates a 5x-10x increase in inference speed compared to GNS and DMCF.\n*   **Memory Efficiency**: NeuralMPM uses less memory than previous methods. The paper states that it reduces memory consumption by 10x-100x compared to GNS and DMCF.\n*   **Generalization**: The model exhibits strong generalisation capabilities. It can generalise to larger systems and different domain shapes. A model trained on a square domain can generalise to larger rectangular domains when using a fully convolutional U-Net. Also, the model is invariant to the number of particles.\n*   **Differentiability**: NeuralMPM is differentiable, which makes it suitable for solving inverse problems and optimisation tasks. This is demonstrated by optimising the direction of a ramp to make particles reach a target location.\n\nIn summary, NeuralMPM introduces a novel approach for emulating particle-based simulations by combining the strengths of MPM with neural networks. This results in a faster, more memory-efficient, and more generalisable model than previous methods. The method's differentiability also makes it applicable to inverse problems.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Here are some proposed adjustments to the NeuralMPM submission, specifying whether they are critical for acceptance or would simply strengthen the work:\n\n**Critical Changes (Essential for Acceptance):**\n\n*   **Address Limitations of Voxelization:** The limitations imposed by voxelization are a significant concern and should be addressed. This is critical for the acceptance of the submission because these limitations can affect the applicability of the method. The authors should:\n    *   Acknowledge and discuss in more detail the inability of NeuralMPM to handle particles that lie outside the defined domain.\n    *   Discuss and justify the limitation to regular grids and how it can affect simulations with complex geometries.\n    *   Provide guidelines or recommendations for selecting appropriate voxel sizes, explaining how the choice of voxel size affects the level of detail and accuracy of the simulation.\n    *   Discuss how the model’s performance can be improved or adapted to avoid performance degradation in sparse domains.\n*   **Investigate Performance on Compressible Fluids**: The authors should thoroughly investigate and discuss NeuralMPM's performance on simulations involving compressible fluids. This is a critical point because the model's ability to handle diverse fluid types is essential for its broader applicability. If the model is not suitable, the authors should acknowledge the limitation clearly.\n\n**Changes That Would Strengthen the Work (Recommended but not Essential for Acceptance):**\n\n*   **Explore Alternative Interpolation Methods**: While voxelization is efficient, the authors should explore and test alternative particle-to-grid and grid-to-particle functions. This could potentially improve the accuracy and flexibility of the model. Specifically, the paper mentions the non-uniform Fourier transform as a potential avenue to explore, as well as more sophisticated interpolation methods from classical MPM literature. While not critical for acceptance, this would enhance the novelty and potential impact of NeuralMPM.\n*   **More Detailed Comparison with Hybrid Models**: The authors should attempt to provide a more detailed comparison against hybrid simulation methods. While it is mentioned that some hybrid models are tailored for specific domains, it would be helpful to either identify benchmarks that could be used, or clearly justify and acknowledge the lack of comparison, perhaps by discussing the differences in methodology that make the comparison impossible. This is important to ensure the method is assessed within the landscape of current techniques.\n*  **Mitigate Error Accumulation**: While the paper acknowledges error accumulation during long rollouts, it does not propose ways to mitigate this issue. The authors should investigate more advanced training strategies or other methods to stabilise the model over longer sequences and reduce error accumulation over time. Although the current performance of NeuralMPM is strong, improving long term accuracy would strengthen the work.\n*   **Elaborate on Generalisation**: While the paper discusses generalisation to different domain shapes and numbers of particles, it could benefit from more in-depth analysis and validation. For example, by including a quantitative analysis of the behaviour of the model at different scales, or by demonstrating the generalisation behaviour with a more diverse array of domain shapes.\n*   **Consider a Probabilistic NeuralMPM**: The authors should further develop their idea of making NeuralMPM probabilistic by encoding richer distributional information about particles in grid nodes. This approach would potentially improve the model's ability to resolve subgrid phenomena and make the model more robust to uncertainty in the simulations. Although this is positioned as future work, a stronger emphasis on this idea as an achievable next step would strengthen the submission.\n*   **Further Explain Time Bundling:** The paper mentions time bundling as one of the reasons for the efficiency of the model, but could further explain how bundling m timesteps influences stability and accuracy, and provide more intuition on how the model manages to predict several timesteps at once.\n*   **Clarify the Role of Autoregressive Training:** The paper mentions autoregressive training is used but could provide more detail about the number of unrolling steps, and how this contributes to the stability of the model, beyond its use for training. A more detailed analysis of how the autoregressive strategy affects training stability will also make the submission stronger.\n\nIn summary, the critical changes focus on addressing the limitations of voxelization and the potential issues with compressible fluids, as these directly affect the applicability of the method and must be resolved for the acceptance of the paper. The changes that would strengthen the work are recommendations for improving the scope, analysis, and potential of the NeuralMPM framework by exploring and comparing to alternative methods, and delving further into the behaviour of the model under various conditions.", "broader_impact_concerns": "Although there are no such broader impacts of concern, I would like to address the following: \n\n*   **Potential Use:** The technology for simulating complex physical systems, such as fluids and solids, could be used for purposes beyond those intended by the researchers. For example, the technology might be adapted for military applications, such as simulating explosions or weapon effects. This is a common concern for simulation technologies, and it should be acknowledged.\n\n*   **Bias in Training Data**: The performance of NeuralMPM is dependent on the training datasets used. If the datasets are not representative of real-world conditions or contain biases, this could lead to inaccurate simulations and potentially harmful outcomes when used in real-world applications. The researchers should state that they used publicly available datasets, but they should acknowledge the fact that these datasets may have limitations. Further, if a dataset is biased or does not represent reality sufficiently, the trained model may not generalise well to unseen scenarios. This is a common concern for machine-learning-based simulation tools.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "LtY8Bc32GG", "reviewer": "Reviewer_P4tj", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- the paper is well-written\n- the training speed and stability of emulator are crucial\n- extensive experiments with comprehensive analysis in accuracy, runtime, memory usage, generalization and its application in inverse problems.\n\nQuestions and suggestions:\n\n1. The algorithm aims to solve a fixed point solution, existing works discuss the convergence, stability in training and acceleration methods for fix point solvers using neural networks, such as \n\n[1] Shaojie Bai, J. Zico Kolter, Vladlen Koltun. Deep Equilibrium Models\n\n[2] Tanya Marwah, Ashwini Pokle, J. Zico Kolter, Zachary Lipton, Jianfeng Lu, Andrej Risteski. Deep Equilibrium Based Neural Operators for Steady-State PDEs\n\nIt may be valuable to address the fixed-point solvers in greater detail and compare the training speed to existing acceleration methods, especially since training efficiency is emphasized as a key contribution of the paper.\n\n2. In the first paragraph of section 4.1, \"We vary the number K of autoregressive steps with and without noise...\" Could the author clarify where the noise are injected? The initial state or intermediate states or something else? How is it different from grid and particle noise?\n\n3. Right before Section 4, the authors suggest that noise injection is a costly method to stabilize training. However, the computational overhead of noise injection is minimal, especially when compared to methods like spectral normalization, which is commonly used to stabilize GAN and DEQ training. It would be interesting to explore how incorporating spectral normalization could further enhance the stability of NeuralMPM.\n\n4. Figure 3 investigates the robustness of NeuralMPM to hyperparameters. However, without comparing to how other architectures (or even NeuralMPM trained with noise injection) respond to the same hyperparameter changes, it is hard to make conclusion about robustness. \n\n5. The paper claims that NeuralMPM is time-efficient during training due to the use of time-bundling. The training speed is compared against GNS and DMCF, which do not incorporate time-bundling. Is it possible to apply time-bundling or equivalent acceleration techniques to GNS and DMCF? If so, including such results would provide a more fair and comprehensive comparison of the methods.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a neural emulator inspired by the Material Point Method (MPM), named NeuralMPM, which takes advantages of hybrid Eulerian-Lagrangian methods to enhance its efficiency and accuracy. NeuralMPM significantly reducing computational time and memory usage while maintaining comparable or even superior accuracy to existing methods.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see suggestions above.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ABD1KbRC8c", "reviewer": "Reviewer_XJx1", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n* The work is well motivated and might be a useful experimental work for the computational fluids community. It seems novel after a brief search.\n* The proposed method is significantly faster than the compared graph-network based GNS and the momentum-conserving DMCF methods, in terms of both training and inference time, as well as efficient scaling to many-particle regime.\n* Detailed qualitative and experimental results comparing the method against other neural methods, as well as the classical MPM method used to generate the dataset.\n* Availability of code suggests reproducibility.\n## Weaknesses\n* Lack of theoretical properties of the proposed neural estimator. For example, momentum conservation, energy preservation(?), convergence (FEM-like?). \n* The proposed estimator requires a fixed grid prior to training and during inference, and the method seems to be fairly sensitive to this hyperparameter. This could limit the transferability, as graph-based methods seem to be grid-independent. This is however pointed out in the concluding section. A failure example would be interesting.\n* The paper's presentation can be a bit difficult to read at times. In particular, there is a mix of , and extremely long paragraphs that could be broken up into multiple ones (there is a paragraph that spans half a page in section 2!). There is a lot of speculative text as well which dampens the nice experimental claims of the paper (e.g. p.1: \"These emulators may also require long...\", p.3: \"hybrid models inherit ...'', p.4: \".. allows the U-Net to efficiently extract both local and global information''). I would remove or rephrase the text to emphasize the informality/heuristic nature of some of these claims. It may also be worth expositing more on the baseline GNS and DMCF methods.\n\n## Questions\n* Differentiability seems to be a major motivation for the proposed NeuralMPM method, cropping up in both introduction sections, however it does not seem to be explored fully in the main text, and is rather left to Appendix B (Fig 14). How many iterations does it take to reach the target $\\alpha$?. It would be interesting to see how the optimized angles perform under classical MPM as well.\n* Is there a reason differentiability is not feasible for fluids MPM? It seems to be doable for shape deformation, e.g. appendix of [1] or DiffTaichi package. \n* Appendix B: what is SEGNN?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work proposes NeuralMPM, which is a data-driven hybrid particle simulation method based on the classical Material Point Method (MPM). The work proposes data-driven learning of an intermediate step of MPM, specifically time evolution on the discretized dynamics grid, which is preceded and followed by particle-grid mappings. The learned map is given by a (convolutional) neural network that maps grids to grids. Extensive computational simulations are carried out to demonstrate the efficacy of the proposed method compared with some other data-driven methods (based on graph neural networks and momentum-preserving learned kernels), which show significant memory and computational requirement reductions.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "## Strengthening\n* Section 2 ends abruptly. Perhaps write something about NeuralMPM falling into the hybrid model category.\n* The algorithm description in Section 3 needs rewriting. There is irrelevant implementation-specific information (such as a pseudo-description of UNets) scattered throughout, and a proper algorithm environment would greatly aid the presentation here.\n* The paragraph describing MPM in the introduction is very cramped, consider an enumerate environment.\n* Please define EMD in the main text, I could only find it in the appendix.\n* 3D examples would be an interesting further work, and might be a regime where classical methods and GNS outperform convolution-based methods. \n\n## Typos?\n* Figure 4 caption seems to be bolded.\n* (p.2) Missing reference for Smoothed Particle Hydrodynamics\n* (p.3) \"Following MPM,...,as illustrated in Figure 1:\"\n* (p.10) \"am example optimization in Figure 8\"", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The work studies a fundamental problem ML on PDEs. This direction yet to be fully studied and is an interesting subject. 
The theory and practice in this work has room for improvement in the future works.
Decision: Accept (poster)