Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamics
Authors: Lukas Prantl
Benjamin Ummenhofer
Vladlen Koltun
Nils Thuerey
Venue: NeurIPS 2022 Accept (2022)
Keywords: particle-based fluids
conservation of momentum
symmetry
antisymmetry
physical simulation

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

Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamics
Lukas Prantl
Benjamin Ummenhofer
Technical University of Munich lukas, Saikas.panttoen. prant1@tum.de
Intel Labs
Vladlen Koltun
Nils Thuerey Technical University of Munich
Apple
Abstract
We present novel method for guaranteeing linear momentum in learned physics simulations. Unlike existing methods, we enforce conservation of momentum with hard constraint, which realize via antisymmetrical continuous convolutional layers. We combine these strict constraints with hierarchical network architecture, carefully constructed resampling scheme, and training approach for temporal coherence. In combination, the proposed method allows us to increase the physical accuracy of the learned simulator substantially. In addition, the induced physical bias leads to significantly better generalization performance and makes our method more reliable in unseen test cases. We evaluate our method challenging fluid scenarios. Among others, we demonstrate that our approach generalizes new scenarios with up one million particles. Our results show that the proposed algorithm can learn complex dynamics while outperforming existing approaches in generalization and training performance. An implementation of our approach is available at https: / //ththub..ttttt/////// com/ hon//uum-ae pbnn-bpbsDCCFFF
range of different,
Introduction
Learning physics simulations with machine learning techniques opens up a wide range of intriguing paths predict the dynamics of complex physical phenomena such as fluids 2
As traditional simulators for fluids employ handerafted simplifications [3]] and require vast amounts of computational resources <4! recent learning-batt based methods have shown highly promising results. In face of the impressive performance and accuracy of these techniques, it is surprising to see that they still neglect some of the most basic physical principles, like the symmetries that arise from Newton's 's third law of motion and the conservation of momentum. This makes learned simulators less predictable and can lead to severe implications when basing decisions on the simulation outcomes.
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4
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Predicting physical properties with neural networks is commonly treated as regression problem 27 51 (54). where the training signal is defined formulation allows to effectively learn and approximate physical processes but also gives way to unwanted shorteuts that deviate from the basic laws of physics. At the same time network architectures are usually designed with gencralizability in mind, e.g., with applications ranging from geometry processing 34 physical simulations 2 contrast, we propose lean and efficient architecture that provides stable and large receptive field, while adhering to the desired physical constraints. At training time, we dynamically take the unrolled evolution of the physical state into account to ensure stability when training with longer sequences.
soft constraint. This simple and desirable
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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For the implementation of the inductive bias, our approach is inspired by Smoothed Particle Hy- drodynamics (SPH) methods [I3
20 where physical properties are preserved by the appropriate
design of the smoothing kernels S Our method uses learned convolution kernels that have inherent
benefits over general graph structures the context of physics simulations, as they can easily process positional information. Instead of using soft constraint, e.g., by integration into the loss function, our method enforces physical properties by using inductive biases while keeping the simple training signal of regression problem. To this end, we integrate symmetries, which are integral part of the underlying physical models, into our neural network. This results guaranteed conservation of momentum for our method.
1.1 Related Work
Numerical computation of fluid-based dynamics is a long and extensively treated topic in research
We discuss the most relevant work here, and refer to surveys 21 for full review. Our approach uses a Lagrangian viewpoint, where fluids are represented as smoothed particles. This viewpoint by design conserves the mass of the fluid, as the number of partieles does not change, and limits valuable computation time to where the fluid is. This idea stems from astrophysics known as Smoothed Purticle Hydrodynamics (SPH). It was followed by a large number of papers that improved the SPH method in terms of accuracy and performance
15 55
[K] and is
K 30 26
In contrast to many classic simulators that use sophisticated and handcrafted model equations to describe the motion of fluids, our method belongs to the class of data-driven solvers, which entirely learns their dynamics from observations. A seminal example of learning fluid simulation from data is Ladicky et al. [22], which uses carefully designed density features with random forests to regress fluid motions. However, the method does not guarantee momentum conservation. In recent work, algorithms have been proposed to use physical priors based on Lagrangians 355
or Hamiltonians 36 improve the physical accuracy of learning ODEs and PDEs, and compliance with conservation laws. Nevertheless, the methods are limited to low-dimensional problems and do not address complex PDEs such those required for fluids. Other works have used convolutional neural networks, or ConvNets, to achieve acceleration for grid-based fluid simulations 44 47 19]
The former proposes ConyNets to accelerate the expensive pressure correction step 46). while the other two propose ConvNets learning corrections to reduce the grid size and, consequently, the runtime of simulations. Although these methods show large speed-ups, they inherently cannot be applied to particle- based simulations.
For processing point clouds, the community has followed multiple strategies address the challenges with this representation. A key challenge is the permutation invariance of particles. An approach to address this uses a combination of point-wise MLPs, order-independent set functions, and farthest point sampling to create a point cloud hierarchy 34 Another line of work processes point clouds with graph neural networks. These methods represent fluid particles with graph nodes and define Here, interaction networks with relation-centri and object-centric functions were developed to predict future object states [2]. Since the framework aims to be general and employs unconstrained MLPs, it cannot be guaranteed that interactions are symmetric. The same holds for Li al. 23 which uses the same framework and builds dynamic graph connecting nearby particles to model fluids. SPH simulations can also be treated as
edges between them [2] 2 3
on graphs These approaches typically decompose simulator into multiple MLPs, which act as encoder, processor, and decoder. The processor performs multiple rounds of general message-passing updates node and edge features, and does not enforce physical properties like momentum conservation. Further, graph-based methods are strongly tied to the chosen particle discretization, and changes to it, like the sampling density, may require non-trivial changes to the models.
37 60
As an alternative to explicit graph representations, convolutional architectures represent an interesting option for particle-based simulation. Schenck et al. [39] implement special layers for particle-particle interactions and boundary interactions. The work does not aim to learn a general fluid simulator but implements a differentiable position- based fluids (PBF) 226] solver, which is used to estimate liquid parameters or train networks to solve control problems by backpropagation through the solver. Moreover, Fourier neural operators 24 or their adaptive variants [4] have been proposed to solve PDEs. Most related to our approach is Ummenhofer et al. 48 which implements a trainable continuous convolution operator resulting in compact networks. The learned kernels describe general continuous functions with a smooth radial falloff but do not account for symmetric interactions of

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particles. We extend the continuous convolutions to enable simulations that guarantee momentum conservation and introduce a particle hierarchy for better accuracy and robustness.
Symmetries play an essential role in physics, as isolated systems are invariant to certain transforma- tions, i.e., the behavior of physical system does not depend on its orientation. For example, Lie point symmetries were used to optimize data augmentation and thus improve the sample efficiency for learning PDEs [6]. Similar observations apply for the structural analysis of molecules [T and
vision applications
ITO 41]. The chemical properties of proteins do not change under rotation, nor should class labels of objects in images. Rotationally equivariant architectures have been proposed to leverage this concept for improved generalization and sample efficiency, e.g., constraining the convolution kernels for the (2) group achieve equivariance [53]] A similar approach by Wang et al. 12 was applied to physics- bascs----deddde problems. Kernel constraints can also be transferred to continuous convolutions on 2D point clouds for improved sample efficiency of traffic trajectory predictions [S0) For 3D problems, spherical harmonics can be used to define rotationally equivariant filters 45 which shows their application to shape classification, molecule generation, and simple problems in classical mechanics. The method guarantees rotational equivariance, but filter evaluations can become expensive for higher degrees. In addition, architectures for symmetries in arbitrary which however require graph-based representations.
dimensions were proposed 49 38
2 Method
Given a physical quantity P. S Rd at a time t in the form of particles, let Ft P. denote the position, i E Rd the corresponding veloc- ity and d the spatial dimension. The central learning objective is to approximate the under- lying physical dynamics predict the state of Pe+se after period of time At. We integrate the velocities i and positions 2; of the phys- ical system in time while taking into account external forces Fext. e.g., gravity, as illustrated in Fig. [ For an initial prediction, an explicit Euler step of the form a' 2 At Feat and o(-at = 2t + At 12+ is used, where m represents the mass of the particles. The resulting physics state passed to a neural network G with trainable parameters O which infers the residual of the position Ae++ such that particles match a set of desired ground-truth positions #t Velocities are obtained with position-based update. This yields:
(x kt)
(f Winad
Ga
Ax,ar
Pest
Figure 1: Each time step of our method performs predictive time integration step (blue) for position T, and velocity u, of set of particles time l. A neural network G (red) provides a position update obtain Z++Ar-
EitAr = a/-st + AZ44A: : U4+
T++Ai It.
At
Where the minimization target of the neural network G is as follows:
min L(E'+At + G(al+se 1/4 Q). M(+ae).
with L denoting distance- based loss function, and V++Ai the ground-truth positions. By separating the external forces and the update inferred by the neural network, the conservation of momentum only depends on the latter. In the next section, derive which properties the network must fulfill to achieve this.
2.1 Conservation of Momentum
The linear momentum of a physical system is given by M = S= Ug where P represents an arbitrary control volume in the system, my the mass, and e₂ the velocity of particle z. It follows that the rate of change of the momentum in the absence of external forces Fext, according to Newton's second law m, Fext Fz, is given by:
M
mg
F
JEEP
JEEP
where Fg. represents the internal forces of the physical system, and ag denotes acceleration. Hence the central condition for preserving linear momentum is
F2 = 0.
(4)
JEP
3

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It is theoretically possible to include this conservation law as minimization goal. However, this would merely impose a soft constraint that needs be weighed against the other terms the learning objective. We instead propose to use the conservation of momentum an inductive bias at training time in the form of hard constraint.
In order to guarantee that momentum is conserved, we work with convolutional layers with specially designed kernels. We define the convolution in the continuous domain as
(f * g)(x)
f(y)g(y - x), z E PD. PE
(5)
where y represents learnable kernel function and f is the feature vector of a quantity that should be processed the convolution operation. Here PD denotes a set of data points, while Po denotes a set of query points which the convolution is performed. The use of convolutional layers already guarantees permutation invariance and translation equivariance.
However, instead of just relying this learned kernel function, we additionally modify the convo- lution to further assist in ensuring conservation momentum. For the deduction, we reformulate the residual position Azt+Ac that the network generates for given position 2; the result of an internal force Eg:
ATH+A F3 m A/2
(6)
In general, the internal force at the position can be expressed as the integral of all pair-wise interaction forces exerted on x, iec Pa = -yep Pay Following Eq. to enforce conservation of momentum we have ensure
Fay O.
JEEP JEP
If it is guaranteed that for every internal force Fay corresponding opposing force exists, i.e.
Fay = = ya.
Vr,yEP,
(8)
then the integral of Eq. evaaluaees to zero, and the condition for the conservation momentum is fulfilled. This corresponds antisymmetry of the internal forces in the fluid.
Applied to convolutional layers, we achieve this property by adjusting the convolution Eq. follows:
as
* (x))))=)= = yePo
(f(x) + f())) x),
x E Pp:
where g, is an antisymmetric kernel with the restriction that
PD = PQ
(10)
and fxx) is the feature vector at the position x, where the convolution is evaluated. The condition TO is necessary to guarantee that there is a bidirectional connection between two points from the two
sets. This only possible if both points exist in both sets. As before, let us consider the relative
features fzy (f(x) + fw))g,, xt), isolated for two points in space 2, y P. The kernel g, is antisymmetric, i.e. go (x) - x). From this follows:
fay (fa + fy)( gs(x v)) - Syz-
(11)
Due to the restriction from Eq. 10
yyu exists for every fay and hence Eq. 8 is satisfied.
2.2 Antisymmetric Continuous Convolution
To implement the convolutional layer fulfilling the constraints above, we make use of continuous convolutions (CConv) 4 While the constraints could likewise be realized with other forms of convolutional layers, CConvs are adapted for efficiently processing unstructured data, such as particle-based data from fluids, and hence provide a suitable basis to demonstrate the impact of the antisymmetric constraints. CConv layers determine the nearest neighbors from given set of data points Pd for given set query points Po and aggregate their features f. Thereby, the features of the neighbors are weighted with a kernel function depending on their relative position. The kernel functions themselves are discretized via a regular grid with a spherical mapping and contain the leamnale parameters.

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fo
10
Fud
type-awars Preproceaaing
Branch 2
Add
Multi-ecate Featurn Aggregation
Figure 3: Neural network architecture: The colored squares symbolize the different CConv blocks, whereas the rotated square in red represents the antisymmetric layer. The color shows the used query point set. Orange corresponds to the original point set, with each set below halving the resolution. Search radii are enlarged accordingly.
The original CConvs defined as: CConv, PD, (f) gx) = E Seglax x)) for
E (P x) and x E Po. This corresponds to a discretized version of Eq. where Nr, represents the nearest neighbor search for fixed radius To conserve momentum, adapt the equation
according to Eq.
Tn.yann as follows:
gSCC,, (f.PD. = (f ( ())
E
(f+ fk)ge(Ix x),
Pq
(12)
kEA-(PD.2)
with the restriction that Po Pq. Here the discrete, antisymmetric kernel 9s consists of a grid with a user- defined size. To obtain the antisymmetric property we halve the learnable kernel parameters along a chosen axis and determine the second half by reflection through the center of the kernel. Additionally, the mirrored values are negated, as shown in Fig. 2 Mathematically, the mirroring fulfills Eq. y and hence the axis of reflection for the kernel weights can chosen freely. The convolution evaluated with the reflexive term f + fu from Eq, to preserve the antisymmetry. In following, we refer to the proposed antisymmetric CConv layer ASCC.
2
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-7
4
3
4
-6
-5
6
5
6
-4
-3
8
7
8
-2
-1
Trainable
Reflected
The trainable variables are negated and mirrored by the center point. This results in antisymmetric kernel.
Figure 2:
2.3 Neural Network Formulation
To construct full neural network model that preserves momentum, we build on a CConv architecture, replacing the output layer with an ASCC layer, as shown Fig. B We deliberately replace only the last layer, this maximizes the flexibility of the previous layers and still guarantees momentum preservation. While antisymmetric kernels have highly attractive physical properties, they intention- ally restrict the scope of actions of neural network. The resulting networks generalize much better, but the initial learning objective more complex than unconstrained networks. Regular networks can take unphysical shortcuts by overfitting problem-specific values. A corresponding evaluation of the generalization capabilities ASCCs is provided Sec.
Layer Hierarchy To support learning generalizable physical dynamics, we employ hierarchical network structure that allows for a large receptive field. Specif- ically, we
architecture inspired by the parallel pro- cessing the multi-scale feature aggregator of HRNet 4 It processes sets of extracted features different spatial scales with separate branches of the network. Akin to pooling layers, the density of the sampling points is reduced successively in each branch. The structure of the network and its branches shown in Fig. 3
Figure 4: Farthest point sampling (left) vs. voxel sampling (right). Data points in black, sampling points in red. The red circles indicate sampling region's 's for feature aggregation.
The main branch of network works on the full set of sample points Poo which in this case equal to the data points Pp. For the secondary branches, new set of sampling points Poi is
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generated from the input points PD with a scaling factor s; for the corresponding branch such that s; z p with p denoting the density function of the given point set. Additionally, points evenly as possible so that the scaling factor remains
in each set should be spatially distributed similar across all partial volumes. To satisfy condition TO the point features are resampled back to original sample points Pqo before processing them in the ASCC layer.
Generating the sampling points Po: of a layer is non-trivial task, for which previous work typically employs farthest-point sampling (FPS) M4 While FPS extracts a uniformly distributed subset from a point cloud, its time complexity of (Nlogg is not favorable. We have found that voxelization approach is preferable, which in a similar form is already used in established computer vision methods We choose the centers of the cells of a regular grid, i.e. voxels, as sampling points, as illustrated in Fig. + The spacing of the grid is determined from the initial particle sampling modified by the scaling factor 8;. The voxelization can be performed in O(N), and results in an evenly distributed set of sampling points.
[56] 12
Feature Propagation To compute the features fiz for level and branch j, we use a set of CConvs that process the features of all branches from the previous pass for each branch. This results in multi-level feature aggregation across all branches. For merging the features, the different features are accumulated via summation:
fij Σ CConv(J(--1).) POR, Pos):
(13)
k
where is denotes all branch indices of the previous layer, as also illustrated in Fig. Here, it is important to choose the sampling radius of the convolution operator based on the scale of the corresponding input branch Sy- This ensures that the convolutions on average always process neighborhood of similar size. Le., for a small input resolution a larger radius used, and vice versa. Finally, the accumulated features of the main branch are processed by ASCC layer.
2.4 Training and Long-term Stability
Following previous work prediction and ground truth weighted by the neighbor count as our loss function:
2
37
use a mean absolute error of the position values between
L(I) P
Σ
Lri yt.
(14)
Here, C denotes the number of neighbors for particle i, and Cavg the average neighbor count. We purposefully formulate the loss work without additional loss terms, such as physical soft constraints, and instead rely on the conservation of momentum from the ASCC.
For temporal stability, we we run the network for time steps, where the input for the next step is based on the result from the previous prediction. The loss is evaluated and averaged across all T frames with L, T Σ L(t). This integrates the temporal behavior the training and stabilizes it for a medium time range.
rollout of T frames training time. That is, for each training iteration,
While larger 7 are preferable for improving long-term predictions, they lead to significantly increased memory requirements for backpropagation. To counteract this effect, we prepend the rollout by sequence of W simulated preprocessing steps using the current training state. Altogether N = W -T frames are generated, but irrespective of the number of preprocessing steps, the loss and hence gradient, is only computed for I' steps after preprocessing. This lets the network learn to correct its own errors that occur over longer time spans and improves long-term stability even in complex scenarios without increasing the memory. During training, we adapt W based on the training progress and the example difficulty. We provide the details of the implementation in App. A.2
3 Results
We perform a series of tests to show the physical accuracy, robustness, and generalizability of our method compared to other baseline methods. We primarily use data from a high-fidelity SPH solver with adaptive time stepping []. The resulting, two-dimensional dataset "WBC-SPH" consists of randomly generated obstacle geometries and fluid regions. Gravity direction and strength are additionally varied across simulations. In addition to this primary dataset, we also use the MPM- based fluid dataset "WaterRamps" from Sanchez-Gonzale et al. [ and the ttree-dimensional

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1.25
Base
1.00
ASCC
Multi Scale
0.50
Voxelize
0.25
Preprocess
Ours
RMSE (per step)
Momentum
Max. Dens.
Average
Figure 6: Quantitative evaluation for the ablation study: We consider a relative accuracy to the final
method (Ours 1.0) with higher values meaning better performance. The metrics computed as described in Sec. S with Vel. Dist. denoting the JSD of the velocity distributions. While RMSE (per step) shows the RMSE after one time step, all other metrics are evaluated over a whole sequence. Average corresponds to the mean of all previous metrics.
liquid dataset "Liquid3d" from Ummenhofer et al. 48 for additional evaluations. Both consist of randomized fluid regions with constant gravity. more detailed description of the datasets is provided in App. A.3
To quantify the accuracy of the methods, we several different metries with respect to the ground truth data from the corresponding dataset. We use the root-mean-squared error (RMSE) for single simulation steps to evaluate short-term accuracy. However, over long sequences, it is common for fluid particles to mix chaotically regardless of whether the general fluid behavior resembles the reference, leading to artificial per-particle errors. Therefore, the earth mover's distance (EMD) is used as an assessment of the long-time accuracy over the fluid volume as a whole I ming:r-t Σg (|x(x)) yll.. where and y are positions from two point sets and denotes a bijective mapping of minimal distances. EMD matches the particles as probability distributions, such that the global distance error is minimized. The deterministic assignment of the particles makes EMD agnostic to the ordering of particles, which counteracts the assessment errors caused by mixing. Intuitively. EMD evaluates the similarity of density- volumes instead of the individual particles. Thus, it quantifies the error of the simulated configuration relative to the reference, neglecting small-scale changes in particle order. In addition, we evaluate the difference between the velocity distributions to evaluate the accuracy of the velocity content of simulation. We compute a histogram of the magnitude of the velocities and compute its Jensen- Shannon divergence to the histogram from ground truth data: JSD(V=.lV,)) = {(DxI. (V2 || V) + D (V, (Vy]IVV), where V = (V2 Vy) and DKL is the Kullback-Leible divergence. We also evaluate deviations in terms of maximum density, [1 - (max, p(z:)/ max, p(y)))l, with p being function to compute the density of particle i. This evaluation correlates with the compressibility of the fluid and gives an indication of the stability of a method. Lastly, also compute the linear momentum change as Σ m.a, FEzt, where a; is the acceleration and m. the mass of a particle, and Fext denotes external forces.
The source code of our approach, together with the associated evaluation tools and links to datasets, available at fttttts:////t //gi httgg/ittuub... wtw. uum Saammmpb/DMMFFFFF
Standing Liquid We first consider one-
No-Sym
dimensional simulations of a hydrostatic col- umn of liquid. The training data consists of different number of fluid particles which, under the effect of gravity, rest on fixed boundary at the bottom. The reference stems from a solver with implicit time stepping [16] that can handle global effects instantaneously. To test the gener- alization, we also evaluate the methods for a free fall scene. Here we generate a certain number of fluid particles in free fall, which substantially deviates from the behavior of the training data. We trained two networks with the architecture from Sec. 2.3 once with the proposed ASCC layer (ASCC) and once with a regular CConv layer instead (No Sym). Despite the seeming simplicity of the test case, even small inaccu- racies can lead to unstable behavior over time.
40
33
10
Column
Fice Fall
Figure 5: RMSE (x10-3) evaluation over 100 steps for a set of Column scenes and Free Fall scenes. A schematic is shown on the left side of the graphs: fluid particles (blue) are affected by grav- ity; solid boundary particles are shown in black. The ASCC network shows the best performance in terms of generalizing to the free fall cases.
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Ponti: Ctat
0.0
6445
120.64
9.
0.19
02
0:00
3.0
RMSE (+1a3)
Figure 7: Excerpts from a test sequence with the high-fidelity WBC-SPH dataset and quantitative evaluation on the right. The frames on the left represent, f.l.t.., (GT) the ground-truth, (a) PointNet, (6) CConv, (c) GNS, and (d) Ours.
Without the antisymmetric constraint, network can simply overfit to the gravity and negate it. This reduces the problem to a local per particle problem. With ASCC, on the other hand, the network relies on particle interactions to solve the problem. This makes it necessary to determine global context, and infer a pressure-like counter force to stabilize the fluid. We compare the performance of both learned versions with the solution obtained by explicit SPH solver with small time step (2) For new column heights, Fig. 5 shows that both learned versions quickly compensate for compression effects in the liquid and yield very small errors, while the SPH solution oscillates in place. Whereas SPH handles the free fall case on the right reasonably well with an error of 19.68, the network without antisymmetry largely fails and produces erroneous motions with an error of 46.36. The antisymmetric version, having learned a physically consistent response, yields an error that improves over SPH by factor of 3.3.
Ablation Study We evaluate the relevance of individual features introduced for our method with the ablation study using the WBC-SPH dataset. We start with a base model (Base) without ASCC layer and gradually add relevant features. While this model fares very well for single steps, as shown in Fig. 6it performs poorly with respect to all other evaluations. Here, the graphs show the performance relative to the full method, i.e., Lours/ L, with L denoting the loss of variant under consideration. For the second version, we add the antisymmetrical layer (ASCC), which ensures the conservation of momentum and significantly improves the motion in terms of velocity distribution. Here the momentum score of 0.0 for Base corresponds to effective error of 0.095, all following variants having a score of 1.0 with zero error. Next, we evaluate the relevance of the multi-scale processing. We tested two different variants, one with FPS (Multi Scale) and one with the voxelization (Voxelize). The performance is very similar for both, with the latter variant having clear advantages in terms of resource usage. While improvements following the multi-scale handling are not obvious first it starts to more clearly pay off when combined with preprocessing for temporal coherence training in (Preprocess). The improved long-term stability and accuracy are clearly visible in the maximum density and EMD metrics. Lastly, our full method (Ours) includes additional input rotation normalization and a more advanced boundary processing step, as explained in more detail in App. A.I This leads to further slight improvements and yields the final algorithm, which we will compare to previous work in the following sections.
Comparisons with Previous Work In the following, we compare our method with several estab- lished methods as baselines: an adapted variant of PointNet [33, the CConv network 4S] and GNS 337. More information about the employed networks and the training procedures are given in App. We train and evaluate networks with our WBC-SPH dataset. For fairness, we additionally evaluate
A.2
the different approaches with their respective data sets, i.., with the WaterRamps dataset of GNS and with the Liquid3d dataset of CConv.
When trained on the WBC-SPH dataset, which consists of sequences with 3200 time steps, neither CConv nor PointNet are able to generate stable results. While the short-term RMSE accuracy is good, both networks fail to stabilize the challenging dynamics of our test sequences. Examples are shown in Fig. 7 The outputs from GNS are stable, but the network seems to maintain stability by smoothing and damping the dynamics. Also, despite receiving multiple frames over time, the GNS model has difficulty predicting the dynamics of changing gravity directions. For this scenario, the antisymmetric ASCC network clearly outperforms GNS and the other variants in quantitative as well as qualitative terms. In particular, its EMD error is 7.22 times smaller than the closest competitor.
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CCome
160
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320
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Figure 8: Snapshots over time from models for the WaterRamps dataset.
0.10
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046
066
Oraniy
No Grunity
Figure 10: Assessment of generalization with droplet collisions: left with, right without gravity. Each side shows EMD evaluation and three centers of mass trajectories with overlaid snapshots over time. The reference trajectories are shown in green. While our approach closely matches these trajectories, hiding them behind the blue lines, GNS noticeably deviates, yielding larger errors in terms of EMD.
For fairness of the comparison, we also retrain our model and the CConv model with the Ramer dataset. While the accuracy of GNS with this data very high, our method nonetheless outperforms GNS with a decrease of 24% in terms of EMD. In comparison, CConv results in a 3.61 times larger EMD. Over short periods the dynamics of GNS be- have similarly to ours. For longer periods of more than 30 frames, the results diverge, shown in
CConv
0.
20
Fig. X Most importantly, our network yields a high accuracy while requiring only a fraction of the com- putational resources that GNS requires (Fig D) the ASCC network has 29% of the weights of GNS, and inference is 2.79 times faster, while yielding the aforementioned improvements in terms of accuracy. In addition to the conservation of momentum, this advantage be explained by the inductive biases of our convolutional architecture: networks can efficiently process positional information, which is difficult for generic graph networks. We discuss the inherent advantages of the convolution architectures over graph networks in more detail in appendix A.2.2 While CConv performs best in terms of resources, it does so at the expense of inference accuracy: its EMD is three times larger than the one obtained with method.
RMSE (+103)
EMD
Param [+fe0]
[ms]
Figure Quantitative evaluations with the WaterRamps dataset. We show the accuracy of the models along with their size and compu- tational performance.
Generalization To assess the generalizing capabilities of our network and GNS trained with the WaterRamps data, we perform tests with droplets colliding under the influence of varying gravity. This case allows for an accurate evaluation terms of the center of mass trajectory of the liquid volume. Considering case with default gravity, the ASCC network already yields EMD error reduced by 57%. When reducing the gravity to zero, the GNS trajectory noticeably deviates from the ground truth trajectory and yields 195% larger eттor than our method. The GNS has encoded the influence of gravity in its learned representation and hence has difficulties transferring the dynamics to the changed physics environment.
3D Simulations Finally, we test our network with 3D data using the Liquid3d dataset. Unlike the other datasets, its simulations use a significantly larger time step (8 times larger than WBC-SPH). Combined with the higher dimensionality. is substantially more difficult obtain stable solutions. Again, our method achieves comparable accuracy with much better generalization. Similar to the previous examples, it becomes apparent that CConv tends to overly dampen acceleration of gravity, which, unlike our method, leads to discrepancies in terms of the motion compared to the ground truth, as shown in Fig. I show the scalability and generalizability of our method, we apply it to multiple large-scale scenes with up to 165x more particles than in the training set. Qualitative results

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GT
CConv
Ours
100
6.10
0.53
0.80
(110-))
Figure 11: Example 3D sequence based on the Liquiddd dataset. CConv visibly dampens the
acceleration by gravity while our method closely matches the dynamics of the ground truth.
Figure 12: Test scene with over one million particles after 800 time steps generated with our method.
can be found in Fig. and App. A.4 (Fig. Our model robustly predicts the dynamies of these scenes using the same time step as for the smaller scenes.
Performance We have also evaluated the computational performance of our method, with detailed measurements given in App. A.4 E.g., for the 2D data WBC-SPH our method requires 25m5 per frame on average. For comparison, the corresponding reference solver requires 10925ms per frame. Thus, our network achieves speed-up of 162 over the reference solver.
4 Conclusion
We have presented method guarantee the conservation of momentum in neural networks and demonstrated the importance of this concept for learning the complex dynamics of liquids. In particular, approach outperforms state-of-the-ar methods in terms of accuracy and generalization. Nonetheless, the limits of symmetric constraints are far from exhausted. E.g., while our network is antisymmetric, permutation invariant, and translation invariant, interesting avenue for future work will be to investigate more generic forms of Ev2) equivariance
50 38
10

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13


Reviews:
[{"review_id": "mEyuNTd61Uw", "reviewer": "Reviewer_qoJP", "summary": "The authors present a physics-guided deep learning model for fluid simulations. The method uses continuous convolutions to process particle representations of fluids where the convolutional kernels in the final layer are constrained to be antisymmetric. The antisymmetric constraint enforces conservation of momentum. The authors show that embedding this constraint results in improved simulation results compared to other methods.", "questions": "- Possible typos:\n  - Line 31: write out SPH\n- Questions:\n  - WBC-SPH: Are the strength and direction of gravity determined randomly?\n  - [261] : [EMD] is especially important to reliably evaluate long simulation sequences. What is the intuition here?\n  - Figure 1: In eq 2 G is a function of x_t and v_t in the figure it is a function of x’_t, v’_t.\n- Comments:\n  - Related work doesn't include other NN + Lagrangian methods [1][2][3][4]\n  - [116] G is used to denote both the linear momentum and the network that predicts the update which is a bit confusing\n  - [31] SPH is not defined until the next section\n  - [219] F is used for features and forces which is a bit confusing\n  - [318] “An adapted variant of PointNet” – how is it adapted?\n  - [320] “For fairness…” It seems more appropriate to use the dataset of CConv since they are using a more similar methodology (CConv vs GNNs)\n\n[1] Saemundsson, Steindor, et al. \"Variational integrator networks for physically structured embeddings.\" International Conference on Artificial Intelligence and Statistics. PMLR, 2020.\n[2] Lutter, Michael, Christian Ritter, and Jan Peters. \"Deep lagrangian networks: Using physics as model prior for deep learning.\" arXiv preprint arXiv:1907.04490 (2019).\n[3] Cranmer, Miles, et al. \"Lagrangian neural networks.\" arXiv preprint arXiv:2003.04630 (2020).\n[4] Sanchez-Gonzalez, Alvaro, et al. \"Hamiltonian graph networks with ode integrators.\" arXiv preprint arXiv:1909.12790 (2019).\n", "limitations": "A limitation implicitly noted in the conclusion is that the learned representations are not rotationally invariant. I suspect rotationally symmetric kernels will substantially reduce expressivity. It may be possible to use something like group convolutions [1].\n\n[1] Cohen, Taco, and Max Welling. \"Group equivariant convolutional networks.\" International conference on machine learning. PMLR, 2016.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "- Originality: The proposed method is an application of continuous convolutions to particle representations of fluids. The proposed intervention is to constrain filters in the final layer to be antisymmetric. The related work is adequately situated in the context of existing literature, I encourage the authors to consider adding the literature on Lagrangian neural network to related work(see questions/comments).\n- Quality: The work is technically sound and the claims are well supported. A limitation implicitly noted in the conclusion is that the learned representations are not rotationally invariant.\n- Clarity: For me the paper was not easy to read. E.g., I think the paragraph at [163] interrupts the flow, perhaps it is better placed at the end of the subsection. It may not be needed since my sense is the same arguments are given in the related work section. There were also notational clashes and abreviations that were never written out (see questions/comments)\n- Significance: The paper addresses the issue of learning physically plausible fluid simulations which is a challenging problem with the potential to facilitate scientific discovery.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "i0JD6SV1NVv", "reviewer": "Reviewer_mZBm", "summary": "This paper proposed the Antisymmetric Continuous Convolution that can guarantee linear momentum in learned physics simulation, and the authors demonstrated the effectiveness of this design on several particle-based simulation datasets. ", "questions": "1. Incorporating inductive biases into neural nets can make them more sample efficient. But I believe if there are enough training data, then the unconstrained networks can also learn the desired symmetry or conservation law from the data. Do you know how much data you need to make the unconstrained model perform as well as the constrained one? A plot of how the prediction performance of both constrained and unconstrained models change across different training set sizes would be helpful.\n\n2. In Figure 6, there is no bar for the base model on the momentum evaluation metric. Does it mean the unconstrained model does not learn the conservation of momentum at all? That seems impossible.\n\n3. Eqn 9: shouldn't it be the integration over the neighbor of x instead of the entire P_Q?", "limitations": " ", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "$$\\textbf{Strengthes:}$$\nOverall, I enjoyed reading this paper. The writing is very clear and the proposed idea is quite neat. The ablation study of the model architecture really strengthens the paper. \n\n$$\\textbf{Weaknesses:}$$\n1. Lack of Baselines: I think the more SOTA models for forecasting fluid dynamics on the irregular meshes are the message-passing neural nets and the neural operator. These models could also be applied to particle-based simulations. It'd be great if the authors can include some of these models as baselines. \n  * Pfaff et.al; Learning Mesh-Based Simulation with Graph Networks\n  * Brandstetter et.al; Message Passing Neural PDE Solvers.\n  * Guibas et.al; Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers\n\n2. Missing related works: I think there are some other associated works about symmetry and fluid dynamics that should also be discussed in the paper such as\n* Brandstetter et.al; Lie Point Symmetry Data Augmentation for Neural PDE Solvers.\n* Wang et.al; Incorporating Symmetry into Deep Dynamics Models for Improved Generalization.\n\n3. It'd be great if the authors can also evaluate the models based on computational cost or inference speed. Because I think one of the biggest advantages of using neural nets for learning complex fluid dynamics is the computation efficiency.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "D8DvCJUHZfN", "reviewer": "Reviewer_bVLD", "summary": "The paper suggests a novel inductive physical bias to guarantee the conservation of momentum. The latter is achieved by reconsidering the resampling strategies and a reformulation of a continuous convolution operation to incorporate the physical knowledge as a hard constraint to respect the symmetry of interaction of particles by construction. ", "questions": "1/ What is alpha in equation (3) ?\n\n2/ Why the restriction of P_D=P_Q ? and what is its implication on the expressive power of the model ?", "limitations": "This work doesn't have societal concerns and the authors are aware of the current limitations of the proposed work.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This work is inspired by Lagrangian mechanics where fluids could be seen as a constellation of particles in interaction. Particles are represented as nodes and their interactions are represented by edges. This construction is a good starting point to make the analogy with Geometric Deep Learning (GDL) models, including Graph Neural Networks (GNNs) and points of cloud. In GDL, we distinguish isotropic and anisotropic convolutions which are widely used and well studied. However, when it comes to design a continuous convolution operation for physical problems one should look carefully at the sampling strategy (dense / sparse connectivity at different local regions), the symmetries present in the data at different physical scales and how to design a smooth kernel that propagates correctly the physical signal while preserving its physical properties. \nIn this study, the authors proposes a novel design of a convolution operation capable at encoding the conservation law as a hard constraint relying on a well studied continuous operation, know as \"CConv\" instead of incorporating this information in the loss function as soft constraint. One of the strength of this hard constraint is to physically guide the neural network in the learning process for better generalization which a crucial topic in physics. \nThe proposed convolutional kernel is smooth by construction and is derived from concepts in Lagrangian discretization, mainly Smoothed Particles Hydrodynamics to integrate in hard the symmetries of the underlying physical problems. This way of designing a kernel allows to conserve the physical properties \"symmetries\" by design while keeping the properties of convolutional operation by design, namely permutation invariance and translation equivariance. In addition to, a careful choice of sampling scheme is taken into consideration which plays also a crucial role as the choice of convolutional operations.\n\nThe weaknesses in this paper are related to the lack of comparison with challenging datasets as done in the related works, including for instance large scale meshes with challenging obstacles and geometries to show the versatility of the proposed model. Moreover,  it could be valuable and insightful to study the robustness of the proposed convolutional operation to different discretization schemes, sampling strategies and noisy data since the goal is generalization.\n\nIn general, the paper is well written, clear and easy to follow. The claims and their structure are quite clear and the mathematical variables are well defined. \nThe study of the related works is well done. Notably, the authors makes a relationship between different concepts in physics, machine learning, signal processing and computer vision. \nThe experimental study is fair and convincing even if it could be extended.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept