Original: Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamics
Revised: Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamics

Original: Author(s)

Original: Abstract
Revised: Abstract

Original: We present a novel method for guaranteeing linear momentum in learned physicssimulations.
Revised: We present a novel method for guaranteeing linear momentum in learned physicssimulations.

Original: Unlike existing methods, we enforce conservation of momentum witha hard constraint, which we realize via antisymmetrical continuous convolutionallayers.
Revised: Unlike existing methods, we enforce conservation of momentum witha hard constraint, which we realize via antisymmetrical continuous convolutionallayers.

Original: We combine these strict constraints with a hierarchical network architecture,a carefully constructed resampling scheme, and a training approach for temporalcoherence.
Revised: We combine these strict constraints with a hierarchical network architecture,a carefully constructed resampling scheme, and a training approach for temporalcoherence.

Original: In combination, the proposed method allows us to substantially increasethe physical accuracy of the learned simulator.
Revised: In combination, the proposed method allows us to substantially increasethe physical accuracy of the learned simulator.

Original: In addition, the induced physicsbias leads to signiﬁcantly better generalization performance and makes our methodmore reliable in unseen test cases.
Revised: In addition, the induced physicsbias leads to significantly better generalization performance and makes our methodmore reliable in unseen test cases.

Original: We evaluate our method on a range of different,challenging ﬂuid scenarios, and show that the proposed algorithm can learn complex dynamics while outperforming existing approaches in terms of generalizationand training performance.
Revised: We evaluate our method on a range of different,challenging fluid scenarios, and show that the proposed algorithm can learn complex dynamics while outperforming existing approaches in terms of generalizationand training performance.

Original:  Introduction
Revised:  Introduction

Original: Learning physics simulations with machine learning techniques opens up a wide range of intriguingpaths to predict the dynamics of complex physical phenomena such as ﬂuids [28, 23, 48, 37].
Revised: Learning physics simulations with machine learning techniques opens up a wide range of intriguingpaths to predict the dynamics of complex physical phenomena such as fluids [19, 16, 35, 24].

Original: Astraditional simulators for ﬂuids employ handcrafted simpliﬁcations [31] and require vast amounts ofcomputational resources [44], recent learning-based methods have shown highly promising results.
Revised: Astraditional simulators for fluids employ handcrafted simplifications [20] and require vast amounts ofcomputational resources [31], recent learning-based methods have shown highly promising results.

Original: In face of the impressive performance and accuracy of these techniques, it is surprising to see thatthey still neglect some of the most basic physical principles, like the symmetries that arise from
Revised: In face of the impressive performance and accuracy of these techniques, it is surprising to see thatthey still neglect some of the most basic physical principles, like the symmetries that arise from

Original: Newton’s third law of motion and the conservation of momentum.
Revised: Newton’s third law of motion and the conservation of momentum.

Original: This makes learned simulators lesspredictable, and can lead to severe implications when basing decisions on the simulation outcomes.
Revised: This makes learned simulators lesspredictable, and can lead to severe implications when basing decisions on the simulation outcomes.

Original: Predicting physical properties with neural networks is commonly treated as a regression problem[27, 51, 54], where the training signal is deﬁned as a soft constraint.
Revised: Predicting physical properties with neural networks is commonly treated as a regression problem[18, 38, 40], where the training signal is defined as a soft constraint.

Original: This simple and desirableformulation allows to effectively learn and approximate physical processes but also gives way tounwanted shortcuts that deviate from basic laws of physics.
Revised: This simple and desirableformulation allows to effectively learn and approximate physical processes but also gives way tounwanted shortcuts that deviate from basic laws of physics.

Original: At the same time network architecturesare usually designed with generalizability in mind, e.g., with applications ranging from geometryprocessing [34] to physical simulations [2].
Revised: At the same time network architecturesare usually designed with generalizability in mind, e.g., with applications ranging from geometryprocessing [23] to physical simulations [2].

Original: In contrast, we propose a lean and efﬁcient architecturethat provides a stable and large receptive ﬁeld, while adhering to the desired physical constraints.
Revised: In contrast, we propose a lean and efficient architecturethat provides a stable and large receptive field, while adhering to the desired physical constraints.

Original: Attraining time, we dynamically take the unrolled evolution of the physical state into account to ensurestability when training with longer sequences.
Revised: Attraining time, we dynamically take the unrolled evolution of the physical state into account to ensurestability when training with longer sequences.

Original: For the implementation of the inductive bias, our approach is inspired by Smoothed Particle Hydrodynamics (SPH) methods [13, 20], where physical properties are preserved by the appropriate design ofthe smoothing kernels [5].
Revised: For the implementation of the inductive bias, our approach is inspired by SPH methods [8, 13], wherephysical properties are preserved by the appropriate design of the smoothing kernels [5].

Original: Instead of using a soft constraint, e.g., by integration into the loss function,our method enforces physical properties by using inductive biases while keeping the simple trainingsignal of a regression problem.
Revised: Insteadof using a soft constraint, e.g., by integration into the loss function, our method enforces physicalproperties by using inductive biases while keeping the simple training signal of a regression problem.

Original: To this end, we integrate symmetries, which are an integral part of the underlying physical models, into our neural network.
Revised: To this end, we integrate symmetries, which are an integral part of the underlying physical models,into our neural network.

Original: This results in a guaranteed conservation ofmomentum for our method.
Revised: This results in a guaranteed conservation of momentum for our method.

Original: 1.1 Related Work
Revised: 1.1 Related Work

Original: Numerical computation of ﬂuid-based dynamics is a long and extensively treated topic in research[15, 55].
Revised: Numerical computation of fluid-based dynamics is a long and extensively treated topic in research[9, 41].

Original: We discuss the most relevant work here, and refer to surveys [21] for a full review.
Revised: We discuss the most relevant work here, and refer to surveys [14] for a full review.

Original: Ourapproach uses a Lagrangian viewpoint, where ﬂuids are represented as smoothed particles.
Revised: Ourapproach uses a Lagrangian viewpoint, where fluids are represented as smoothed particles.

Original: Thisviewpoint by design conserves the mass of the ﬂuid, as the number of particles does not change, andlimits valuable computation time to where the ﬂuid is.
Revised: Thisviewpoint by design conserves the mass of the fluid, as the number of particles does not change, andlimits valuable computation time to where the fluid is.

Original: This idea stems from astrophysics [13], and isknown as Smoothed Particle Hydrodynamics or SPH.
Revised: This idea stems from astrophysics [8], and isknown as Smoothed Particle Hydrodynamics or SPH.

Original: It was followed by a large number of papersthat improved the SPH method in terms of accuracy and performance [3, 40, 26, 4, 17,  1].
Revised: It was followed by a large number of papersthat improved the SPH method in terms of accuracy and performance [3, 27, 17, 4, 11, 1].

Original: In contrast to many classic simulators that use sophisticated and handcrafted model equations todescribe the motion of ﬂuids, our method belongs to the class of data-driven solvers, which entirelylearns their dynamics from observations.
Revised: In contrast to many classic simulators that use sophisticated and handcrafted model equations todescribe the motion of fluids, our method belongs to the class of data-driven solvers, which entirelylearns their dynamics from observations.

Original: A seminal example of learning ﬂuid simulation from data is
Revised: A seminal example of learning fluid simulation from data is

Original: Ladicky et al.
Revised: Ladicky et al.

Original: [22], which uses carefully designed density features with random forests to regress ﬂuidmotions.
Revised: [15], which uses carefully designed density features with random forests to regress fluidmotions.

Original: However, the method does not guarantee momentum conservation.
Revised: However, the method does not guarantee momentum conservation.

Original: Other works have usedconvolutional neural networks or ConvNets, to achieve acceleration for grid-based ﬂuid simulations[46, 47, 19].
Revised: Other works have usedconvolutional neural networks or ConvNets, to achieve acceleration for grid-based fluid simulations[33, 34, 12].

Original: The former proposes ConvNets to accelerate the expensive pressure correction step [46],while the other two propose a ConvNet learning corrections to reduce the grid size and consequentlythe runtime of simulations.
Revised: The former proposes ConvNets to accelerate the expensive pressure correction step [33],while the other two propose a ConvNet learning corrections to reduce the grid size and consequentlythe runtime of simulations.

Original: Although these methods show large speed-ups, they inherently cannot beapplied to particle-based simulations.
Revised: Although these methods show large speed-ups, they inherently cannot beapplied to particle-based simulations.

Original: However, learning and leveraging Lagrangian mechanics is notlimited to ﬂuid systems.

Original: E.g., in recent work, algorithms have been proposed to use physical priorsbased on Lagrangians [35, 25, 9] or Hamiltonians [36] to improve the physical accuracy of learning

Original: ODEs and PDEs, and compliance with conservation laws.

Original: For processing point clouds, the community has followed multiple strategies to address challengesthat come with this representation.
Revised: For processing point clouds, the community has followed multiple strategies to address challengesthat come with this representation.

Original: A key challenge is the permutation invariance of particles.
Revised: A key challenge is the permutation invariance of particles.

Original: Anapproach to address this uses a combination of point-wise MLPs, order-independent set functions, andfarthest point sampling to create a point cloud hierarchy [34].
Revised: Anapproach to address this uses a combination of point-wise MLPs, order-independent set functions,and farthest point sampling to create a point cloud hierarchy [23].

Original: Another line of work processes pointclouds with graph neural networks.
Revised: Another line of work processespoint clouds with graph neural networks.

Original: These methods represent ﬂuid particles with graph nodes anddeﬁne edges between them [2, 23, 37].
Revised: These methods represent fluid particles with graph nodesand define edges between them [2, 16, 24].

Original: Here, interaction networks with relation-centric and objectcentric functions were developed to predict future object states [2].
Revised: Here, interaction networks with relation-centric andobject-centric functions were developed to predict future object states [2].

Original: Since the framework aims to begeneral and employs unconstrained MLPs, it cannot be guaranteed that interactions are symmetric.
Revised: Since the frameworkaims to be general and employs unconstrained MLPs, it cannot be guaranteed that interactions aresymmetric.

Original: The same holds true for Li et al.
Revised: The same holds true for Li et al.

Original: [23], which uses the same framework and builds a dynamic graphconnecting nearby particles to model ﬂuids.
Revised: [16], which uses the same framework and builds adynamic graph connecting nearby particles to model fluids.

Original: SPH simulations can also be treated as message-passingon graphs [37, 30].
Revised: SPH simulations can also be treated asmessage-passing on graphs [24].

Original: Following a similar concept, Brandstetter et al. [7] have presented a generalizablemethod that prevent temporal distribution shifts from the temporal accumulation of errors.
Revised: This divides the approach into multiple MLPs, which act as encoder,processor, and decoder. The processor, which 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 tothe discretization, like the sampling density, may require non-trivial changes to the models.

Original: As an alternative to explicit graph representations, convolutional architectures represent an interestingoption for particle-based simulation.
Revised: As an alternative to explicit graph representations, convolutional architectures represent an alternativefor particle-based simulation.

Original: Schenck et al.
Revised: Schenck et al.

Original: [39] implement special layers for particle-particleinteractions and particle-boundary interactions.
Revised: [26] implement special layers for particle-particleinteractions and particle-boundary interactions.

Original: The work does not aim to learn a general ﬂuidsimulator but implements a differentiable position-based ﬂuids (PBF) [26] solver, which is used toestimate liquid parameters or train networks to solve control problems by backpropagation throughthe solver.
Revised: The work does not aim to learn a general fluidsimulator but implements a differentiable position-based fluids (PBF) [17] solver, which is used toestimate liquid parameters or train networks to solve control problems by backpropagation throughthe solver.

Original: Moreover, Fourier neural operators [24] or their adaptive variants [14] have been proposedto solve PDEs.

Original: Most related to our approach is Ummenhofer et al.
Revised: Most related to our approach is Ummenhofer et al.

Original: [48], which implements a trainablecontinuous convolution operator resulting in compact networks.
Revised: [35], which implements a trainablecontinuous convolution operator resulting in compact networks.

Original: The learned kernels describe generalcontinuous functions with a smooth radial falloff, but do not account for symmetric interactions ofparticles.
Revised: The learned kernels describe generalcontinuous functions with a smooth radial falloff, but do not account for symmetric interactions ofparticles.

Original: We extend the continuous convolutions to enable simulations that guarantee momentumconservation and introduce a particle hierarchy for better accuracy and robustness.
Revised: We extend the continuous convolutions to enable simulations that guarantee momentumconservation and introduce a particle hierarchy for better accuracy and robustness.

Original: Symmetries play an important role in physics, as isolated systems are invariant with respect tocertain transformations, i.e., the behavior of a physical system does not depend on its orientation.
Revised: Symmetries play an important role in physics, as isolated systems are invariant with respect tocertain transformations, i.e., the behavior of a physical system does not depend on its orientation.

Original: For example, Lie point symmetries were used to optimize data augmentation and thus improve thesample efﬁciency for learning PDEs [6].

Original: Similar observations apply for the structural analysis ofmolecules [11] and vision applications [10, 41].
Revised: Similar observations apply for the structural analysis of molecules [7] and vision applications [6, 28].

Original: Chemical properties of proteins do not change underrotation and neither should class labels of objects in images.
Revised: Chemical properties of proteins do not change under rotation and neither should class labels of objectsin images.

Original: Rotationally equivariant architectureshave been proposed to leverage this concept for improved generalization and sample efﬁciency, e.g.,constraining the convolution kernels for theE (2) group to achieve equivariance [53]. A similarapproach by Wang et al.
Revised: Rotationally equivariant architectures have been proposed to leverage this concept forimproved generalization and sampling efficiency, e.g., constraining the convolution kernels for the

Original: [52] was applied to physics-based problems.
Revised: E (2) group to achieve equivariance [39].

Original: Kernel constraints can also be transferred to continuous convolutions on 2D point clouds for improved sample efﬁciency of trafﬁctrajectory predictions [50].
Revised: Kernel constraints can also be transferred to continuousconvolutions on 2D point clouds for improved sample efficiency of traffic trajectory predictions [37].

Original: For 3D problems, spherical harmonics can be used to deﬁne rotationallyequivariant ﬁlters [45], which shows their application to shape classiﬁcation, molecule generation,and simple problems in classical mechanics.
Revised: For 3D problems, spherical harmonics can be used to define rotationally equivariant filters [32],which shows their application to shape classification, molecule generation, and simple problems inclassical mechanics.

Original: The method guarantees rotational equivariance, but ﬁlterevaluations can become expensive for higher degrees.
Revised: The method guarantees rotational equivariance, but filter evaluations can becomeexpensive for higher degrees.

Original: In addition, architectures for symmetries inarbitrary dimensions were proposed [49, 38], which however require graph-based representations. 
Revised: In addition, architectures for symmetries in arbitrary dimensions wereproposed [36, 25], which however require graph-based representations. [Figure] [Equation] x

Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original: Given a physical quantity P t ✓ R d at a timet in the form of particles, let x t 2 P t denoteposition, v t 2 R d the corresponding velocityand d the spatial dimension.
Revised: Given a physical quantity P t ⊆ R d at a timet in the form of particles, let x t ∈ P t denoteposition, v t ∈ R d the corresponding velocityand d the spatial dimension.

Original: The central learning objective is to approximate the underlyingphysical dynamics to predict the state of P t +  tafter a period of time  t .
Revised: The central learning objective is to approximate the underlyingphysical dynamics to predict the state of P t +∆ tafter a period of time ∆ t .

Original: We integrate thevelocities v t and positions x t of the physicalsystem in time while taking into account external forces F ext , e.g., gravity, as illustrated in
Revised: We integrate thevelocities v t and positions x t of the physicalsystem in time while taking into account external forces F ext , e.g., gravity, as illustrated in

Original: Fig.
Revised: Fig.

Original: For an initial prediction, an explicit Euler step of the form v 0 t +  t = v t +  t F ext m andx [-
-]
Revised: For an initial prediction, an explicit Eu- [Figure] ler step of the form v ′ t +∆ t = v t + ∆ t F ext m andx [-
-]′

Original: t +  t = x t +  t v 0 t +  t , is used, where m represents the mass of the particles.
Revised: t +∆ t = x t + ∆ t v ′ t +∆ t , is used, where m represents the mass of the particles.

Original: The resulting physicsstate is passed to a neural network G with trainable parameters ⇥ , which infers the residual of theposition  x t +  t such that particles match a set of desired ground-truth positions y t +  t .
Revised: The resulting physicsstate is passed to a neural network G with trainable parameters Θ , which infers the residual of theposition ∆ x t +∆ t such that particles match a set of desired ground-truth positions y t +∆ t .

Original: Velocitiesare obtained with a position-based update.
Revised: Velocitiesare obtained with a position-based update.

Original: This yields:
Revised: This yields:

Original: [Equation]
Revised: [Equation]

Original: Where the minimization target of the neural network G is as follows:
Revised: Where the minimization target of the neural network G is as follows:

Original: [Equation]
Revised: [Equation]

Original: with L denoting a distance-based loss function, and y t +  t the ground-truth positions.
Revised: with L denoting a distance-based loss function, and y t +∆ t the ground-truth positions.

Original: By separatingthe external forces and the update inferred by the neural network, the conservation of momentumonly depends on the latter.
Revised: By separatingthe external forces and the update inferred by the neural network, the conservation of momentumonly depends on the latter.

Original: In the next section, we derive which properties the network must fulﬁll toachieve this.
Revised: In the next section, we derive which properties the network must fulfill toachieve this.

Original: 2.1 Conservation of Momentum
Revised: 2.1 Conservation of Momentum

Original: The linear momentum of a physical system is given by M = R x 2 P m x v x , , where P represents an 120 arbitrary control volume in the system, m x the mass, and v x the velocity of particle x .
Revised: The linear momentum of a physical system is given by G = (cid:82) x ∈ P m x v x , , where P represents anarbitrary control volume in the system.

Original: It follows thatthe rate of change of the momentum in the absence of external forces, according to Newton’s secondlaw, is given by:
Revised: It follows that the rate of change of the momentum in theabsence of external forces, according to Newton’s second law, is given by:

Original: [Equation]
Revised: [Equation]

Original: where F x represents the internal forces of the physical system, and a x denotes acceleration.
Revised: where F x represents the internal forces of the physical system.

Original: Hencethe central condition for preserving linear momentum is
Revised: Hence the central condition forpreserving linear momentum is

Original: [Equation]
Revised: [Equation]

Original: It is theoretically possible to include this conservation law as a minimization goal.
Revised: It is theoretically possible to include this conservation law as a minimization goal.

Original: However, thiswould merely impose a soft-constraint that needs to be weighted against the other terms of thelearning objective.
Revised: However, thiswould merely impose a soft-constraint that needs to be weighted against the other terms of thelearning objective.

Original: We instead propose to use conservation of momentum as an inductive bias attraining time in the form of a hard constraint.
Revised: We instead propose to use conservation of momentum as an inductive bias attraining time in the form of a hard constraint.

Original: In order to guarantee that momentum is conserved, we work with convolutional layers with speciallydesigned kernels.
Revised: In order to guarantee that momentum is conserved, we work with convolutional layers with speciallydesigned kernels.

Original: We deﬁne the convolution in the continuous domain as
Revised: We define the convolution in the continuous domain as

Original: [Equation]
Revised: [Equation]

Original: where g represents a learnable kernel function and f is the feature vector of a quantity which shouldbe processed in the convolution operation.
Revised: where g represents a learnable kernel function and f is the feature vector of a quantity which shouldbe processed in the convolution operation.

Original: Here P D denotes a set of data points, while P Q denotes aset of query points on which the convolution is performed.
Revised: Here P D denotes a set of data points, while P Q denotes aset of query points on which the convolution is performed.

Original: The use of convolutional layers alreadyguarantees permutation invariance and translation equivariance.
Revised: The use of convolutional layers alreadyguarantees permutation invariance and translation equivariance.

Original: However, instead of just relying on this learned kernel function, we additionally modify the convolution to further assist in ensuring conservation of momentum.
Revised: However, instead of just relying on this learned kernel function, we additionally modify the convolution to further assist in ensuring conservation of momentum.

Original: For the deduction, we reformulatethe residual position  x t +  t that the network generates for a given position x t as the result of aninternal force F x :
Revised: For the deduction, we reformulate the residual position ∆ x t +∆ t that the network generates for a given position x t as the result of aninternal force F x :

Original: [Equation]
Revised: [Equation]

Original: In general, the internal force at the position x can be expressed as the integral of all pair-wiseinteraction forces exerted on x , i.e. F x = R y 2 P F xy .
Revised: In general, the internal force at the position x can be expressed as the integral of all pair-wiseinteraction forces exerted on x , i.e. F x = (cid:82) y ∈ P F xy .

Original: Following Eq.
Revised: Following Eq.

Original: (4), to enforce conservation ofmomentum we have to ensure
Revised: (4), to enforce conservation ofmomentum we have to ensure

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: If it is guaranteed that for every internal force F xy a corresponding opposing force exists, i.e.
Revised: If it is guaranteed that for every internal force F xy a corresponding opposing force exists, i.e.

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: then the integral of Eq.
Revised: then the integral of Eq.

Original: (7) evaluates to zero, and the condition for the conservation of momentum isfulﬁlled.
Revised: (7) evaluates to zero, and the condition for the conservation of momentum isfulfilled.

Original: This corresponds to an antisymmetry of the internal forces in the ﬂuid.
Revised: This corresponds to an antisymmetry of the internal forces in the fluid.

Original: Applied to convolutional layers, we achieve this property by adjusting the convolution Eq.
Revised: Applied to convolutional layers, we achieve this property by adjusting the convolution Eq.

Original: (5) asfollows:
Revised: (5) asfollows:

Original: [Equation]
Revised: [Equation]

Original: where g s is an antisymmetric kernel with the restriction that [Equation] and f ( x ) is the feature vector at the position x , where the convolution is evaluated.
Revised: where g s is an antisymmetric kernel with the restriction that P D = P Q and f ( x ) is the feature vectorat the position x , where the convolution is evaluated.

Original: The condition(10) is necessary to guarantee that there is a bidirectional connection between two points from thetwo sets.

Original: This is only possible if both points exist in both sets. As before, let us consider the relativefeatures f xy =
Revised: As before, let us consider the relative featuresisolated for two points in space x, y ∈ P :

Original: ( f ( x ) + f ( y ))
Revised: [Equation]

Original: g s ( y  x ) , isolated for two points in space x, y 2 P .
Revised: [Equation]

Original: The kernel g s isantisymmetric, i.e. g s ( x ) =  g s (  x ) .
Revised: The kernel g s is antisymmetric, i.e. g s ( x ) = − g s ( − x ) .

Original: From this follows:
Revised: From this follows:

Original: [Equation]
Revised: [Equation]

Original: Due to the restriction from Eq.(10) f yx exists for every f xy and hence Eq. (8) is satisﬁed.
Revised: fulfilling the condition of Eq.

Original: 2.2 AntisymmetricContinuous Convolution
Revised: 2.2 Antisymmetric

Original: [Figure]{+Continuous Convolution1 2 -83 4 -65 6 -47 8 -
+}
Revised: {+Continuous Convolution1 2 -83 4 -65 6 -47 8 -
+}

Original: Trainable
Revised: Trainable

Original: Reﬂected
Revised: Reﬂected

Original: To implement the convolutional layer fulﬁlling the constraints above, we make use ofcontinuous convolutions (CConv) [48].
Revised: To implement the convolutional layer fulfilling the constraints above, we make use ofcontinuous convolutions (CConv) [35].

Original: Whilethe constraints could likewise be realized withother forms of convolutional layers, CConvs areadapted for efﬁciently processing unstructureddata, such as particle-based data from ﬂuids, andhence provide a suitable basis to demonstrate theimpact of the antisymmetric constraints.
Revised: Whilethe constraints could likewise be realized withother forms of convolutional layers, CConvs areadapted for efficiently processing unstructureddata, such as particle-based data from fluids, andhence provide a suitable basis to demonstrate theimpact of the antisymmetric constraints.

Original: CConvlayers determine the nearest neighbors from agiven set of data points P D for a given set ofquery points P Q and aggregate their features f .
Revised: CConvlayers determine the nearest neighbors from agiven set of data points P D for a given set ofquery points P Q and aggregate their features f .

Original: Thereby, the features of the neighbors are weightedwith a kernel function depending on their relative position.
Revised: Thereby, the features of the neighbors are weightedwith a kernel function depending on their relative position.

Original: The kernel functions themselves arediscretized via a regular grid with a spherical mapping, and contain the learnable parameters.
Revised: The kernel functions themselves arediscretized via a regular grid with a spherical mapping, and contain the learnable parameters.

Original: The original CConvs are deﬁned as:
Revised: Alternatively, Graph Neural Networks (GNNs) have been established for processing unstructureddata. Even if the data is not given in the form of an explicit graph structure, a graph can be createddynamically by creating new edges based on the proximity of particles and a distance threshold. Whilegraphs and spatial convolutions can be seen as equivalent representations that can be transformed intoone another for a given discretization with particles, CConvs contain an explicit inductive bias in theform of positional information of the convolution kernel. The relative positions of query points aredirectly put into correlation.

Revised: This is a crucial operation of classical Lagrangian discretizations suchas SPH, and supported by CConvs without having to be learned and encoded in parameters.

Original: CConv
Revised: As aconsequence, CConvs yield leaner networks with correspondingly faster evaluation and training times. Additionally, in contrast to graphs, the kernels describe a continuous function, which is independentof the problem discretization and simplifies changing the density having to [Figure] +

Original: [Equation]
Revised: [Figure]

Original:  =
Revised: + + [Figure] The original CConvs are defined as: CConv g ( f, P D , P Q ) = ( f ∗ g )( x ) =

Original: P k f k g ( x k  x ) for 170 k 2 N r ( P D , x ) and x 2 P Q .
Revised: (cid:80) k f k g ( x k − x ) fork ∈ N r ( P D , x ) and x ∈ P Q .

Original: This corresponds to a discretized version of Eq.
Revised: This corresponds to a discretized version of Eq.

Original: (5), where N r 171 represents the nearest neighbor search for a ﬁxed radius r .
Revised: (5), where N rrepresents the nearest neighbor search for a fixed radius r .

Original: To conserve momentum, we adapt theequation according to Eq.
Revised: To conserve momentum, we adapt theequation according to Eq.

Original: (9) as follows:
Revised: (9) as follows:

Original: [Equation]
Revised: [Equation]

Original: [Figure]

Original: +

Original: [Figure]

Original: with the restriction that P D = P Q .
Revised: with the restriction that P D = P Q .

Original: Here the discrete, antisymmetric kernel g s consists of a grid witha user-deﬁned size.
Revised: Here the discrete, antisymmetric kernel g s consists of a grid witha user-defined size.

Original: To obtain the antisymmetric property we halve the learnable kernel parametersalong a chosen axis, and determine the second half by reﬂection through the center of the kernel.
Revised: To obtain the antisymmetric property we halve the learnable kernel parametersalong a chosen axis, and determine the second half by reflection through the center of the kernel.

Original: Additionally, the mirrored values are negated, as shown in Fig.
Revised: Additionally, the mirrored values are negated, as shown in Fig.

Original: Mathematically, the mirroringfulﬁlls Eq.
Revised: Mathematically, the mirroringfulfills Eq.

Original: (9), and hence the axis of reﬂection for the kernel weights can be chosen freely.
Revised: (9), and hence the axis of reflection for the kernel weights can be chosen freely.

Original: Theconvolution is evaluated with the reﬂexive term f + f k from Eq.
Revised: Theconvolution is evaluated with the reflexive term f + f k from Eq.

Original: (9) to preserve the antisymmetry.
Revised: (9) to preserve the antisymmetry.

Original: Inthe following, we refer to the proposed antisymmetric CConv layer as ASCC .
Revised: Inthe following, we refer to the proposed antisymmetric CConv layer as ASCC .

Original: 2.3 Neural Network Formulation
Revised: 2.3 Neural Network Formulation

Original: To construct a full neural network model that preserves momentum, we build on a CConv architecture,replacing the output layer with an ASCC layer, as shown in Fig.
Revised: To construct a full neural network model that preserves momentum, we build on a CConv architecture,replacing the output layer with an ASCC layer, as shown in Fig.

Original: We deliberately replace only thelast layer, as this maximizes the ﬂexibility of the previous layers and still guarantees momentumpreservation.
Revised: We deliberately replace only thelast layer, as this maximizes the flexibility of the previous layers and still guarantees momentumpreservation.

Original: While antisymmetric kernels have highly attractive physical properties, they intentionally restrict the scope of actions of a neural network.
Revised: While antisymmetric kernels have highly attractive physical properties, they intentionally restrict the scope of actions of a neural network.

Original: The resulting networks generalize much better,but the initial learning objective is more complex than for unconstrained networks.
Revised: The resulting networks generalize much better,but the initial learning objective is more complex than for unconstrained networks.

Original: Regular networkscan take unphysical shortcuts by overﬁtting to problem-speciﬁc values.
Revised: Regular networkscan take unphysical shortcuts by overfitting to problem-specific values.

Original: A corresponding evaluationof the generalization capabilities of ASCCs is provided in Sec.
Revised: A corresponding evaluationof the generalization capabilities of ASCCs is provided in Sec.

Original: Layer Hierarchy To support learning generalizablephysical dynamics, we employ a hierarchical networkstructure that allows for a large receptive ﬁeld.
Revised: Layer Hierarchy To support learning generalizable physical dynamics, we employ a hierarchicalnetwork structure that allows for a large receptive field.

Original: Specifically, we use an architecture inspired by the parallel processing of the multi-scale feature aggregator of HRNet[43].
Revised: Specifically, we use an architecture inspiredby the parallel processing of the multi-scale feature aggregator of HRNet [30].

Original: It processes sets of extracted features at differentspatial scales with separate branches of the network.
Revised: It processes sets ofextracted features at different spatial scales with separate branches of the network.

Original: Akinto pooling layers, the density of the sampling points isreduced successively in each branch.
Revised: Akin to poolinglayers, the density of the sampling points is reduced successively in each branch.

Original: The structure of thenetwork and its branches are shown in Fig.
Revised: The structure of thenetwork and its branches are shown in Fig.

Original: The main branch of the network works on the full set of sample points P Q 0 which is in this caseequal to the data points P D .
Revised: The main branch of the network works on the full setof sample points P Q 0 which is in this case equal to thedata points P D .

Original: For the secondary branches, a new set of sampling points P Qi isgenerated from the input points P D with a scaling factor s i for the corresponding branch i such thats i ⇡ ⇢ ( P Q 0 ) / ⇢ ( P Qi ) , with ⇢ denoting the density function of the given point set.
Revised: For the secondary branches, a new set ofsampling points P Qi is generated from the input points P Dwith a scaling factor s i for the corresponding branch i suchthat s i ≈ ρ ( P Q 0 ) /ρ ( P Qi ) , with ρ denoting the densityfunction of the given point set.

Original: Additionally, pointsin each set should be spatially distributed as evenly as possible so that the scaling factor remainssimilar across all partial volumes.
Revised: Additionally, points in eachset should be spatially distributed as evenly as possibleso that the scaling factor remains similar across all partialvolumes.

Original: To satisfy condition (10), the point features are resampled back tooriginal sample points P Q 0 before processing them in the ASCC layer.

Original: Generating the sampling points P Qi of a layer is a non-trivial task, for which previous work typicallyemploys farthest-point sampling (FPS) [34].
Revised: Generating the sampling points P Qi of a layer is a non-trivial task, for which previous work typicallyemploys farthest-point sampling (FPS) [23].

Original: While FPS extracts a uniformly distributed subset froma point cloud, its time complexity of O ( NlogN ) is not favorable.
Revised: While FPS extracts a uniformly distributed subset froma point cloud, its time complexity of O ( NlogN ) is not favorable.

Original: We have found that a voxelizationapproach is preferable, which in a similar form is already used in established computer vision methods[56, 42].
Revised: We have found that a voxelization approach is preferable, which in a similar form is already used in established computer vision methods[42, 29].

Original: We choose the centers of the cells of a regular grid, i.e. voxels, as sampling points, asillustrated in Fig. 4.
Revised: We choose the centers of the cells of a regular grid, i.e. voxels, as sampling points, asillustrated in Fig. 4.

Original: The spacing of the grid is determined from the initial particle sampling modiﬁedby the scaling factor s i  voxelization can be performed in O ( N ) , and results  an evenlydistributed set of 
Revised: The spacing of the grid is determined from the initial particle sampling modifiedby the scaling factor s i . The voxelization can be performed in O ( N ) , and results in an evenlydistributed set of sampling points.

Original: Feature Propagation To compute the features f ij for level i and branch j , we use a set of CConvsthat process the features of all branches from the previous pass for each branch.
Revised: Feature Propagation To compute the features F ij for level i and branch j , we use a set of CConvsthat process the features of all branches from the previous pass for each branch.

Original: This results in amulti-level feature aggregation across all branches.
Revised: This results in amulti-level feature aggregation across all branches.

Original: For merging the features, the different featuresare accumulated via summation:
Revised: For merging the features, the different featuresare accumulated via summation:

Original: [Equation]
Revised: [Equation]

Original: (13) where k denotes all branch indices of the previous layer, as also illustrated in Fig.
Revised: [Equation] where k denotes all branch indices of the previous layer, as also illustrated in Fig.

Original: Here, itis important to choose the sampling radius of the convolution operator based on the scale of thecorresponding input branch s k .
Revised: Here, itis important to choose the sampling radius of the convolution operator based on the scale of thecorresponding input branch s k .

Original: This ensures that the convolutions on average always process aneighborhood of similar size.
Revised: This ensures that the convolutions on average always process aneighborhood of similar size.

Original: I.e., for a small input resolution a larger radius is used, and vice versa.
Revised: I.e., for a small input resolution a larger radius is used, and vice versa.

Original: Finally, the accumulated features of the main branch are processed by an ASCC layer.
Revised: Finally, the accumulated features of the main branch are processed by an ASCC layer to produce thedesired output.

Original: 2.4 Training and Long-term Stability
Revised: 2.4 Training and Long-term Stability

Original: Following previous work [23, 48, 37], we use a mean absolute error of the position values betweenprediction and ground truth weighted by the neighbor count as our loss function:
Revised: Following previous work [16, 35, 24], we use a mean absolute error of the position values betweenprediction and ground truth weighted by the neighbor count as our loss function:

Original: [Equation]
Revised: [Equation]

Original: Here, c i denotes the number of neighbors for particle i , and c avg the average neighbor count.
Revised: Here, c i denotes the number of neighbors for particle i , and c avg the average neighbor count.

Original: Wepurposefully formulate the loss to work without additional loss terms, such as physical soft constraints,and instead rely on the conservation of momentum from the ASCC.
Revised: Wepurposefully formulate the loss to work without additional loss terms, such as physical soft constraints,and instead rely on the conservation of momentum from the ASCC.

Original: For temporal stability we use a rollout of T frames at training time.
Revised: For temporal stability we use a rollout of T frames at training time.

Original: That is, for each training iteration,we run the network for T time steps, where the input for the next step is based on the result from theprevious prediction.
Revised: That is, for each training iteration,we run the network for T time steps, where the input for the next step is based on the result from theprevious prediction.

Original: The loss is evaluated and averaged across all T frames with L r = 1 T P Tt [-=
-]
Revised: The loss is evaluated and averaged across all T frames with L r = 1 T (cid:80) Tt [-=
-]=

Original: L ( t ) .
Revised: L ( t ) .

Original: This integrates the temporal behavior in the training and stabilizes it for a medium time range.
Revised: This integrates the temporal behavior in the training and stabilizes it for a medium time range.

Original: While larger T are preferable for improving long-term predictions, they lead to signiﬁcantly increasedmemory requirements for backpropagation.
Revised: While larger T are preferable for improving long-term predictions, they lead to significantly increasedmemory requirements for backpropagation.

Original: To counteract this effect, we prepend the rollout by asequence of W simulated preprocessing steps using the current training state.
Revised: To counteract this effect, we prepend the rollout by asequence of W simulated preprocessing steps using the current training state.

Original: Altogether N = W + Tframes are generated, but irrespective of the number of preprocessing steps, the loss and hencegradient, is only computed for T steps after preprocessing.
Revised: Altogether N = W + Tframes are generated, but irrespective of the number of preprocessing steps, the loss and hencegradient, is only computed for T steps after preprocessing.

Original: This lets the network learn to correctits own errors that occur over longer time spans and improves long-term stability even in complexscenarios without increasing the memory.
Revised: This lets the network learn to correctits own errors that occur over longer time spans and improves long-term stability even in complexscenarios without increasing the memory.

Original: During training we adapt W based on the training progressand the example difﬁculty.
Revised: During training we adapt W based on the training progressand the example difficulty.

Original: We provide the details of the implementation in App.
Revised: We provide the details of the implementation in App.

Original: A.2.
Revised: A.2.

Original:  Results
Revised:  Results

Original: We perform a series of tests to show the physical accuracy, robustness and generalizability ofour method compared to other baseline methods.
Revised: We perform a series of tests to show the physical accuracy, robustness and generalizability ofour method compared to other baseline methods.

Original: We primarily use data from a high-ﬁdelty SPHsolver with adaptive time stepping [1].
Revised: We primarily use data from a high-fidelty SPHsolver with adaptive time stepping [1].

Original: The resulting, two-dimensional dataset "WBC-SPH" consistsof randomly generated obstacle geometries and ﬂuid regions.
Revised: The resulting, two-dimensional dataset "WBC-SPH" consistsof randomly generated obstacle geometries and fluid regions.

Original: Gravity direction and strength areadditionally varied across simulations.
Revised: Gravity direction and strength areadditionally varied across simulations.

Original: In addition to this primary dataset, we also use the MPMbased ﬂuid dataset "WaterRamps" from Sanchez-Gonzalez et al.
Revised: In addition to this primary dataset, we also use the MPMbased fluid dataset "WaterRamps" from Sanchez-Gonzalez et al.

Original: [37], and the three-dimensionalliquid dataset "Liquid3d" from Ummenhofer et al.
Revised: [24], and the three-dimensionalliquid dataset "Liquid3d" from Ummenhofer et al.

Original: [48] for additional evaluations.
Revised: [35] for additional evaluations.

Original: Both consistof randomized ﬂuid regions with constant gravity.
Revised: Both consist ofrandomized fluid regions with constant gravity.

Original: A more detailed description of the datasets isprovided in App.
Revised: A more detailed description of the datasets is providedin App.

Original: A.3.
Revised: A.3.

Original: To quantify the accuracy of the methods, we use several different metricswith respect to the ground truth data from the corresponding dataset.
Revised: To quantify the accuracy of the methods, we use several different metrics with respect tothe ground truth data from the corresponding dataset.

Original: We use the root mean-squarederror (RMSE) for single simulation steps to evaluate short-term accuracy.
Revised: We use the root mean-squared error (RMSE) forsingle simulation steps to evaluate short-term accuracy.

Original: However, over longsequences, it is common for ﬂuid particles to mix chaotically regardless of whether the generalﬂuid behavior resembles the reference, leading to artiﬁcial per-particle errors.

Original: Therefore, the earthmover’s distance (EMD) is used as an assessment of the long-time accuracy over the ﬂuid volumeas a whole [12]: min  : x ! y
Revised: The earth mover’s distance (EMD) is used asa value for the long-time accuracy for the fluid volume as a whole:

Original: P x k x   ( x ) k 2 2 , where x and y are positions from two point sets and 257  denotes a bijective mapping of minimal distances.
Revised: min ϕ : x → y (cid:80) x

Original: EMD matches the particles as probabilitydistributions, such that the global distance error is minimized. The deterministic assignment of theparticles makes EMD agnostic to the ordering of particles, which counteracts the assessment errorscaused by mixing. Intuitively, EMD evaluates the similarity of density-weighted volumes insteadof the individual particles. Thus, it quantiﬁes the error of the simulated conﬁguration relative to the
Revised: ∥ x − ϕ ( x ) ∥ 22 , where

Original: [Figure]
Revised: [Figure]

Original: Momentum

Original: Max.
Revised: Max.

Original: Dens.
Revised: Dens.

Original: EMD
Revised: [Figure]

Original: Average
Revised: Average

Original: Base
Revised: Base

Original: ASCC
Revised: ASCC

Original: Multi Scale
Revised: Multi Scale

Original: Voxelize
Revised: Voxelize

Original: Preprocess
Revised: Preprocess

Original: Ours reference, neglecting small-scale changes in particle ordering.
Revised: Ours x and y are positions from two point sets and ϕ denotes a bijective mapping of minimal distances. This is especially important to reliably evaluate long simulation sequences.

Original: In addition, we evaluate the differencebetween the velocity distributions to evaluate the accuracy of the velocity content of a simulation.
Revised: In addition, we evaluatethe difference between the velocity distributions to evaluate the accuracy of the velocity content ofa simulation.

Original: Wecompute a histogram of the magnitude of the velocities and compute its Jensen-Shannon divergenceto the histogram from ground truth data: JSD ( V x k V y )
Revised: We compute a histogram of the magnitude of the velocities and compute its JensenShannon divergence to the histogram from ground truth data: JSD ( V x ∥ V y )

Original: =
Revised: =

Original: 12 ( D KL ( V x k V )
Revised: 12 ( D KL ( V x ∥ V )

Original: +D KL ( V y k V )) , where
Revised: +

Original:  V = 12 ( V x + V y ) and D KL is the Kullback-Leibler divergence.
Revised: D KL ( V y ∥ V )) , where V = 12 ( V x + V y ) and D KL is the Kullback-Leibler divergence.

Original: We also evaluate deviations in termsof maximum density, | 1  (max i ⇢ ( x i ) / max i ⇢ ( y i )) | , with ⇢ being a function to compute the densityof particle i .
Revised: We alsoevaluate deviations in terms of maximum density, | 1 − (max i ρ ( x i ) / max i ρ ( y i )) | , with ρ being afunction to compute the density of particle i .

Original: This evaluation correlates with the compressibility of the ﬂuid, and gives an indicationof the stability of a method.
Revised: This evaluation correlates with the compressibility ofthe fluid, and gives an indication of the stability of a method.

Original: Lastly, we also compute the linear momentum change as [Figure] m 
Revised: Lastly, we also compute the linearmomentum as (cid:80) m i a

Original: [Equation] a i [Equation] F ext ,270 where a i is the acceleration and m i the mass of a particle, and F ext denotes external forces.
Revised: i − F ext , where a i is the acceleration and m i the mass of a particle, and F extdenotes external forces.

Original: [Figure]
Revised: [Figure]

Revised: FreeFall

Original: Standing Liquid
Revised: Standing Liquid

Original: We ﬁrst consider onedimensional simulations of a hydrostatic column of liquid.
Revised: We first consider onedimensional simulations of a hydrostatic column of liquid.

Original: The training data consists of adifferent number of ﬂuid particles which, underthe effect of gravity, rest on a ﬁxed boundary atthe bottom.
Revised: The training data consists of adifferent number of fluid particles which, underthe effect of gravity, rest on a fixed boundary atthe bottom.

Original: The reference stems from a solverwith implicit time stepping [16] that can correctglobal effects instantaneously.
Revised: The reference stems from a solverwith implicit time stepping [10] that can correctglobal effects instantaneously.

Original: To test the generalization, we also evaluate the methods for a freefall scene.
Revised: To test the generalization, we also evaluate the methods for a freefall scene.

Original: Here we generate a certain numberof ﬂuid particles in free fall,which substantiallydeviates from the behavior of the training data.
Revised: Here we generate a of fluid particles in free fall, from the behavior of the training data

Original: We trained two networks with the architecturefrom Sec. 2.3, once with the proposed ASCClayer (ASCC) and once with a regular CConvlayer instead (No Sym).
Revised: We trained two networks with the architecturefrom Sec. 2.3, once with the proposed ASCClayer (ASCC) and once with a regular CConvlayer instead (No Sym).

Original: Despite the seemingsimplicity of the test case, even small inaccuracies can lead to unstable behavior over time.
Revised: Despite the seemingsimplicity of the test case, even small inaccuracies can lead to unstable behavior over time.

Original: Withoutthe antisymmetric constraint, a network can simply overﬁt to the gravity, and negate it.
Revised: Without the antisymmetric constraint, a network can simply overfit to the gravity, and negate it.

Original: This reducesthe problem to a local per particle problem.
Revised: Thisreduces the problem to a local per particle problem.

Original: With ASCC, on the other hand, the network relies onparticle interactions to solve the problem.
Revised: With ASCC, on the other hand, the networkrelies on particle interactions to solve the problem.

Original: This makes it necessary to determine a global context,and infer a pressure-like counter force to stabilize the ﬂuid.
Revised: This makes it necessary to determine a globalcontext, and infer a pressure-like counter force to stabilize the fluid.

Original: We compare the performance of bothlearned versions with the solution obtained by an explicit SPH solver with a small time step [32].
Revised: We compare the performanceof both learned versions with the solution obtained by an explicit SPH solver with a small timestep [21].

Original: Fornew column heights, Fig.
Revised: For new column heights, Fig.

Original: 5 shows that both learned versions quickly compensate for compressioneffects in the liquid and yield very small errors, while the SPH solution oscillates in place.
Revised: 5 shows that both learned versions quickly compensate forcompression effects in the liquid and yield very small errors, while the SPH solution oscillates inplace.

Original: Whereas SPH handles the free fall case on the right reasonably well with an error of 19 . 68 , the networkwithout antisymmetry largely fails, and produces erroneous motions with an error of 46 .
Revised: 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 of46 .

Original: Theantisymmetric version, having learned a physically consistent response, yields an error that improvesover SPH by a factor of 3 . 3 .
Revised: The antisymmetric version, having learned a physically consistent response, yields an errorthat improves over SPH by a factor of 3 . 3 .

Original: Ablation Study We evaluate relevance of individual features introduced for our method with theablation study using the WBC-SPH dataset.
Revised: Ablation Study We evaluate relevance of individual features introduced for our method with theablation study using the WBC-SPH dataset.

Original: We start with a base model ( Base ) without an ASCC layerand gradually add relevant features.
Revised: We start with a base model ( Base ) without an ASCC layerand gradually add relevant features.

Original: While this model fares very well for single steps, as shown in
Revised: While this model fares very well for single steps, as shown in

Revised: [Figure]

Revised: [Figure]

Original: Fig.
Revised: Fig.

Original: 6 it performs badly with respect to all other evaluations.
Revised: 6 it performs badly with respect to all other evaluations.

Original: Here, the graphs show the performancerelative to the full method, i.e. L Ours /L , with L denoting the loss of a variant under consideration.
Revised: Here, the graphs show the performancerelative to the full method, i.e. L Ours /L , with L denoting the loss of a variant under consideration.

Original: For the second version we add the antisymmetrical layer ( ASCC ), which ensures the conservation ofmomentum and signiﬁcantly improves the motion in terms of velocity distribution.Here the score of GT a c d [Table]
Revised: For the second version we add the antisymmetrical layer ( ASCC ), which ensures the conservation ofmomentum and significantly improves the motion in terms of velocity distribution.

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Table]

Original: [Figure]

Original: for Base corresponds to a momentum error of 0 .

Original: 095 , all following variants having a score of 1 .

Original: 0with zero error.

Original: Next, we evaluate the relevance of the multi-scale processing.
Revised: Next, we evaluatethe relevance of the multi-scale processing.

Original: We tested two differentvariants, one with FPS ( Multi Scale ) and one with the voxelization ( Voxelize ).
Revised: We tested two different variants, one with FPS ( Multi Scale ) and one with the voxelization ( Voxelize ).

Original: The performanceis very similar for both, with the latter variant having clear advantages in terms of resource usage.
Revised: The performance is very similar for both, with thelatter variant having clear advantages in terms of resource usage.

Original: While improvements multi-scale aspect is not obvious, but combined with the preprocessing themulti-scale handling starts to more clearly pay off when introducing the preprocessing for temporalcoherence training in ( Preprocess ).
Revised: While improvements multi-scaleaspect is not obvious, but combined with the preprocessing the multi-scale handling starts to moreclearly pay off when introducing the preprocessing for temporal coherence training in ( Preprocess ).

Original: The improved long-term stability and accuracy is clearly visiblein the maximum density and EMD metrics.
Revised: The improved long-term stability and accuracy is clearly visible in the maximum density and EMDmetrics.

Original: Lastly, our full method ( Ours ) includes additional inputrotation normalization and a more advanced boundary processing step, as explained in more detail in App. A.1.
Revised: Lastly, our full method ( Ours ) includes additional input rotation normalization and a moreadvanced boundary processing step, as explained in more detail in App. A.1.

Original: This leads to further slight improvements and yields the ﬁnal algorithm which we willcompare to previous work in the following sections.
Revised: This leads to furtherslight improvements and yields the final algorithm which we will compare to previous work in thefollowing sections.

Original:  Work In the following, we compare our method with several established methods as baselines: an adapted variant of PointNet [33], the CConv network [48] and GNS[37].
Revised: Comparisons with Previous Work In the following, we compare our method with several established methods as baselines: an adapted variant of PointNet [22], the CConv network [35] and GNS[24].

Original: More information about the employed networks and the training procedures are given in App.
Revised: More information about the employed networks and the training procedures are given in App.

Original: A.2.
Revised: A.2.

Original: We train and evaluate networks with our WBC-SPH dataset.
Revised: For fairness, we train and evaluate networks with our WBC-SPH dataset, as well as with the

Original: For fairness, we additionally evaluatethe different approaches with their respective data sets, i.e., with the WaterRamps dataset of GNS,and with the Liquid3d dataset of CConv.
Revised: WaterRamps dataset of GNS.

Original: When trained on the WBC-SPH dataset, which consists of sequences with 3200 time steps, neither
Revised: When trained on the WBC-SPH dataset, which consists of sequences with 3200 time steps, neither

Original: CConv nor PointNet are able to generate stable results.
Revised: CConv nor PointNet are able to generate stable results.

Original: While the short-term RMSE accuracy isgood, both networks fail to stabilize the challenging dynamics of our test sequences.
Revised: While the short-term RMSE accuracy isgood, both networks fail to stabilize the challenging dynamics of our test sequences.

Original: Examples areshown in Fig.
Revised: Examples areshown in Fig.

Original: The outputs from GNS are stable, but the network seems to maintain stability bysmoothing and damping the dynamics.
Revised: The outputs from GNS are stable, but the network seems to maintain stability bysmoothing and damping the dynamics.

Original: Also, despite receiving multiple frames over time, the GNSmodel has difﬁculties to predict the dynamics for changing gravity directions.
Revised: Also, despite receiving multiple frames over time, the GNSmodel has difficulties to predict the dynamics for changing gravity directions.

Original: For this scenario, theantisymmetric ASCC network clearly outperforms GNS and the other variants in quantitative, as wellas qualitative terms.
Revised: For this scenario, theantisymmetric ASCC network clearly outperforms GNS and the other variants in quantitative, as wellas qualitative terms.

Original: In particular, its EMD error is 7 .
Revised: In particular, its EMD error is 7 .

Original: 22 times smaller than the closest competitor.
Revised: 22 times smaller than the closest competitor.

Original: Forfairness of the comparison, we also retrain our model and the CConv model with the WaterRampsdataset.
Revised: Forfairness of the comparison, we also retrain our model and the CConv model with the WaterRampsdataset.

Original: While the accuracy of GNS with this data is very high, our method nonetheless yields acompetitive accuracy with an increase of only 13% in terms of EMD.
Revised: While the accuracy of GNS with this data is very high, our method nonetheless yields acompetitive accuracy with an increase of only 13% in terms of EMD.

Original: In comparison, CConv resultsin a 3 .
Revised: In comparison, CConv resultsin a 3 .

Original: 61 times larger EMD.
Revised: 61 times larger EMD.

Original: Over short periods the dynamics of GNS behave similarly to ours.
Revised: Over short periods the dynamics of GNS behave similarly to ours.

Original: Forlonger periods of more than 30 frames, the results diverge, as shown in Fig. 8.
Revised: Forlonger periods of more than 30 frames, the results diverge, as shown in Fig. 8.

Original: Most importantly, ournetwork yields a high accuracy while requiring only a fraction of the computational resources
Revised: Most importantly, ournetwork yields a high accuracy while requiring only a fraction of the computational resources

Original: that
Revised: that

Original: GNS requires: training is 4 .
Revised: GNS requires: training is 4 .

Original: 52 time faster, and the ASCC network only requires 29% of the weightsof GNS.
Revised: 52 time faster, and the ASCC network only requires 29% of the weights

Original: While CConv performs best in terms of resources, it does so at the expense

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Revised: of GNS

Revised: While CConv performs best in terms of resources, it does so at the expense of inferenceaccuracy: its EMD is three times larger than the one obtained with our method.

Original: Generalization To assess the generalizing capabilities of our network and GNS trained with the
Revised: Generalization To assess the generalizing capabilities of our network and GNS trained with the

Original: WaterRamps data, we perform tests with droplets colliding under the inﬂuence of varying gravity.
Revised: WaterRamps data, we perform tests with droplets colliding under the influence of varying gravity.

Original: This case allows for an accurate evaluation in terms of the center of mass trajectory of the liquidvolume.
Revised: This case allows for an accurate evaluation in terms of the center of mass trajectory of the liquidvolume.

Original: Considering a case with default gravity, the ASCC network already yields an EMD errorreduced by 57% .
Revised: Considering a case with default gravity, the ASCC network already yields an EMD errorreduced by 57% .

Original: When reducing the gravity to zero, the GNS trajectory noticeably deviates from theground truth trajectory and yields a 195% larger error than our method.
Revised: When reducing the gravity to zero, the GNS trajectory noticeably deviates from theground truth trajectory and yields a 195% larger error than our method.

Original: The GNS has encoded theinﬂuence of gravity in its learned representation, and hence has difﬁculties to transfer the dynamicsto the changed physics environment.3D
Revised: The GNS has encoded theinfluence of gravity in its learned representation, and hence has difficulties to transfer the dynamicsto the changed physics environment.3D

Original: Simulations Finally, we test our network with 3D data using the Liquid3d dataset.
Revised: Simulations Finally, we test our network with 3D data using the Liquid3d dataset.

Original: Unlike theother datasets, its simulations use a signiﬁcantly larger time step ( 8 times larger than WBC-SPH ).
Revised: Unlike theother datasets, its simulations use a significantly larger time step ( 8 times larger than WBC-SPH ).

Original: Incombination with the higher dimensionality this makes it substantially more difﬁcult to obtain stablesolutions.
Revised: Incombination with the higher dimensionality this makes it substantially more difficult to obtain stablesolutions.

Original: Again, our method achieves comparable accuracy, with much better generalization.
Revised: Again, our method achieves comparable accuracy, with much better generalization.

Original: Similarto the previous examples, it becomes apparent that CConv tends to overly dampen the acceleration ofgravity, which, unlike with our method, leads to discrepancies in terms of the motion compared tothe ground truth, as shown in Fig.
Revised: Similarto the previous examples, it becomes apparent that CConv tends to overly dampen the acceleration ofgravity, which, unlike with our method, leads to discrepancies in terms of the motion compared to theground truth, as shown in Fig.

Original: To show the scalability and generalizability of our method,we have applied it to a large-scale scene with 5 ⇥ more particles than in the training set.

Original: Qualitativeresults can be found in App.

Original: A.4 (Fig. 21) and the supplementary video.

Original: Performance We have also evaluated the computational performance of our method, with detailedmeasurements given in App.
Revised: Performance We have also evaluated the computational performance of our method, with detailedmeasurements given in App.

Original: A.4.
Revised: A.4.

Original: E.g., for the 2D data WBC-SPH our method requires 67 . 25 ms perframe on average.
Revised: E.g., for the 2D data WBC-SPH our method requires 195 ms perframe on average.

Original: For comparison, the corresponding reference solver on avg. requires 10925 ms perframe.
Revised: For comparison, the corresponding reference solver on avg. requires 10925 ms perframe.

Original: Thus, our network achieves a speed-up of 162 over the reference solver.
Revised: Thus, our network achieves a speed-up of 56 over the reference solver.

Original:  Conclusion
Revised:  Conclusion

Original: We have presented a method to guarantee the conservation of momentum in neural networks, anddemonstrated the importance of this concept for learning the complex dynamics of liquids.
Revised: We have presented a method to guarantee the conservation of momentum in neural networks, anddemonstrated the importance of this concept for learning the complex dynamics of liquids.

Original: Inparticular, our approach outperforms state-of-the-art methods in terms of accuracy and generalization.
Revised: Inparticular, our approach outperforms state-of-the-art methods in terms of accuracy and generalization.

Original: Nonetheless, the limits of symmetric constraints are far from exhausted.
Revised: Nonetheless, the limits of symmetric constraints are far from exhausted.

Original: E.g., while our network isantisymmetric, permutation invariant and translation invariant,  an interesting avenue for future workwill be to investigate more generic forms of E ( n ) equivariance [8, 50, 38].
Revised: E.g., while our network isantisymmetric, permutation invariant and translation invariant, it will be an interesting avenue forfuture work to investigate more general forms of E ( n ) equivariance [25, 37]. Checklist For all authors...

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