Neural Fluid Simulation on Geometric Surfaces
Authors: Haoxiang Wang
Tao Yu
Hui Qiao
Qionghai Dai
Venue: ICLR 2025 Poster (2025)
Keywords: Fluid simulation
Implicit Neural Representation
Exterior Calculus
Primary area: applications to computer vision, audio, language, and other modalities
TL;DR: An implicit neural representation based physical simulator for fluid dynamics on surfaces

Abstract: Incompressible fluid on the surface is an interesting research area in the fluid simulation, which is the fundamental building block in visual effects, design of liquid crystal films, scientific analyses of atmospheric and oceanic phenomena, etc. The task brings two key challenges: the extension of the physical laws on 3D surfaces and the preservation of the energy and volume. Traditional methods rely on grids or meshes for spatial discretization, which leads to high memory consumption and a lack of robustness and adaptivity for various mesh qualities and representations. Many implicit representations based simulators like INSR are proposed for the storage efficiency and continuity, but they face challenges in the surface simulation and the energy dissipation. We propose a neural physical simulation framework on the surface with the implicit neural representation. Our method constructs a parameterized vector field with the exterior calculus and Closest Point Method on the surfaces, which guarantees the divergence-free property and enables the simulation on different surface representations (e.g. implicit neural represented surfaces). We further adopt a corresponding covariant derivative based advection process for surface flow dynamics and energy preservation. Our method shows higher accuracy, flexibility and memory-efficiency in the simulations of various surfaces with low energy dissipation. Numerical studies also highlight the potential of our framework across different practical applications such as vorticity shape generation and vector field Helmholtz decomposition.

Published as a conference paper at ICLR 2025
NEURAL FLUID SIMULATION ON GEOMETRIC SUR- FACES
Haoxiang Wang' Tao ²u²- Hui Qiao Qionghai Dail
'Department of Automation, Tsinghua University, ²BNRist, Tsinghua University, whx228mails, tsinghua. edu. en, ytrock@12 66.. com, {qiachui, da daiqionghai @etsinghuaa eew.huuuaedu...mm edu.cn .n
ABSTRACT
Incompressible fluid on the surface is an interesting research area the fluid sim- ulation, which is the fundamental building block in visual effects, design of liquid crystal films, scientific analyses of atmospheric and oceanic phenomena, etc. The task brings two key challenges: the extension of the physical laws on 3D surfaces and the preservation of the energy and volume. Traditional methods rely on grids meshes for spatial discretization, which leads to high memory consumption and a lack of robustness and adaptivity for various mesh qualities and representations. Many implicit representations based simulators like INSR are proposed for the storage efficiency and continuity, but they face challenges in the surface simula- tion and the energy dissipation. We propose neural physical simulation frame- work on the surface with the implicit neural representation. Our method constructs a parameterized vector field with the exterior calculus and Closest Point Method on the surfaces, which guarantees the divergence-free property and enables the simulation on different surface representations (c.g. implicit neural represented surfaces). We further adopt a corresponding covariant derivative based advection process for surface flow dynamics and energy preservation. Our method shows higher accuracy, flexibility and memory-efficiency in the simulations of various surfaces with low energy dissipation. Numerical studies also highlight the poten- tial of our framework across different practical applications such as vorticity shape generation and vector field Helmholtz. decomposition.
INTRODUCTION
Fluids are fascinating but complex simulate, with applications from aerodynamics and hydro- dynamics to special effects in computer animation. Flow on the surface is a challenging problem
while the practical usages are essential on the visual effects with foam or bubble (Da et al. 2015
Deng al. 2022), studies of liquid crystal films (Crowdy & Marshall 2005 Turner et al.) 2010),
atmosphere/oceen occan evolution (Miller et al. 1992. Nüler 2001) and fluid-solic interaction in robotics (Ruan et al., 2021). The incompressible Euler flow model serves
valuable simplification of real-world fluid dynamics. This model, characterized by a vector field w(x, t) representing velocity,
along with pressure p(x,,) t) and density Py, adheres to the following equations:
Ov Ps( 01
+ Vv) -Vp
(la)
V
0,
(lb)
where & in a surface S.
Two main challenges in solving Eqs. D puzzle the researchers: one is enforcing the governing equation on the surface and the other is developing efficient approaches for the time integration (advection) and the divergence-free constraint in Eq. Tp. which is critical to ensure the conservation of fluid volume and energy.
Classical approaches often utilize grids or meshes on surfaces and reduce the problem to 2D sce- narios. However, they encounter significant challenges in the geometry and the differential operator
"Correpooniigg authors.

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computation. Accurate calculation surfaces relies high mesh/grid quality, leading lim- ited robustness and flexibility on different geometry representations. Moreover, the introduction of mesh grid through spatial discretization also hinders simulation in continuous spatio- temporal domain owing to the limited memory usage. Finally, the traditional methods need to conduct advec- tion and pressure projection for the divergence-free
field, causing the energy dissipation problem. While many alternatives are proposed to solve the problems (Qu et al. 2019 Elcott et al.] 2007b] Nabizadeh al., 2022 Yin et al., 2023). they often come with implementation complexities and lack adaptability to different geometries.
As promising alternative, simulations based on the neural implicit representations are proposed in recent years Stichter-Powell et al. [2022] Chen et al. (023). Unlike other data-driven simulation methods (Morimoto et al., 2021 Pfaff et al., 2020 that have limited generalization ability, these methods leverage neural networks to parameterize spatial functions and support the simulation on the continuous domain with limited storage. However, existing methods (Raissi et al.) 20199] Chen et al. 2023) can not guarantee the divergence-free property and suffer from the advection error, which leads to the energy dissipation problem. Furthermore, extending these methods to surfaces presents additional challenges for practitioners.
To tackle the challenges, we propose a neural flow on surfaces method based on the neural implicit representation. Neural implicit representation keeps high memory efficiency and supports robust and accurate differential operator computation for the continuous simulation across various geometry representations. Our method leverages construction (Ruuth & Merriman 2008) and differential forms, and automatically satisfies the divergence-fiee constraint, assisting
the surface with Closest Point Method
enforce the constitutive laws on the surface. To mitigate the challenges of energy dissipation encountered in both classical and advanced methods, we adopt a covariant- derivative based advection to enforce the dynamics of the incompressible fluid. By integrating this process with our divergence-fre free field construction, our framework eliminates the need for velocity advection and pressure projection, thus minimizing energy dissipation. Furthermore, our framework is versatile and applicable to various tasks such as generation and field decomposition, offering an end-end solution that capitalizes on the advantages of neural representation. In summary, we make the following contributions:
We present a novel neural physical simulator for surface flow, named NFFS (Neural Func- tional Flow Surface), leveraging the Closest Point Method and exterior calculus in the neural implicit representation. Our approach ensures divergence- freee properties and adapt- ability across various geometric surface representations. Notably, it is the first study to present simulation results of incompressible fluid flow on implicitly neural-represeened sur- faces (Sitzmann al. 2020) with a guarantee of divergence-free behavior.
We design a complementary advection process based on the covariant derivatives for fluid dynamics with low energy dissipation.
We conduct comprehensive numerical studies to verify the correctness, energy preserva- tion, memory efficiency and geometry adaptivity for our proposed framework. Benefitting from the advantages of compact representation, also highlighted in Chen et al.) (2023), our results show that our method achieves approximately 15 times higher accuracy than other methods with the same storage cost and provides 5 times memory savings compared to the classic method while accurately describing the phenomenon of fluid dynamics on the surface. Additionally, we demonstrate the conditioning ability of our simulator through an end-to-end generation task and apply it to a real-world velocity field decomposition task.
2 RELATED WORKS
Flow on two-dimensional surface. The main stream of the fluid simulation consists of the La- grangian methods like Smoothed Particle Hydrodynamics (SPH) (Gingold & Monaghan 1977
Monaghan| 1992) and the Euler methods such as stable fluid with Mark-and-Celll grid (Stam) 1999).
On surfaces, particle-based methods have been studied extensively in the past decade. The pri-
mary focus is the differential operator with SPH-style. Many approximators (Petronetto et al. 2013
Belkin & Niyogi 2008 Cheung et al. 2015 Nealen 2004) are proposed and adopted in fluid dy- namics (Auer & Westermann, Leung et al. [201] Wang et al. 22200] Tao et al., 2022 Suchde
2021). However, the particle-basee methods suffer from high computation cost for the comparable
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Figure I: The paradigm of our paper. Left: the pipeline of our proposed methods. Our method uti- lizes the implicit neural representation to construct a divergence free field (showed in Sec. [.1).. We employ exterior calculus and Closest Point Method to construct the surface velocity field vo and vor- ticity field w (showed in Sec. (.2). Subsequently, we adopt the covariant derivatives based advection to calculate the flow in each discretized time iteratively (showed in Sec. 4.1 Right: the potential applications of our proposed method: (a) Our method supports simulation on different surface rep- resentation, like analytic surfaces (showed in Sec. [.T]]. explicitly represented mesh surface (showed in Sec. 5.2) and implicitly represented surface (showed in Sec. 5.2). (b) Our method also enjoys the ability of conditioning and generation benefitting from our network architecture (showed in Sec 4.2 and 5.3 (c) We demonstrate the effectiveness our method for the Helmholtz decomposition and
analyze the potential usage for scientific research (showed in Sec. 5.4
accuracy. Dealing with differential forms on surfaces for divergence-fre projection also remains challenging, particularly without correspondence between particles.
We next focus our discussions on Euler methods, which can be categorized into velocity- based methods and vorticity-basee methods. Velocity-based methods, following Stam (1999). utilize the global or local surface parameterizations (Lui et al.] 22055] [eeemaan et al., 2009: Hill & Henderson 2016 Yang et al., 2019), which might introduce undesired distortion. Methods like Stam (2003)
restrict the problem to the subdivision surfaces to mitigate this issue. Shi & Yu (2004) tackles the
problem by directly simulating general triangle meshes but entails explicit complex computation of flow lines. More recently. [Bhattacharya et al. (2019) extends these to unstructured quadrilateral surface meshes. However, these methods require the advection and divergence-
projection, lead- ing to energy dissipation (Nabizadeh et al. 2022). While there are approaches to address the energy problem such as Muullen et al. (2009); ; Pavlov al. (201T) and recent Qu al. (2019): Deng et al. 2023), they do not primarily focus on the surface dynamics and often entail additional complex computations (such as accurate characteristic mapping (Wiggert& & Wylie 1976)).
For the stream of vorticity-based approaches, methods are grounded in differential forms and ex- terior calculus on surfaces, circumventing surface parameterization and naturally enforcing the divergence-fiee free property. One of the pioneering methods in this domain was proposed by |Ecoot jet al. (2007b). While their method preserves circulation (vorticity), it suffers from the issues of nu- merical instability. Another notable vorticity-based approach is presented by De Witt et al. (0012). leveraging eigenfunctions of the Laplacian operator to reduce the computational cost of the Pois- son solver when computing velocity from vorticity. This approach has been further extended in works such as CCu et al. (2018) 2021) for spectral-based simulations. However, such methods can be computationally expensive and require numerous eigenvectors for flows with high spatial fre- quencies. A method closely related to ours is Functional Fluids on Surfaces (Azencot et al., 2014)) which employs the Discrete Exterior Calculus on the surface to derive the vorticity and advect with covariant derivatives. This approach achieves convenient computation on surfaces with the energy preservation. Our method builds upon this by employing continuous exterior calculus to generate divergence- frree free fields and vorticity functions on surfaces. The theoretical accuracy of our method higher with a sufficient number of surface samples. Moreover, we incorporate the neural implicit representation to alleviate the high memory burden associated with high smoothness and accuracy requirements.
Physical Simulation based on Neural Network. Neural physical simulation can be divided into two main streams. The first one is the data-driven simulation. This type of methods often aims at solving simulation problems based more on data but less the governing equation. They often adopt the training data from the classical solvers or the real world observation to make the neural network leam the physical rules and generalize it to other scenarios. Some convolution network approaches (Morimoto et al. 202T) and designed U-Net approaches (Lu et al. 2019) show higher efficiency than the classical solvers. Neural operator approach (Li et al. 2020 makes full use of
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the Fourier layer and becomes important milestone for the type of methods. Recently, a newly proposed Lagrangian Flow Networks (Torres et al. 2023) embeds the idea of the characteristic map- ping and design data-driven PDE solver. However, these methods may struggle to generalize well to different initial/boundary conditions, material parameters, or geometries. The training data ac- quisition and time-consuming training processes also block their wide applications. Another stream of the line is to embed the governing equations into the network. For this type. one representative direction we have mention is Physical-Infone Neural Network (PINN) (Raissi et al. 2019). The method designs the physical loss term according to the governing equation and the neural network trained extract the features of the spatiotemporal correlation and fit the target field via the phys- ical loss. However, the ways to directly force the neural network to fit all the physical rules make training process difficult. The training will cost a very long time and often can not achieve the required accuracy. To fulfill this issue, the implicit neural representation Chen et al, (2023) is also introduced to better describe the spatiotemporal dependencies and reduce the burden of network training. Nevertheless, this method enforces physical laws via operator splitting manner, leading to the energy dissipation problem. Kim et al. (2019); Rao et al. (2020); |Rchter--owwer et al. (2022) propose divergence-free neural field construction approaches and optimize the advection process. These methods, while effective in guaranteeing the divergence-free property, face challenges in di- rect application surface flows and encounter difficulties in optimization processes.
Hence, in this work, we propose a novel framework that constructs a neural-represented divergence- free vector field on surfaces to embed the efficient spatial representation with the physical prior. We also design the corresponding covariant derivative based advection process for the fluid dynamics computation. Our method can achieve a memory-cfficiincyy accurate and energy-preseee peeggy-preserving simula- tion on different surfaces representation, even for the implicit neural represented surfaces. We show the paradigm of our paper in Fig. !
3 NEURAL FLOW ON SURFACES
In this section, we present a framework designed to enforce the divergence- characteristics for vector fields on surfaces. We first present the general philosophy construct the divergence- freee field then we explain how we apply it to the different surface representations, especially in implicit neural represented surfaces.
3.1 CONSTRUCTION OF THE DIVERGENCE- FREE VECTOR FIELD
We adopt the terms of differential forms to derive the divergence- free vector field. More preliminar- ies are shown Do Carmo (1998) and Appendix A. in Bocchter-Powelle et al. (022). We first discuss the divergence-free free vector field on a Riemannian manifold M. (e.g. R*).
Let (MM) as the space of formmss. The operator d: A (M) - A A (M) is the exterior derivative while *: A (M)
(M**(M), where is the dimension of M, denotes the Hodge star operator mapping each k-form to an (n fo-form. Our vector field can be represented as 1-form, v Sz-rudd;;
With the notation of the differential forms and the computation tools of the differential manifold, we can define the divergence div(v) *d v. We can observe that div(u) can be expressed with differential k-forms, but after computation it reduce to a 0-form, resulting in a scalar function.
A fundamental property of the exterior calculus need to be proposed that for arbitrary (n 2)- form E (M)) we have
d d(dp) 0,
(2)
then it follows that
1 zdu
(3)
divergence-free since ** yields a sign function. Consequently, our objective is to construct a parametric (n 2)-form μ that enforces a divergence-free v. We can construct a network param- eterization to construct f and derive the required velocity field a with the divergence eeeerennce-free property for the incompressibilityy via Eq. 3 Therefore, the main issue comes to the computation of the op- erator * and on our provided surface S. We calculate it by Closest Point Method in the following subsection.
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3.2 CONSTRUCTION OF THE NEURAL FLOW ON SURFACES
On the surfaces S embedded in Rd, the analysis differs. To facilitate a more convenient analysis of differential forms on the surface without specific surface parameterization, we consider studying S in R2 instead of R2. The Closest Point Method (CPM) (Ruuth & Merriman 2008 Li et al. 2023a) serves as a tool transform differential forms on surfaces into ones defined in R using the closest point on the surface. We define the inclusion map j : S + N C. R2, where N C R is a neighborhood of jS) The closest point function op : + S takess a point in the neighborhood and returns the closest point on the surface. Then we can define the pullback operator j AA (N) - (S) and cp" (S) + (N) to map the tangential vector the neighborhood and on the surfaces. The endomorphism (j o cp)" cp"""" (N) (N) replaces neighbor k-form by its extension from its value at the surface j(S).
With the Closest Point Method, we construct the divergence-free field on the surfaces S in the following theorem.
Theorem 3.1. Given a parameterized scalar function (stream function) S - R, one can construct divergence-free v S S - R2, for x E S:
v(x) j ((V (cp'o j(x)) x n(x)),
(4)
where n : S + R represents the normal of the surfaces, and the corresponding vorticity function S + R can be defined as
w = (V x v(x)) n(x).
(5)
Equation 4 defines stream function and employs the gradient operator to transform into a vector field. The operator cp" pulls the vector field back into the ambient space, while and j* ensure that the field lies on the tangent plane and is restricted to the surface. The vorticity can be interpreted as having rotation axis aligned with the surface normal, it is evaluated directly on the surface, making it a scalar field. To provide a clearer understanding of the computation described in Theorem we include illustration in Fig. 2
3.1
Proof sketch: Our goal is to construct a divergence-free field. We adopt the form in Sec. 3.1 +dw the basis. Then calculate the value of this differential form for a surface function. We utilize the Closest Point Method and pullback the surface field v into the ambient space. We extend the differential form with surfaces normals to R by the pullback and derive the parameterization on R3 that preserves the divergence- free. Moreover, the pullback utilizes the closest point and shares the property that the closest point of the surface point is the point itself. Therefore, we can simply constrain this parameterization for v on the surface and derive our required surface field that satisfies the divergence- free property on the surface. We can also derive vorticity function as *dv through the similar process.
P(rp"))
Figure 2: Illustration for divergence-free field.
The construction of the divergence-fres field in Eq. 4 related to the term "surface curl" in the electromagnetic's ' Helmholtz decomposition (Scharstein 1991). However, our theorem provides more formal formulation indicating why it is divergence- free from the perspective of the Closest Point Method and adopts it as a neural parametric function for surface vector field dynamics. Building on our analysis, we can further derive scalar vorticity function on the surface support the fluid dynamics simulation.
Remark:
Following the conclusion of Theorem 3.1 we can construct our neural flow (a parametric vector field based on a and n) for different representations of the surfaces. For the explicit representations like sphere, plane (analytical), and mesh (discretized), our neural flow can be constructed with sam- pling on the surfaces and computing/queryl our construction lies in its applicability to implicit representations, particularly the implicit neural representation (INR), such as DeepSDF (Park et al. (201)) and siren (Sitzmann et al. 2020). These methods take x E R as input and return the sign distance function s(x). In the application on this
the normal. Moreover, the most significant aspect of
Herre we follow the similar discussions as Richter-Powell et al. (2022) and omit the case for non-zero homology for clear and concise in theoretical analysis. We actually can address the issue as we state in the following sections.
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type of surface, points S can be simply sampled by s(x) Va(x)
[ leveraging the character-
istics of the sign distance function, while the normal on the surface can be computed using Vs(x). Then, for the sampled points on the surface, can create parametric form o (0,z) and finalize the divergence-free construction with the normals following Eq. 4 The function can be also derived
via Eq.
Our construction of v directly enables tasks like Hodge-Helmhol decomposition (extracting the divergence-free component given vector field). It can be simply achieved with our construction with the mean square error loss for v. In addition, the construction of w can be utilized to conduct advection serving for the vortices dynamics in fluid simulation, as we will introduce subsequently.
4 APPLICATIONS WITH NEURAL FLOW ON SURFACES
In this section, we put our construction of neural flow on surfaces into practice. We first discuss the advection for the simulation of the real fluid dynamics. Then we delve into applications involving the conditioning.
4.1 ADVECTION NEURAL FLOW ON SURFACES
Our vector field inherently satisfies the incompressibility condition with the divergence-free con- struction, eliminating the need for pressure projection to enforce the constraint and associated errors. However, our proposed field, while possessing this advantageous property, introduces new challenge: how can we advect this neural field over time to adhere to physical laws? The incompressible Euler equation can be written with the vortex form DDvidson 2015):
Ow: = (w; Vlwi, 0t
(6)
Po(al
where v, is the divergence- velocity field and the vorticity field.
Mol
(o(((
Remark: A direct sequence of Eq. 6 is that wi is transported in the same manner as fluid particles. This implies that we can advect/diffuse wi similarly to scalar property carried by the particles, such tem- perature.
Figure 3: Illustration for covariant derivative.
For the advection process, instead of the semi Lagrangagnn scheme (Staniforth & Côté 1991) that requires the discretization, we opt for functional way fully enjoy the continuity of our neural field construction (Azencot al. [014)) As indicated in the remark above, we define the flow Φe(p) that denotes the particle position after time for the particle that starts at time O. That is,
dor(p) = (((((()), @o(p) =
Øl
for p E S. Then we can observe de is invertible self-map on and can be adopted to transport wy. Define di acts on the smooth function f S + R through the push forward:
(8)
We can find out our vorticity mapped by satisfies:
We (wo).
(9)
Next we implement the advection of w. We discretize time and consider time i, w; with the time
step h. Then we can impose the following requirement, by assuming advecting for time h - t and 1 advecting for time
W1+1 = ( = "
4h h-tt
(10)
as t E [O, hl, which is similar with the implicit Euler scheme (Butcher 2016). This enables us to derive the equivalence for the forward advected w; (along vz) and backward advected Wi+ (along 1;+1) by taking / = h/2 as follows:
4 1/2 (W;-1) D (w,))
(11)

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Then our goal turns to compute Dy. We first define the covariant derivative Do( as function g, which measures the change in f w.r.t the flow under v:
f(o.(p)) - f(p)
g D.(f)p) = lim
(12)
(-0
A classic result in Riemannian geometry is that the covariant derivative be computed Morita 200T (as shown in Fig.
(o( = s(p) = ((V))))) v(p)),
(13)
With the conclusion in Azencot et 2013) (Lemma 2.5), we can derive that
(IDo)
$ (w) exp(ID.)w Σ
(14)
kd
with respect to v. Using Eq. and Eq. abovve, along with the first-order approximation estimated by inner product, we can derive the following expression for each time step:
h (Wi, v.) + (VW++1, vz+1) 0.
C.
wi+1
(15)
2
Then our advection schemes involve iteratively minimizing the loss function C; with respect to parametric w;+1 and 0;+1 through Theorem 3.1 and preparing them for the advection for next time i + 2. More specifically, for the parameters 6; of the parametric function w; and Us we seek to optimize
0,+1 = argming
Σ
fi ((,),, (0:1),
(16)
REMCS
where M is the sample set from the surface and { represents the vector field for 7 time steps.
For the initial time G. we utilize the given velocity field vo vorticity field wp and conduct fitting for initialization. For the case that vn and Wp are all provided, these exists the harmonic component that does not contribute the vorticity but influences the velocity field. The term is often associated with the topological structure and modeled as time-invariant (Azencot et al. 2014). Therefore, we need to construct another time-invariant field parameterization (MLP) fit the harmonic term for the non-zero homology. We can follow Aichter-Powell et al. (2022): Azencot et al. (2014) and simply add another parameterized vector field 7) Eq. 4 Specifically, at the initial time, we employ another MLP y to learn the residual in the initial velocity after fitting the vorticity, treating this residual as the harmonic components. Similar with [Azencot et al. (2014), remains time-invariant in subsequent velocity computations and not trained in the following iterative computation. More discussions about the topology are included in Appendix F.I
For all the examples presented in this work, we solve this time-integration optimization problem via Adam (Kingma & Bal 2014), a first-order stochastic gradient descent method. The computational
process is showed in the pseudocode Algorithm T in Appendix D
4.2 CONDITIONING PROPERTY IN NEURAL FLOW ON SURFACES
As mentioned in Theorem 3.1 we need to construct parametric af,, x) for x E and parameters
0. The formulation also allows to conduct conditioning, i.e. function a (00, xr =), where is the features of the conditioning for the divergence-free field. This property can be leveraged in different tasks. For instance, in Sec. 4.1 the feature can represent time Moreover, for the tasks involving the vector field generation, encode Z can represent semantic features extracted from natural scenes or images, thereby enabling control over the shape, scale or other more semantic information of the field. We further present an example utilizing the variation auto-encoder for the vorticity field generation (settings Appendix [[and results in Sec. (.3), showing the capability of our framework to involve the semantic prior.
5 NUMERICAL STUDIES
In this section, conduct numerical studies for our proposed framework. Our primary emphasis lies in verifying the efficacy of our framework in fluid dynamics across various surface represen- tations, exploring conditioning characteristics, and demonstrating practical applications such as the

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Tec
Tesh
Texk:
JAM
OCB
Dnd
Figure 4: Comparison results for sphere jet flow.
Helmholtz decomposition using real-world data. worth noting that our method does not rely on any training data, distinguishing it from certain neural network waaowork-bseddd simulation methods (Mo-
rimoto et al. 2021 Lu et al. 2019 Pfaff et al. 2020 Torres et al. 2023
Methods
Вrror
Time
Storage
PINN
1.73e5
12.1 h
568.IKB
INSR
8.63e4
20.2 h
516.3KB
Small-F.S.
5.34e3
0.8h
583.8KB
Ours
2.89e2
16.5 h
532.8KB
GT
N/A
8.3
2643.0 KB
Table 1:
Quantitative results for the sphere jet flow. Error: mean square error (MSE) averaged by 100 time steps on 81924 mesh vertices.
5.1 FLOW ON THE ANALYTICAL SURFACES
In this subsection, we present the benchmark studies for our proposed framework on the analytical sphere and inclined plane, which allows us to easily derive samples and normal. We design the sphere jet flow and inclined plane to assess the correctness of our methods, compared against func- tional fluid dynamics and other baseline methods. Additional, also utilize sphere rot case to further validate the mmergy-preservation characteristics of our method Appendix E.1
SwCFS
204
XSX
Teedeppri
Figure 5: Vorticity comparison for inclined planar Taylor vortices on the 40th time step. HOLA; Pseudospectral; Elcott et al 2007 results are quoted from McKenzie (2007).
Sphere Jet.
the sphere, we also simulate a jet flow as illustrated Fig. We initialize two opposite vortices the sphere to generate the jet. Our vorticity results are compared with those from PINN (Raissi et al. 2019). Implicit Neural Spatial Representations (INSR) (Chen et al. 2023) and Functional Fluid on Surfaces with the same storage cost. We use higher- higher-resolution Functional Fluid on Surfaces as the reference ground truth, whose vector field storage is times than ours. The results method better characterizes the jet phenomenon of the flow vortices
depicted in Fig. indicate compared with other methods. Our method allows optimization on the subspace of divergence- free functionals via the physical constraint in network design while others need projection to the divergence-free functional subspace via an extra-designed loss function, which brings the error in each discretized time step, leading to the significant inaccuracy with the cascading effects in Fig. Additionally, our method produces smoother results since we sample across the entire sphere rather than only solving for the mesh vertices as in high/low-resolutin Functional Fluid on Surfaces. In other words, for higher resolution and smoother results, the Functional Fluid on Surfaces method requires more storage, whereas our memory cost remains constant. We provide the quantitative comparison for the vorticity values the reference ground truth method's mesh vertices. The mean square error in 100 time steps is shown in Tab. The results demonstrate that at the same memory cost, ours achieve the highest accuracy compared to other methods. More empirical results and comparison with both classical and advanced methods are included in Appendix E.3 to further verify the effectiveness of the proposed framework.
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Thea
Tima
Vorke Par
Jei
IFT
Referece
Figure 6: Results for flow explicit meshes.
Taylor vortices on inclined plane. We also simulate Taylor vortices on an inclined plane with known normal. The phenomena observed should be similar to those on 2D plane. The initialization of the vortices referred to McKenzie (2007). The quantitative results are displayed in Fig. [ Ours capture the details of vortices phenomenon with high smoothness and low alias. The classical methods with the close memory cost lost some details or leaves aliasing artifacts. The advanced method INSR shows similar result but still fails in details and vortices energy preservation due to the energy dissipation. PINN suffers from the largest energy dissipation and the result is the least accurate due the enforcement of the divergence- results and more comparison with classic solvers in Appendix B22 to further show the memory efficiency and accuracy of our proposed method.
constraint. We also include the quantitative
5.2
FLOW ON THE EXPLICIT AND IMPLICIT MESHES
Tiee
Timed
Vodklty
Vetaoyy
Welcdt
Figure 7: Results for flow implicit neural representation.
In this subsection, to further demonstrate the generality and robustness our methods, we present the flow results for both explicit and implicit geometry. For these geometries, the samples and normal can not be analytically derived, resulting in inherent errors. Nevertheless, we demonstrate that our method still performs effectively and captures the flow phenomenon accurately"
Flow on the explicit meshes. We initialize the Taylor vortices pair and jet flow on the explicit hand (Jacobson et al. 2018) and spot model (Crane et al.] 2013)). The results are depicted Fig. 6 They show that albeit less smooth results, due to the relatively low mesh resolution, our flows exhibit the similar behavior the reference GT and effectively capture the vortices and jet phenomenon.
Flow on the implicit neural representation. We also simulate a jet flow on the implicit neural rep- resented Armadillo (Krishnamurthy & Levoy 1996) and Lucy (Turk & Levoy 1994). The results in Fig. 7 accurately capture the smooth jet flow phenomenon on different implicit surfaces. Sur- prisingly, the classic functional flow on surface method fails to converge for both the mesh obtained from marching cubes ton solver). To substantiate our claim regarding non-convergence, we include more studies in the supplementary (Appendix E.4) that lists the simulation crash time steps across different marching cube resolutions for the traditional methods. The outcome indicates that our method keeps higher robustness than the traditional method which depends on mesh quality and exhibits instability but consumes high memory when applied to the implicit neural representations. Instead, our method
implicit neural representation and even the original mesh (using New-
Note that we do not compared with INSR and PINN in these cases, since the advection and projection in the two methods can not be simply adopted to the flow on the various surfaces without the surface parameterization R² and the continuous pullback R. which beyond the scope of the work.

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only needs the samples in 3D rather than meshes, thereby avoiding the subtle in the geometry and topology inherent in the complex shape, and therefore derive the smooth jet flow results. More- over, the learnable neural divergence-free field is capable of tolerating errors and supports more robust simulation in the long term. Generally speaking, our method supports wider applications on different geometry representations with high memory efficiency in practice.
Dioegeoos Vogritsde
Mayiitale
Magrital
(a) Vorticity fields gemeration.
(b) Helmholtz decomposition for o0-metre wind data.
Figure 8: Vorticity fields generation and decomposition tasks.
5.3 FLOW WITH CONDITIONING
To verify the conditioning property for our proposed neural-based framework, design the gen- eration task as stated in Sec. 4.2 and Appendix C
We take EMNIST dataset (Cohen et al., (017) as the image imput and generation the divergence-free velocity fields with the vorticity imitating the silhouettes of alphabets.
We construct variational auto-encoder for the vorticity fields representing different alphabets in Fig. [8 (a). The numerical studies demonstrate the feasibility of our proposed framework, benefitting from the neural network, to effectively utilize conditioning encodes from other modalities. Ours operates in more end-to-end manner compared to approaches that first generate and then fit with classic simulators, thereby avoiding the repeated fitting process for each generation.
5.4 FLOW FOR HELMHOLTZ DECOMPOSITION
Finally, apply method to the real-world atmosphere dataset (Raoult al.) 2017). The Helmholtz decomposition performed on the 100-metre wind velocity data with our proposed framework. In this study, we make the sphere assumption for the latitude and longitude coordi- nates the dataset and derive the normal analytically. The results, shown in Fig. g (b), reveal identifiable vortices after the decomposition. Further inference can be made regarding atmosphere information from the results (Cao et al. 2014 Hammond & Lewis 2021).
DISCUSSION AND CONCLUSION
This work adopts the neural representation to construct the parametric divergence-free vector field based on the Closest Point Method that supports exterior calculus the surfaces. Our framework facilitates field construction along with oovariant-derivative based advection directly on various sur- face representations, especially on the implicit neural representation, bypassing the need for march- ing cubes or meshing functions. aligns well with current trends in neural implicit representation methods like DeepSDF (Park al.) 2019) and NeuS (Wang et al. (02I) from this point. The experiment results validate the correctness, and memory-eflicien of our method. Furthermore. our framework shows high robustness and flexibility and also supports various conditioning tasks for further applications.
not well-supported by classic simulators. Our framework
While offering important benefits, our method also suffers from limitations. Our main limitations stem from topology, geometry and the time efficiency. and we have discussed the details in Ap- pendix F For more future extensions. theoretical analysis of convergence and stability of our method would be valuable. Also, more efforts in advection can be made, including high order ap- proximation (Suzuki. 1985), combination with other advanced methods like reference mapping (Li al. 2023c) and data-driven network (Liu et al. 2021). Expanding support for boundary conditions and flow viscosity p on the surfaces are also vital, for more realistic and practical applications. is also promising direction integrate struction pipeline like NeRF (Mildenhall et al. 2021 Wang et al.) (202). enabling more improved performances in dynamic reconstruction and inverse physics with the vision input.
proposed framework into the existing neural recon-
'The viscosity term can be also supported by applying the Laplacian A to the vorticity function w and ((Vw), v). We can derive the calculation on the surface using Theorem 3.1 by substituting the stream function with and ((Vw),
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ACKNOWLEDGEMENTS
This work was supported by National Key R&D Program of China 2N020023Y03300970000 and No.2024YFB2809101), National Natural Science Foundation of China (No.62322110 and No.62171255).
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A PRELIMINARIES FOR THE MATHEMATICAL TOOLS
A. THE DIFFERENTIAL GEOMETRY IN R
We provide an in-depth discussion of our field construction which follows the introduction in Ap- pendix A of Richter- Powell et al. (2022). Please refer to the work for more details. Readers with a background in differential geometry can skip this section. We first discussion the basic concept in the differential geometry for the introduction of differential form that supports us to derive and prove our theorem in the paper.
We take local coordinate chart for I E RD as x = (#1, (#1,,..,, Inn) and dæ dx, denotes the coordinate differentials, i.e. dr,(()) = Ty, i [ (1...,.) n). which is also the co-vector field of the local coordinates. Note that discuss R extended for smooth manifold and well-defined but needs the construction the local chart or other mathematical manipulations. For more extensive introduction see Do Carmo (1998); Morita (001))
example and all the definition can be
Define the linear vector space * (R") for R as the space of the k-linear alternating map:
k times
Φ R:R"
R R.
(17)
k-linear alternating map 6 is linear each coordinate and satisfies the alternating property:
( ((1,.... Ue Mj: ........ Un) - (V11, Uj Us , Un.)
(18)
The basis of the space A (RR) can be denoted with the differentials by dai, .... A doi. The way of the basis act on k-vectors, f1; ..... Uk E R as:
dr,, A AAdi (Vi, .... 0k)
sgn(T)ddr,,)) A.. A-daa,,,,--,,,,,,,,,,, (MT()) det dr (U=))nee(k)
TESx
(19)
where S₁ is permutation for ii, in. More specifically, for X E A (RR") can be represented by the basis as: =Earda' aadd'
X
Oi dx dx,,
dra E
(20)
where i,.....;4 E [n]] and 1= (f....., ix) determining scalars and dxf = daa A ... A da
The space of differential k-form A²t (R") (k-forms in the main content) is defined by the smooth function wy RD as E A (R") RD A (RR):
w
wydx'.
(21)
Then the differential operator can be viewed d (R)) A (RR)) as:
df(x) = Σ 0s (r)dr;-
(22)
Dx:
The exterior derivative d (RR)) + A++ (R))) is defined linear operator and can be calcu- lated by:
dw(x)= Edw dwi A dz
(23)
where the calculating rules for the exterior product A for two forms Σ, wydx' and En dfd can be derived by (with Eq. 19):
A
Σ
wry dx!
(24)
1.J
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For the exterior derivative, important property is dd 0. It can be checked with the definition It can be also extended to the general manifold M with the local chart
of² 0z Ja,ōmi'
af²
Eq. 23 and
and the similar proof.
The hodge operator (R*) + An-*AR""" maps each k-form to (n k)-ffrmm by functioning as:
*wi(dx') = ( (-1)oom(T)w wieeewyy (dx'),
(25)
where Y denotes the permutation (1,J) = (i1, -n. on [n)]. We can also derive the
** (-1))((nn
(26)
in R" (the similar sign function result also holds in the general M). We can also define the codif-
ferential operator ò (RR) + An=* (R") as:
8 (-1)a(k+)++1 d+
(27)
Then for vector field (V1,... .... Un), we can represent it as 1-form = Ei= dadr, Then the divergence can be calculated as follows:
des Ozi
Om Dr,
*d
=d)
Us * dr:
Σ 1=1
dz1 A ... A dxn = Σ
(28)
1=1
where the equation holds with the fact dæ; A dr; = O.
Following the discussion above, also simply prove that our Eq. 3 divergence free:
*d v = ad x #dp = (-1)" * ddd O,
(29)
where r (n 1) for (n form du in R".
There are various approaches to parameterize μ. One alternative involves directly parameterizing using an antisymmetric matrix-valued function PRichter-Powar al., 2022) characterize the anti- symmetric property in hodge star and wedge product computation. Also, the codifferential operator ju 5v can also employed keep the properties required for v E EA""-((R") and
+dor.
(30)
This form can be written as a vector representation with v and simplify the parameterization.
We also propose specific example to explore the relationship between the divergence-free field construction through the differential operator and the classic approach involving the curl operator (vector potential in fluid simulation) to enforce divergence-free.
We take Rdd l dy + vydy + vzdz for u = (Vay Uys Uz) and the differentials d.X = (dx, dy, dz).
Then we can derive
Qv: Ouy oy Fx
Ovy Oz
Ov:
Ovv,
Ou:
dz +
dr +
+dp = (V x v) dx dy
oy
Oz
Ox
(31)
(curl v) d.X.
With Eq. we can deduce that v = #dp is divergence-free. Additionally, it coincides with the curl operator of the vector μ. which happens to be a vector field in R3. Consequently, the curl of a vector field possesses zero divergence (div curl) in R23 This conclusion is frequently utilized in simulations, known as the vector potential design (Elcott et al., 2007a| Chang et al.) 2022), where the divergence-free field construction leverages curl operator. However, for the general manifold, it might be tricky to define each "curl" operator and the classical vector potential can not even be simply designed vector, while the differential operator computation for the divergence- free field construction can still work. We will next focus on the calculation *d operator for the divergence- free parameterization.
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A.2 INTRODUCTION TO THE CLOSEST POINT METHOD
We also discuss some preliminaries for the Closest Point Method. We follow the introduction in King et al. (2023). For more details, please refer King al. 2023). Consider S embedded in Ra. The Closest Point Method utilizes a closest point surface representation, which a mapping from S. The point ep is defined as the closest point on S in Euclidean
RD to the point ep(x) distance, i.c.
cp(x) argmin yll-
(32)
For the smooth surfaces, ep is unique and well-defined in sufficiently narrow tubular neighborhood W(S) C R2 surrounding S (Marz & Macdonald 2012) as an embedding in R3,, which can be formally described by:
((S) = fi E R| sr).
(33)
where r is called the tube-radius. The method designed for solving PDE on the surface. The former definition enables us to formulate embedding PDE on (SS). whose solution agrees with the solution the surface PDE at the points y E S. More specifically, let i(y) for E and u(x) for E (S) denote the solution to the surface PDE and the embedding PDE, respectively. Fundamentally, the Closest Point Method (CPM) is based on extending surface from S onto N (S) such that the data is constant in the normal direction of S. The task can be accomplished by the closest point extension, i.e. we take u(x) = u(c((=)) for all x E (S)). We can observe that the CP extension assigns surface data at the closest point of x to x itself. Then solve the embedding PDE u(x) ) and constrain the point on the surface to derive the solution. This extension also allows the differential forms the surface be replaced with the differential forms on the Cartesian differential forms (Ruuth & Merriman 2008).
Hence, we can utilize the transform of differential forms on the surface to the differential forms on the neighborhood, parameterizing with it to preserve the certain property (divergence-free). Then we use the parametric functionals solve of PDEs on the Cartesian neighborhood and pull back to the surface with vlz)) u(c((z)). Hence, our objective is to calculate the transformation of differential form that is required in PDE. The basic differential operator transformations such as exterior derivative, hodge star, wedge product d, *, A are showed in Li et al. (2023a) and Tab. 2 We next employ them to calculate the differential operator that we need to construct the corresponding divergence free field.
B PROOFS AND DERIVATIONS
B. PROOF FOR THEOREM 3.
We first propose the exterior calculus rules on 3D in Tab. where f(a) means the k-form of f
and g represent the scalar function and u, V, represent vector functions.
Output type
Wedge product A
Interior product i in (1) = w)(0) in W2) = (w u)(1)
Exterior derivative d Hodge star
(orrrmm
f1o) A g(o) = =(g))00)
N/A
*5(3) = f(0)
1-form
f(o) A u(1) = fu(1)
df = (V))
#U(2) = u(1)
2-form
u(1) A V(1) = (uu (2)
fuf(3) fu())
du(1) (V x ()
*u(1) u(2))
3-form
U(1 A V(2) = -(u-w (u)v)(3)
N/A
dugz (V ()
* *f(0) = f(3)
Table 2: Exterior calculus operators in 3D.
Based on Li al. (023a), we can further write down the exterior calculus with the Closest Point Method via cp". It allows to emulate the operators on S (denoted with superscript S) using the operators on R (denoted with a superscript R²):
CP-wedge product: cp" (ax ^ B) = (cp' A RR3
3),
(34a)
CP-interior product: op" (1S,a) a) = 1 cp'a, a,
(34b)
CP-exterior derivative: cp" (dS d cp'
(34c)
CP-hodge star: cp"(x (**8)))((s) in * R²
(cp (ap"a) ())| ((())
(34d)
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where F d(j cp) denotes the Jacobian and CP-hodge star is only applicable directly at the surface.
Proof for Theorem 3.1 We prove theorem 3.1 by mathematical manipulations. As state in Sec. 3.1 we need to construct v * dd a via σ on the surface. Then we apply ep" on both side, derive
cp e = cp (+ d
in RD (cp' o)
in * d cp'o
(35)
in R V(cp" Vicp'0)))
V(cp' 0) xn.
Then on the surface, we have (cp'))(((z)) v(cpoj(z)) v(z) (the closest point for the surface point is itself). Hence, we can derive
v = (Vicp' o j(z) x n.
(36)
Then for the vorticity function in 2D, defined by curl scalar. For the surface, derive the vorticity function via the hodge decomposition. We need to derive the divergence-free
component of v (as "curl" surfaces), which actually is +dv. Then can derive
cp w cp' (=
in * Rd
(cp'd v)
= in Ra R
(cp ()
(37)
in (V x (cp (2)
in R² (V x (cp" ())(
(cpp" v)
Similar with Eq.
35
we achieve
w V x v) n.
(38)
Note that on the surface, the rules work like ones in 2D: the stream and the vorticity functions are both scalars (compared with the vector potential functions in R3 (Elcott al. 2007a Chang al.
). It can also be verified with the exterior calculus. a is not (-form but 1-form instead, we can not derive
1-form but 0-form, which brings trouble for our neural representation.
Remark:
We can next parameterize the (with NN) and compute v and w by sampling points and do calculus with Eqs. 36 and 38 The theorem provides continuous and analytic formulation on the vorticity function on the surfaces with the help of the Closest Point Method in Tab. 2 while the classic method (Azencot et al.) 2014) needs compute the discretized differential form with the triangle meshes. CPM enables us to parameterize the surface field on the ambient space and facilitate the neural representation.
C CONDITIONING PROPERTY OF OUR FRAMEWORK
For the generation problems, verify the capability, we mirror the architecture of variation auto- encoder. Let's consider images an example input, and our objective is to provide model that re- ceive the coordinate and image conditions as an input and generate a divergence-free field which vorticity w resembles the corresponding conditioned images silhouette in visualization. Images & encoded by the parametric encoder with 6g and the features are provided with the reparameteriza- tion trick 20(0g. g) ~ (V( 9). (()g q)) (Kingma & Welling 2013), where E(0g. g). T(g, q) is the mean and variance output by the encoder. The encoded feature z (žn Zaux)) where 2noz auxiliary information such the image class, is input together with positional embedding (like
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siren) from Then the decoder translates the feature to function and further derive with The- orem 3.1 The loss function consists of two parts: One is ([ x(f,)) G(g))l2. where G is mapping from the image color space to vorticity space. This term supervises the vorticity field to shape as the silhouette of the corresponding images. The other term Kullhack-Leiin
divergence constrain the distribution of zy be normal, given by ||5² + log 72 + + with &, R', where r is the dimension of žy and log operation is computed clement-wise. With such design, we could derive simple conditioning framework, which enables more data prior into the simulation via a more end-to-end manner. More details are also provided in Appendix [ Our neural parame- terization provides more direct and effective approach involve the semantic information and show more potential to combine with the advanced generation methods in 3D or language model.
The idea of the conditioning resonates with the eigen-decomposition of the fluid fields, as discussed in Cui et al. (202T). This method decomposes the vector field into several divergence- free basis u4 i.e. v Ei=1 w;u, and u, the eigenfunction of the Laplacian operator. It's noteworthy that w;; can serve the encoding in our formulation. If strengthen our neural represented velocity field to become eigenfunctions of the Laplacian operator on surfaces, we can derive similar decomposition weights through the conditioning process described above.
Remark:
D PSEUDOCODE FOR ADVECTION
Algorithm Advection for Neural Flow on Surfaces.
Input:
Initial velocity field vo. vorticity field wo. timestep size h, number of timesteps N,, surface S, training steps E, sample size k., learning rate Fitting the initial network weight 0g and non-zero harmonic term ry with vy and wip for to N do
0 + O, for 1 to E do
Sample & point on S as sample set M.
Compute the stream function a (0xx+1) on M.
Compute the velocity v(0,), v(0,+1) and vorticity w(On). w(+++1) with Eqs. and for
time and n + with (8x). (6,++) and n)
Construct loss function Lo..1 with Eq. 15 for M.
0,+1 + 0x+1 - a VL
end for
end for
E ADDITIONAL EXPERIMENTAL RESULTS
We include additional experiments to verify the performances of our method as a supplementary. We provide the quantitative study for the flow the inclined plane, the ablation and comparison on the sphere jet flow and the convergence study for our flow on the implicit neural representation.
1.3033
3.ours:
Ie-b-ttuun:: Eiroe:
33632
2e hu-ee-.ifncctoons Erron 1e-6
he hetteetiinicioonn frnn I--6
h-Se 4 RK4, Emon: 5
1--S-tihhiit Emnr: le.3
Time
(a)
(b)
Figure 9; Results for rotating sphere flow. (a) Qualitative results for rotating sphere. (b) Quantitative results for energy preservation.
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E. RESULTS FOR ROTATING SPHERE FLOW
We verify the energy conservation property of our method by examining the rotating sphere flow, us- ing analytic solution validated by Azencot et al. (014)) This approach allows rigorously test our simulator's reliability in preserving energy over time. The initial flow conditions combine Killing vector field with a rotated gradient of an cigenfunction of the Laplace-Beltramm operator on the surface. Killing vector filed is vector field whose Lie derivative of the metric vanishes, mean- ing that the flow generated by the vector field remains constant in advection (Jost & Jost 2008). A classic example on the sphere is the vector field (-y, z, 0) around z-axis. The eigenfunctions of the Laplace-Beltte Beltrami operator on the sphere are well-known as Spherical Harmonics functions. apply the rotation them for better identification. Actually, this rotating sphere flow appears as if the sphere with the vector field is rotating over time when observed from fixed point. It can be demonstrated that the energy of inviscid flow with these initial conditions remains constant, making this configuration an excellent test case for energy conservation. As the results showed Fig. 9 (a), the results exhibit periodic patterns resembling the sphere rotating.
also
We also plot the energy change across the entire sphere, comparing it with Functional Fluid on Surfaces (Azencot et al.) 2014), the classic method with Runge-Kutta (RK) time integrator and Implicit Neural Spatial Representations (INSR) (Chen et al. (2023) with semi-Lagrangian advection as the loss function. The results indicate that our method achieves a relative change in energy on the order of 10 10m5, which is comparable to the results from the Functional Fluid on Surfaces method for both larger and smaller time steps. Conversely, the RK method and INSR suffers from larger energy losses, even for smaller time steps.
E.2 MORE RESULTS FOR THE TAYLOR VORTICES ON THE INCLINED PLANE
Time Os
Figure 10: Dynamic Results for Taylor vortices on inclined plane.
We include the qualitative results of our simulation dynamics for Taylor vortices in Fig. and more comparison results are provided in the supplementary video. We can observe clear vortex pair and the phenomenon of separation.
Methods
Error
Time
Storage
PINN
4.16e5
9.6 h
568.IKI
INSR
3.45e3
16.8
516.3KB
Small-F.S.
3.21e2
0.2h
535.6KB
HomoLBM
8.92e1
3.8
4.3MB
Small-F.S.
3.21e2
0.2h
535.6KB
Ours
1.71el
12.7 h
521.3KB
GT
N/A
2.3
3964.0KB
Table 3: Quantitative results for the Taylor vortices on inclined plane. Error: mean square error (MSE) averaged by 120 time steps with resolution of 400. The storage of our high resolution ground truth is 7.6 times than ours.
We also include the quantitative results for the Taylor vortices Tab. 3 together with more com- parison with classic methods the 2D plane for benchmark studies, including Stable Fluid (with a resolution of 1024x1024) (Stam 1999) and high-order Lattice Boltzmann methods (not parallel version with resolution of 5T2x512) (Li et al., 2023b). The corresponding qualitative results
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presented in Fig. II The results demonstrate that method enables high accuracy while show- ing high meemory-effiiiene compared with both classical and recent advanced methods under the similar memory cost.
$
S
Iine
Time
G1:lligh-ms Faretinal Frul)
Duss
USma-LUN
Stale Hurd
Figure 11: Qualitative results for the Taylor vortices compared with 2D classical methods.
E.3 MORE COMPARISON RESULTS ON SPHERE JET FLOW
Nap Nodidme
drim
Nedidin
INSR
On
Figure 12: Qualitative results for the sphere jet flow compared with eigen-net and NSF.
We compare with other surface field representation methods to verify the effectiveness of our posed framework. We try to adapt two methods into our framework. One is Koestler al. (Koestler et al. 2022), which proposes surface field representation via the eigenfunction of the Laplacian- beltrami operator (cigen-net) on the surface and the other is Xue et al. (Xue et al. 2023) that utilizes the MIP and projection operator (NSF). We implement them the sphere jet flow case for compar- ison since the Laplacian-beltrami operator can be analytically computed by Spherical Harmonics. To further verify the effectiveness of our field function design, for eigen-net, we implemented two variations as ablations: one that incorporates our divergence-free design along with the covariant derivative advection, and another that avoids our divergence-free approach in favor of the traditional advection and divergence projection; for NSF we adopted the method without our divergence free design, as incorporating would result in configuration similar to our own framework, with the primary distinction being the the Tab. and the corresponding qualitative results Fig. T2
of the MI. or siren. We show the quantitative results
Both quantitative and qualitative results demonstrate that our proposed framework achieves higher accuracy and memory-efficiinccy in the incompressible Euler flow simulation, compared with pure eigen-net or NSF surface field representations. The ablation results also indicate that that our pro- posed divergence-Fo Ours avoid the divergence- ffre projection, which reduces the extra fitting error and cascading error effects. The results also illustrate that our proposed divergence-frre free design is robust with the ways of parameterization (MLP, siren eigen-net).
field design framework significantly improves the accuracy the eigen-mi net.
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Methods
Error
Time
Storage
Eigen-net without divergence-free free design NSF without divergence- free design Eigen-net with divergence-free Ours
6.82e6
13.5 h
632.5KB
9.76e4
12.8 h
501.3 KB
1.13e3
16.7h
622.4KB
2.89e2
16.5 h
532.8KB
Table Quantitative results for the sphere jet flow compared with eigen-net and NSF, Error: mean square error (MSE) averaged by 100 time steps on 81924 mesh vertices.
Methods
Error
Time
Storage
Elcott al. 2007
1.27e4
6.8h
2643.0K
Stable Fluid
8.62e5
0.2h
972.8KB
Small-F.S.
5.34e3
0.8h
583.8KB
Ours
2.89e2
16.5 h
532.8KB
GT
N/A
8.3 h
2643.0 KB
Table 5: More Quantitative results for the sphere jet flow with classic methods.
We also include results of classical solvers for reference, including Stable Fluid (with reso- lution of 256x256) (Stam 1999) and Elcott et al. (Elcott et al. (007a) (using the same mesh as GT). The quantitative results are shown Tab. and the qualitative results are presented in Fig.
The results show that our method achieves high performance and low energy dissipation compared classic methods. However, this improvement comes with the trade-off of higher time consumption as discuss in Appendix F33
E.4
CONVERGENCE VERIFICATION FOR THE FLOW ON THE IMPLICIT REPRESENTED
SURFACES
Marching Cubes resolution
Average steps of Crashing
Storage
64
36.5
117.18 KB
128
53.4
768.32 KB
240
68.7
1123,15 KB
GT Mesh
88.6
1419.48 KB
Ours
N/A
523.4 KB
Table Time steps that the classic method (Azencot al. 2014) with Marching Cubes on implicit neural represented surfaces crashes. We repeat the process with 10 times and derive the averaged results. However, ours can work well with the implicit neural representation.
For the case of implicit represented surfaces, as previously stated in the main context (Sec. 5.2 for the classical method it is not convergent, leading to non-referable results. To substantiate our claim regarding non-convergence, we include new Tab. 66 that lists the simulation crashing time steps across different marching cube resolutions for the traditional methods. This data will offer a quantitative perspective on the limitations of the classic methods robustness. We also include the reference qualitative results in the supplementary video to exhibit the simulation process and describe the non convergence on different resolutions. The results demonstrate that our meshless and end-to-end method can achieve high adaptability and robustness for the simulation the implicit neural representation, while the classical method fails in marching cube meshes and needs further complex geometry processing schemes to improve the mesh quality.
E.5 ABLATION STUDIES FOR THE NETWORK AND TRAINING DESIGN
We also include the ablation studies on the network size and sample count sphere jet case. The results are exhibited in Tab. and Tab. 8 which indicate that our method is relatively insensitive to these settings unless the width extremely small.
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Hat
Figure 13: Qualitative results for the sphere jet flow compared with more classical methods.
Network Size
Error
4 layers, 128
2.89e2
4 layers, 256
2.21e2
4 layers, 64
4.57e2
Table 7: Ablation studies for network size for the sphere jet flow.
F DISCUSSIONS AND LIMITATIONS
F.1 DISCUSSIONS AND LIMITATIONS ON THE TOPOLOGY PROBLEMS
The first limitation is about the singularity problem. don't focus heavily on the singularity issues related to topology, such as those highlighted by the Poincaré- -orpff Theorem, which is complex challenge. Our main interest lies in using neural approaches to simulate fluid dynamics and model vortex dynamics surfaces for visual effects (as described in Eq. [6. rather than dealing with arbitrary vector fields. The scenes chosen in both Azencot et al. (2014); Ando et al. (2015) (which do not mention the issue) and ours typically maintain non-zero measure zero velocity and avoid poles to simplify the analysis. We initialize the simulation with finite vorticity, which makes hard for advection generate infinite vorticity (or singularities) within our settings. Furthermore, we employ the stream and curl regularization to avoid extreme values in the singularity points and preserve system stability. According to Sard's theorem, the number of poles has zero measure in our simple cases, we maintain a very small probability of sampling at the poles, which improves the stability of simulation. Though the velocity field near singularities will be underfitted and approximated by the network, potentially leading to inaccuracies and numerical dissipation near the "cyclone", the overall results provide empirically reasonable visual effects, which is the primary goal of our application.
To further validate the effectiveness of our regularization and sampling, we applied our framework to fit the velocity (- (-sin(0), cos( on sphere with spherical coordinates (1, 0, Φ), which keeps singularity at the northerm/southern pole ((1, O, 0). (1, π, 0)). The errors and velocity magnitude are shown in Fig. [4] where we achieve the desired velocity pattern. Although some errors appear in the initial epochs, they become negligible over time, which sufficient for graphics and visual effects.
Training Samples (millions)
Error
60
2.89e2
40
3.12e2
80
2.79e2
Table 8: Ablation studies for sample number for the sphere jet flow.
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Valodty Macathuiia at opoda
Valddty Magaitato 120 spoch
20goch
Eme's 120
Figure 14: Qualitative results of the numerical studies (Velocity magnitude and error) for the con- stant sphere flow. (Views from the northern and southern poles)
Ce
Cheuakee Sutaes
T
Tea
(iota
Tex
Thx
Figure 15: Qualitative results for the sphere jet flow the explicit torus and double torus.
Another problem related with the topology is the cohomology term. Our method mainly adopted for the topology for genius zero, where the velocity can be estimated by the stream function with- out more consideration of the time-variant cohomology component (as stated in Yin et al. (2023) Proposition 5 for the surfaces removing several poles). For the surface with high genius, insufficient for the time-invariant cohomology term to handle highly turbulent flow and generate complex visual effects. The problem be further addressed by incorporating the solver in Yin let al. (2023) with the neural network, which will be our future work. However, only for the visual effects. our method still performs well for the high genius surfaces and shows the correct jet flow behavior compared to the reference classic method. We plot the results in Fig. 15 with simple torus and double torus cases with explicit torus and double torus meshes. They do demonstrate reasonable visual effects on high-genus surfaces even without considering the time-varian cohomology.
find
F.2 DISCUSSIONS AND LIMITATIONS ON THE GEOMETRY PROBLEMS
As stated in Sec. 3.2 our method needs SDF for the flow on the implicit surfaces. This can pose challenges when calculating normals for open surfaces. To overcome this, one can use unsigned distance field (UDF) to extend our framework to open surfaces, which is well applied in Chibanc al. (2020): Yang al. (2023) Long al. (023).
Additionally, our framework is based on the Closest Point Method (CPM), which, due to its reliance on ambient space, faces challenges when dealing with narrow geometric features. This reliance can lead to ambiguities when processing narrow or thin features, resulting in inefficiency and inaccuracy due to increased difficulties in convergence. To mitigate this issue, we can follow the approaches in King et al.) (2023); Marz & Macdonald (2012) and sample the neighborhood within small tube radius adaptively. In parctice, the sampling distance threshold to the surface can be determined following the method outlined in King et al. (2023) (Sec. 8), guided by the estimation of the surface reach distance as described by Aamari et al. (201).. This estimation can be locally constructed using simple mesh extractor from the signed distance function (SDF) if mesh is available. This allows for rough detection of thin regions and adaptive adjustment the sampling distance, thereby improving both efficiency and accuracy.
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Moreover, in the practical scenarios, the non non-smooth surface, imperfect SDF (common problems in implicit neural representations such as Yifan et al. (202I)) and the complex surfaces (like unoriented surfaces) can also limit our performances. Noisy surfaces and normals introduce numerical viscosity, slowing down the simulation. But fortunately, we show that they do not drive our simulation to crash, the neural representation provides smoother and more robust results (as demonstrated in Appendix[E.Ae (iiixxEEwwttt with GT mesh data) compared to traditional methods. To further solve these issues, we can construct smooth approximator and utilize large number of samples from smaller tube radius (as described in King et al. (2023) above) for improved performance. Additionally, the orientation problem addressed in King al. (2023) and the extension for our method с an be similarly constructed.
F.3 DISCUSSIONS AND LIMITATIONS ON THE TIME AND MEMORY CONSUMPTION
Our method actually keeps large advantages at the memory consumption albeit with the increased time requirements. The low memory usage is crucial for handling and analyzing high-resolution data. For example, 2D simulation at a 4096x4096 resolution requires about 200MB per frame, resulting in considerable memory demands for extended simulations. Similarly, high-resolution 3D grid simulations consume even more memory. Although mesh-based simulations can be more efficient, they encounter mesh quality issues, as shown in the robustness tests in Appendix [E44 The time cost remains high, as methods like siren or simple MLPs are too global and inefficient for optimization, and inference sampling (particularly for implicit neural representations) is time- consuming. To address the time consumption issue, a hybrid simulator to achieve both high speed and performances is necessary. Fortunately, recent advances in hybrid representations (Müller al. 2022 Huang et al. 2023) have shown promising results in reducing training time from hours to seconds while maintaining the expressiveness. Employing these more efficient representations hold great promise for improvements of our method. In the inference time, a sampler (Sharp & Jacobson 2022) with high efficiency can be adopted, which will provide huge saving for the inference time with KD hierarchies.
G MORE IMPLEMENTATION DETAILS
We provide our implementation details for our numerical studies. Our experiments are all imple- mented with Jax library (Bradbury et al. 2018) on an NVIDIA GeForce RTX 3090 GPU.
Sphere Jet: We adopt the 4-layers MLP (for shaper simulation results compared with siren) with 128 units for our implementation. The learning rate is set with the exponential decay from le to le 7 with 60000 steps and batch size 1000 for each time step. The time step is chosen as 5e 2. The initialization is the same as Azencot et al. (2014) and the initial vorticity is kept for the whole simulation time. For the comparison. INSR for advection, projection and correction respectively. The learning rate is set as 1e - 6. For each process, the siren iterates for 40000 steps with batch size 5000. For the original PINN, the MLP with 4-layers 128 units. The loss function is to enforce the incompressible Euler equation directly, as showed in Appendix A.3 of Chen et al. (2023). The learning rate is set by le 5 5 with 60000 steps and batch size 2000.
the siren function with 4-layers 128 units
adopt
Taylor vortices on inclined plane: We adopt 4-layer siren with 128 units representation to conduct the positional encoding with the first layer frequency as 30. The time step is chosen as 5e - 3. The initialization is set the same as McKenzie (007). and we rotate the plane and make the normal be (0.3, -0.5, 0.8). The domain size is set |T, π. with the periodic boundary condition. The time step set as 0.05. The learning rate is set with the piece-wise constant from le 5 with a decay factor 0.1 on 40000 and 60000 steps for total 80000 steps and batch size 1000. For comparison, INSR adopts the siren MLP with layers and 128 units per layer for advection, projection and correction respectively. The learning rate is set as le - 5. For each process, the siren iterates for 20000 steps with batch size 1000. For the original PINN, we adopt MLP with layers and 128 units. The loss function consists of the governing equation part which is the same the one in sphere jet case and the periodic boundary condition part. The learning rate is set by le with 60000 iterations and batch size 1000.
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Rotating sphere flow: For the construction of o, adopt 4-layer siren with 128 units represen- tation (Sitzmann et al. 2020 with the first layer frequency as 30. The learning rate is set with the exponential decay from 1c- to 1e 8 with 40000 steps and batch size 1000 for each time step. The time step is chosen as 5e - 4 and 2e 3. The initialization of the velocity is set as the Killing field (-y,20 x;, 0) with the rotated 4-degree Spherical Harmonics functions with order and 5. For INSR for our comparison, we adopt the siren function with 4-layers 64 units for advection, projection and correction respectively. The learning rate set 5e For each process, the siren iterates for 10000 steps with batch size 1000.
Flow on the explicit meshes. We adopt 4-layer siren with 128 units representation for both models with the first layer frequency as 30. The learning rate is set with the exponential decay from 1e - 6 to le with 40000 steps and batch size 2400 for each time step. The time step is chosen as 5e and 8e 2 for the hand and spot respectively. The hand model is initialized by two vortices with the geodesic 0.41 by Crane et al. (2013b) and the spot model is set with 0.3.
Flow on the implicit mesh. First, we adopt 4-layer siren with 256 units to reconstruct the im- plicit neural representation of the SDF function based on Sitzmann et al. (2020) with the uniform sampling. The in the simulation, we take the rejection rules in Yang et al. (2021) to complete the uniformly sampling for the simulation function and avoid the samples concentrating near the high curvature area. The jet vortices are initialized by given two points the mesh as opposite pair. Our simulation neural field is also implemented as 4-layer siren with 128 units representation with the first layer frequency as 30. The learning rate is set with the exponential decay from le 5 to le with 40000 steps and batch size 1000 for each time step. The time step is set by 5e - 2 and 2e for Armadillo and Lucy respectively.
We adopt the encoder consists of 2-layer feature extraction MLP to reduce image data 32-dimension feature space; siren network to generate 228-dimensional feature with the input position r as neural field representation and an one-hot class encoder for image categorization. Note that the first layer frequency for the siren network above is also 30. The decoder is a 2-layer MLP that transforms the concatenated features from the encoders to the stream function a. Then following our Theorem 3.1 we can also derive the corresponding v and w taking the derivative with respect to In the inference time 32-dimension random Gaussian vector and the corresponding image class are input to concatenate with the positional encoding to generate the field value at the spatial point. The learning rate for the training process is set with the exponential decay from 1e - 4 to le - with 400000 steps and batch size 1000. We map the color of the input images to the vorticity function and try to make the velocity field preserving the vortices shape as alphahets via Mean Least Square loss and KI. loss.
Flow with conditioning:
Flow for Helmholtz decomposition: We adopt the atmosphere data (100-metre wind velocity) on Jan, 2024 (Raoult et al. 2017). We adopt 4-layer siren with 128 units representation for both models with the first layer frequency as 100. The learning rate is with the exponential decay from le to le 5 with 200000 steps and batch size 1000. We try to make our velocity results closer to the given data and derive the divergence- component, similar as Richter-Powee al. (2022).
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Reviews:
[{"review_id": "VuX4HNTZpW", "reviewer": "Reviewer_Y5ci", "summary": "This paper introduces a novel framework for simulating incompressible Eulerian fluid flow on 3D surfaces using neural implicit representations. This method leverages the Closest Point Method (CPM) and exterior calculus to parameterize the fluid’s velocity and vorticity fields directly on the surface without relying on discretization, which reduces memory costs and bypasses the need for conventional spatial discretization. The framework introduces a covariant-derivative-based advection process, which integrates surface flow dynamics while minimizing energy dissipation. Notably, this work is among the first to simulate incompressible fluid dynamics on neural surfaces, achieving enhanced accuracy and energy preservation across various geometric representations.", "questions": "### Questions\n1. **Use of DEC Language**: The paper’s use of Discrete Exterior Calculus (DEC) is rigorous and suits the formal approach taken. However, many in the ML and physics communities might be more accustomed to traditional differential or vector calculus, so DEC may require more adjustment for those readers. Adding intuitive explanations alongside the DEC formalism could enhance accessibility, although this may vary depending on the preferences of other reviewers.\n2. **Handling Narrow Geometric Features in CPM**: The reliance on ambient space in CPM may lead to ambiguities when processing narrow or thin features. Clarifying whether this dependency impacts stability or accuracy for such geometries would enhance the framework’s applicability and inform potential adaptations to handle such cases.\n---\n### Suggestions\n1. **Missing Citations**\n    1. For by construction divergence-free field with neural network, maybe also cite [Deep Fluids](https://onlinelibrary.wiley.com/doi/10.1111/cgf.13619).\n2. **Clarifying Performance Gains Over INSR**:Intuitive explanation of why your method is > INSR > PINN when constrained by storage size. Intuitively, INSR is superior to PINN because it doesn’t record time in the neural field, so, given the same storage budget, INSR should and must outperform PINN. However, your method doesn’t gain from saving less information in the neural field to achieve higher accuracy (i.e., it doesn’t concentrate model expressiveness on specific features to achieve this). So, what is the intuitive reason behind your method’s improved results over INSR? Is it due to the CPM formulation or the Helmholtz decomposition? An “ablation” would be helpful here.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "### CPM Formulation\nThe math formulation is clean and concise. It is quite apparent that the authors are coming from a graphics background and I love this clean DDG writing style.\n- The Closest Point Method (CPM) is relatively new in visual computing, yet its integration with neural fields here aligns with my belief in CPM’s potential for solving PDEs on surfaces. Compared to surface sampling techniques (as seen in Geometry Processing with Neural Fields [Yang et al., 2021] and similar studies), CPM offers a structured way to define differential operators in volumetric data by rigorously establishing value transfer in the ambient space embedding the surface.\n- A persistent challenge in neural implicit representations is that, while data is represented volumetrically (e.g., through neural SDFs), the actual solutions are constrained to the 0-level isosurface. Sampling on this isosurface can be inefficient, but CPM provides an effective alternative by leveraging the ambient space, enhancing both efficiency and rigor.\n\nOverall, I would love to see this line of work being continued and the math formulation should be shared and seen within the ML community.\n\n---\nSome misc comments:\n- The related works section is thoughtfully composed, with necessary references cited and no excess, reflecting high-quality citation practices.\n- The choice of ground truth in this paper is well-justified and suitable for the presented comparisons.", "weaknesses": "I have two concerns, regarding the claimed first and third contributions.\n\n---\n\n### Performance vs. Storage vs. Accuracy\nThe storage and accuracy benefits presented as a core contribution appear somewhat overstated since these gains stem from the inherent compact representation of neural fields, as noted in prior works like INSR-PDE (Chen et al., 2023). The neural network, here largely a standard MLP, serves as a model reduction tool or compressed parameter space. However, the substantial cost is slower simulation speeds, particularly noticeable in evolving the PDE on a neural representation, and this tradeoff is well-documented in the field, tracing back to foundational work like *Geometry Processing with Neural Fields* (Yang et al., 2021). Additionally, working with surface PDEs inherently mitigates spatial complexity compared to volumetric Eulerian approaches, further diluting the impact of memory savings in this context. Unless optimized network designs or implementation techniques were used, this contribution may feel more like a tradeoff typical of neural fields than a novel improvement.\n\n**TL;DR:** Without unique implementation optimizations, this tradeoff doesn’t stand out as an independent contribution, as neural networks naturally offer compact representations at the expense of computational speed.\n\n---\n\n### First to Simulate on Neural Implicit Surface Representation\nThe claim of being the first to simulate incompressible fluid flow on neural implicit surfaces is somewhat uncertain, as prior work using sampling techniques, like *Geometry Processing with Neural Fields* (Yang et al., 2021) or INSR-PDE, could also solve surface PDE like Laplace Equation by sampling on the surface. While it’s conceivable that these methods struggle with incompressibility when applied to Navier-Stokes, demonstrating their limitations would highlight the advantages of the Closest Point Method (CPM) for ensuring divergence-free constraints on neural surfaces. Including such comparative results, even as failure cases, could effectively underscore this paper’s unique approach.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}, {"review_id": "BPUpKwHw1I", "reviewer": "Reviewer_Gjy9", "summary": "This paper builds on the recently introduced Closest Point Exterior Calculus (CP-EC) to propose a novel method for preserving the divergence-free property of vector fields on surfaces. By leveraging the closest point map, this approach seamlessly extends computations from the surface to the surrounding Euclidean space. At the core of the paper, Theorem 3.1 presents a specific construction for generating a divergence-free vector field on a surface using the CP-EC framework. This framework enables the calculation of gradient, divergence, and curl in a way that respects the intrinsic geometry of the surface, ensuring that the velocity field remains divergence-free when constrained to the surface. A key advantage of this method is its flexibility, as it supports simulations on various surface representations, including analytic surfaces, explicitly defined mesh surfaces, and, notably, neural implicit surfaces. The paper introduces a complementary advection process based on covariant derivatives for fluid dynamics, designed to minimize energy dissipation. Numerical studies confirm the framework’s accuracy, energy preservation, memory efficiency, and adaptability to geometry. Results show it achieves about 15 times higher accuracy than other methods with similar storage, offers 5 times memory savings over classic methods, and effectively models fluid dynamics. Additionally, the simulator's robustness is demonstrated through an end-to-end generation task and a real-world velocity field decomposition.", "questions": "How do you do the interpolation of the pulled forms? In the CP-EC poster the authors recommended the Cubic Lagrangian.\nHow does this interpolation affect the divergence-free property of the velocity field? Were there any numerical problems?\nIs it possible to do an ablation study on this point?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The paper shows that the recently introduced Closest Point Exterior Calculus (CP-EC) is very well suited to simulate fluid simulation on neural implicitly defined surfaces in 3D. The CP-EC allows to automatically guarantee the divergence-free properties of the vector field. The method achieves up to 15 times higher accuracy than previously used discretization methods on the surface with the same memory requirements, which is confirmed by extensive numerical simulations of different applications.", "weaknesses": "The English in the current version of the paper needs to be improved. Numerous articles are missing and sometimes the wrong words are used (subtle instead of subleties, divergence free instead of divergence free property, etc).\n\nCompared to the actual straigth forward application of the CP-EC to the case of flow simulation on surfaces, the paper seems cumbersomely long and is also not as clear to read as the recent papers on the topic referenced in the paper, whose presentation is clearer and more concise.  Maybe the authors can try to improve on that.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}, {"review_id": "zGMaqPxEgd", "reviewer": "Reviewer_dCvp", "summary": "The paper proposes a framework for fluid simulation on surfaces that is divergence-free by construction. This is done by using exterior calculus tools, in special the definition of the divergence based on the Hodge star operator and the exterior derivative, the property that the Hodge star is (up to a sign) its own inverse and the nilpotent property of the exterior derivative. The Closest Point Method is used to apply those tools on generalized surfaces, making a natural link with Riemannian geometry, and enabling the evaluation in the tangent space around samples in the surface. With those tools it is possible to transit between the surface and $R^3$ as needed, in special for advection which can be done considering the Riemannian metric of the manifold.", "questions": "(1) Why introducing $f$ in equation (8) instead of using $\\sigma$ directly?\n\n(2) I think a $t$ subscript is missing in equation (8) ($\\Phi_t$).", "limitations": "", "rating": 10, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "To be honest, I had a lot of fun reviewing this paper. Unless other reviewers flag major flaws that I could not find, I think it is ready for acceptance.\n\n* It is a very good example of when a simple and elegant core idea based on strong guarantees enables a lot of very interesting questions and consequences. The core idea of using the nilpotent property of the external derivative and the self inverse property of the Hodge star operator to force a divergence free vector field on a surface is elegant and foment all the paper discussion.\n\n* The loss formulation is as expected, very intuitive.\n\n* As far as I know, It is the first method that converges in implicit surfaces.\n\n* The method does not rely on training data.\n\n* Evaluation is robust. The idea of starting with analytic examples where ground truth is easier to evaluate is good.\n\n* Related work section cites every paper I could think of. The care with the citation of classic papers (even for datasets) is notable.\n\n* Mathematical notation is very clean. It easy to see that there is a lot of effort with notation polishing.\n\n* The paper makes use of very good references for background Math.", "weaknesses": "I will point some minor weaknesses that could be fixed to improve the paper.\n\n(1) An image depicting equation (4) and another showing the advection process would greatly improve the friendliness of the paper since both processes are very geometric. That would make the paper be appreciated by a broader audience.\n\n* In the image for equation (4) it is sufficient to show the neighborhood of the surface, the mapping $j$, the mapping $cp*$, the vector resulting from the gradient and the tangential vector acquired from cross product of the gradient with the normal.\n\n* For the advection image it is sufficient to depict the push forward (pull back) function in action and the inner product using the Riemannian metric.\n\n(2) The presentation could be more friendly by giving some intuition along the text. I will point some places I think this kind of intuition would be beneficial.\n\n* Line 199: could say that the even though the divergence may be expressed using different k-forms, the definition of div(v) is the 0-form version resulting in a scalar function.\n\n* Line 299 (equation 4): could say that $cp^*\\sigma$ is a notation abuse because $cp^*$ expect a k-form but $\\sigma$ is a 0-form. Also that the composition with $j(x)$ is to restrict the computation to the surface, the gradient is to acquire a vector field and the cross product is to acquire a tangent vector field. A reference to the proposed image would also be good here.\n\n* Line 234 (equation 5): could say that that vorticity expression considers the rotation axis equals to the normal because it is evaluated on the surface. Then the vorticity may be represented as a scalar field.\n\n* Line 257: could say that the expression is a neighborhood extension of the surface along the normal field.\n\n* Line 320 (equation 13): could say that the $<. , .>_p$ notation is an inner product considering the Riemannian metric of the manifold of the tangent space at point p. A reference to the proposed image would be good here.\n\n* Line 327 (equation 15): could say that the inner products are the first-order approximation of the push forward function.\n\n* Line 344: that paragraph could say that the harmonic components do not contribute to the vorticity and that is the reason why the additional harmonic network is needed. Could also say that it is constant along the simulation because it is associated with the topological structure of the surface, which does not change over time.\n\n(3) This paper deserves an acronym so it may be more easily referenced in the future by other researchers. I advise the authors to think about changing the title to include a creative acronym.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}, {"review_id": "r0ezDk7iz3", "reviewer": "Reviewer_V8TK", "summary": "This paper proposes an implicit neural representation to improve solvers that simulate flows on geometric surfaces through geometric adaptivity. The authors propose a neural physical simulation framework to construct a parameterized vector field on surfaces using exterior calculus formalism. Through a Closest Point Method, it is proposed an implicit neural network representation that is able to maintain a divergence-free property intrinsically. Divergence-free is an important property of Navier-Stokes solvers, and strictly enforcing them is a challenging task. Furthermore, the authors claim that the proposed approach is able to accurately preserve the energy of the flow as time advances.", "questions": "- Did the authors explore alternative network designs for representing the implicit neural fields (such as \"Instant Neural Graphics Primitives with a Multiresolution Hash Encoding”) ?\n- How does the method fares in simulations where regions of turbulence are highly concentrated? Is the proposed adaptivity property working as expected?", "limitations": "", "rating": 1, "confidence": 5, "soundness": 1, "presentation": 1, "contribution": 1, "strengths": "Proposing alternative representations to the standard discretizations (e.g., grid/meshes) for solving PDEs is a very interesting and challenging topic of research. The authors propose a method that considers specific intricacies of the PDE solution when employing neural representations, along with desired properties that can potentially be satisfied in a continuous fashion (e.g., the divergence-free condition).", "weaknesses": "Unfortunately this paper is clearly below the ICLR quality acceptance bar. My major concerns are as follows:\n- Poor exposition along with several typos which make the paper hard to understand. For example. the structure of Section 3.1 is composed of fragmented phrases forming very short paragraphs, making it hard to follow. Several typos and confusing phrasal structures (L11: “Incompressible Euler fluid on the surface”, L20: “We contribute a neural physical simulation framework on the surface with the implicit neural”, L240: “In the meanwhile\" to name a few) greatly compromise the quality of the paper.  \n- The main idea of the paper is based on wrong assumptions. The poster “Closest Point Exterior Calculus”, in which the paper is heavily based, already offers a solution that is independent of the mesh quality. This invalidates one of the main motivations of this submission that previous approaches are dependent on mesh quality, and thus an implicit neural representation is required. Moreover, the assumption that storage is a limiting factor on solvers is also incorrect, since a solver usually has to store a single time-step of the represented variables for advancing the simulation state. The presented results also show very modest resolutions.  \n- There are missing references and/or previous methods are not thoroughly considered, leading to an outdated methodology proposition. Recent approaches (“Covector Fluids”, “Impulse Particle In Cell”, “Fluid Simulation on Neural Flow Maps”, “Eulerian-Lagrangian Fluid Simulation on Particle Flow Maps” and “Lagrangian Covector Fluid with Free Surface” to name a few) adopt structure preserving integrators by considering the deformation of the flow map during advection. This is ignored by the proposed advection method, which has a rather lengthy description in the paper. Lastly, Elcott et al, 2007b does not suffer from instabilities as it is mentioned in the manuscript and modern structure preserving solvers (\"Impulse Particle In Cell”) are able to accurately advect velocities without major stability issues. \n- The paper partially focuses on showing mathematical proofs that are known by the exterior calculus community (divergence free vector fields on surfaces), which make the described theory not so relevant as new theoretical contributions. The authors could just reference relevant discrete exterior calculus material or move the lengthy mathematical descriptions to the Appendix. \n- The paper should have been focused on more relevant aspects of the implicit neural representation, such as network structure, how to properly tackle high-frequencies of the implicit neural field, how to make the training/evaluation process efficient (e.g., check “Instant Neural Graphics Primitives with a Multiresolution Hash Encoding”), etc.\n- The authors mention that pressure projection (usually the most expensive part of a fluid solver) is not required by their approach. However, they solve a non-linear optimization problem iteratively with a simple ADAM gradient descent approach. This approach is way less efficient than traditional operator splitting, as evidenced by the timings shown in Table 1 (16h for 80k vertices is a very inefficient timing for the considered resolution). Lastly, there seems to be some high-frequency “ringing” artifacts generated by the proposed method in Figure 3 which are not present in ground truth or in the HOLA-7 results. \n\nThese are some of the reasons that justify my low score for this paper. I suggest the authors to rethink their approach before resubmitting the manuscript.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)