Neural Fluid Simulation on Geometric Surfaces

We thank the reviewer for the suggestion. To addreses your questions and concerns:

Q1: Poor exposition along with several typos.

R1: For any writing issues are present, we assure that we will conduct a thorough proof reading and optimize the structure and descriptions throughout the paper. (L11: revised Incompressible Euler fluid to Incompressible fluid; L20: corrected to We propose a neural physical simulation framework on the surface with the implicit neural representation; L240 and others: rephased to improve transitions and ensure smoother readability).

Q2: The main idea of the paper is based on wrong assumptions ... 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.

R2:

1. Motivation Clarification. To clarify, we do not attribute the independence from mesh quality solely to the implicit neural representation. In fact, there exist many methods that do not incorporate the Closest Point Method face challenges related to mesh quality, and our goal is to improve these approaches. Even when using the Closest Point Method with a discretized grid, discretization errors can still occur in operator calculations. By leveraging neural parameterization, we achieve analytically accurate operator computation, which enhances robustness and addresses these challenges more effectively.

2. Storage Problem. Regarding the storage concern, we acknowledge that one-step computation is often sufficient for fast local result previews. However, for studies involving long-term effects or rendering on arbitrary paths (such as in scientific research or visual effects), storage becomes a significant consideration. Additionally, for scenarios involving data sharing or long-term analysis, storage efficiency cannot be overlooked. Our model addresses these challenges by supporting continuous input and enabling to produce very high-resolution results. For comparison in the content, we believe on the current resolution is adequate to show the effectiveness and robustness of our results while higher resolution simulation for the classic method will suffer from storage and robustness problem.

Q3: There are missing references and/or previous methods are not thoroughly considered ... modern structure preserving solver accurately advect velocities without major stability issues.

R3: The paper referred here is mainly designed to deal with the numerical dissaption caused by the adveciton and projection. But none of them are designed for the surface flow (even not with a surface flow demo). While their methods could theoretically be extended to surface flows, the additional operators they propose would require careful adaptation and design for surface-specific contexts, which is beyond the scope of this work. To be more specific, Covector Fluid is relatively suitable for surface flows, but it still needs to specify and compute the co-vector form directly on the surface. Methods like Fluid Simulation on Neural Flow Maps, Eulerian-Lagrangian Fluid Simulation on Particle Flow Maps, and Impulse Particle In Cell rely on flow map forms, which could introduce possibility of instability due to surface operator estimation error when applied to surfaces. Lagrangian Covector Fluid with Free Surface differs slightly due to its use of power graph discretization, which also presents extra challenges for extension to surfaces.

Regarding Elcott et al. 2007 [5], the computation of flow lines on triangle meshes is known to introduce numerical instability, as discussed in Anzenot et al. 2014 [6]. Although Impulse Particle In Cell shows potential to solve the problem on the surface but still needs further verfication, which is beyond our scope of the paper.

Q4: The paper partially focuses on showing mathematical proofs that are known by the exterior calculus community ... move the lengthy mathematical descriptions to the Appendix.

R4: We believe this is not a direct consequence of solely the divergence-free field construction using exterior calculus and discrete differential geometry as cited. Our approach requires the use of an additional tool based on the Closest Point Method (CPM), as described in [7], to theoretically identify the stream function and other necessary differential forms. Our theoretical formulation provides a more rigorous construction of the "surface curl" and enables the application of neural parametric functions for surface vector field dynamics. To assist readers who may not be familiar with these techniques, we aim to include a comprehensive sketch in the appendix. We believe this addition will provide valuable insights and improve the accessibility of our methodology.

We want to clarify that our goal is not to propose a general neural-based solver with streamfunctions for all flow construction and energy preservation in 2D/3D simulations. Our focus is specifically on embedded surface flow. While time consumption is noted as a potential limitation (similar to the time in Chen et al.), it can be mitigated through a hybrid design or more efficient representation within our framework.

Presentation has been revised according to the suggestions from Reviewers Y5Ci and dCvp. Evaluation might be influenced by the non-fully convergence of the network (but still better than other baseline methods). In terms of contribution, while the curl of the streamfunction is not news, estimating it on a surface using a continuous functional (neural representation) could be of interest.

For the related work section, we have included works for simulations on surfaces, which align with our previous statements, but not all the fluid simulation frameworks. Regarding boundary conditions for stream functions, our simulations are primarily on closed surfaces, and this issue does not arise. More complex surfaces (open surfaces) are discussed in Appendix F.2 as a limitation and a future research direction.

The specfic problems:

Q1: There are no examples that support the claim that this paper improves previous approaches.

R1: At least, for the functional fluid on the surface (which is a classic solver), we can improve on its robustness on mesh quality as showed in Appendix. E.4. We admit the classic solver benefit much from the speed, but more efforts need to be taken on its surface implementation (including the estimation of the surface differential form) and no guarantee for the robustness of the methods on the surfaces.

Q2: Covector fluids would work on a surface mesh, they have 2D examples. That's the beauty of DEC: it really abstracts the domain representation, one has only to implement the exterior derivative properly (which is easy). I suppose that Covector Fluids did not show examples on a surface because of the limited application of such flows.

R2: It is not exactly the same case for 2D and 2D surface embeded in 3D. The implementation for covector fluid employ the staggered grid but not mesh for the operator estimation. Actually, accurately estimating the differential form is not a direct consequence and can introduce possible instability on the low-quality mesh (like statements in Li et al.'s CPM poster).

Q3: CPM is not necessary to identify the streamfunction as a differential form.

R3: The differential form for the streamfunction does actually not need CPM but its estimation, especially continously, needs CPM (Li et al's) as a support.

Q4: Problems for impulse-based formulation.

R4: In our reply, we primarily address your question regarding traditional operator splitting on surfaces. We acknowledge that impulse-based method offers significant improvements in energy preservation and efficiency. But our main focus is the simulation on the surface with different representations. The discussion about impluse-based method in the scope needs further work on implementing differential operators and impulse gauge variables on surfaces. Actually, we hope our method to serve as a start point for integrating differential form, CPM and neural functionals (as stated in Reviewer Y5ci) via stream-function. Exploring impulse-based methods with neural representations presents an exciting future direction.

Q5: Storage Problem

R5: We benefit from the compact representation for the storage consumption (as noted by Reviewer Y5ci and Chen et al.). Storage can be a challenge because, even with meshes, high-resolution texture maps are needed to store simulation variables, which can be costly. In our paper, we actually aim to construct a continuous representation of surface flow, where compactness naturally reduces consumption. Several CG papers, such as those by Anzenot et al. and Elcott et al., focus on surface flow, and we follow their lead for surface visual effects. Our methods can be further optimized for time efficiency and applied into the practical scenes.

Honestly I’m a bit surprised by the other reviews, specially the one that rated this submission a 10. I strongly believe that this manuscript, in its current form, is not ready for publication. I think this paper would benefit from another submission cycle, carefully reassessing the underlying assumptions about the method. I restate that my major concerns are related to paper usefulness (the proposed method is extremely slow (17 hours for a 2D mesh!!) and has little application), exposition (poor English, several typos and I just noticed a few on my original rebuttal), lack of proper evaluation (few examples, some of them with noticeable artifacts, quantitative evaluation is poor), little contributions (its widely known that taking the curl of the streamfunction yields a div-free velocity field, plugging CPM into a neural representation is not a major breakthrough) and several important related works are missing. Moreover, employing streamfunctions to solve flow equations difficult boundary conditions (not addressed at all in the paper), and it is a reason why the majority of recent approaches in Computer Graphics are using impulse-based formulations for energy preservation.

Let me address a few inconsistencies on the rebuttal:

"To clarify, we do not attribute the independence from mesh quality solely to the implicit neural representation. In fact, there exist many methods that do not incorporate the Closest Point Method face challenges related to mesh quality, and our goal is to improve these approaches."

There are no examples that support the claim that this paper improves previous approaches. A simple solver (e.g. Covector Fluids) implemented on a GPU would yield much faster results. To properly compare against previous approaches, the authors should increase the resolution of a baseline solver until it reaches a performance that is comparable to the proposed approach. I bet that a common solver would require so many variables to be that slow (on a GPU) that the application would crash with out of memory issues before reaching that point.

"The paper referred here is mainly designed to deal with the numerical dissaption caused by the adveciton and projection. But none of them are designed for the surface flow (even not with a surface flow demo). While their methods could theoretically be extended to surface flows, the additional operators they propose would require careful adaptation and design for surface-specific contexts, which is beyond the scope of this work."

Covector fluids would work on a surface mesh, they have 2D examples. That's the beauty of DEC: it really abstracts the domain representation, one has only to implement the exterior derivative properly (which is easy). I suppose that Covector Fluids did not show examples on a surface because of the limited application of such flows.

"Our approach requires the use of an additional tool based on the Closest Point Method (CPM), as described in [7], to theoretically identify the stream function and other necessary differential forms. "

CPM is not necessary to identify the streamfunction as a differential form. CPM is a way of discretizing variables, it has nothing to do with the divergence-free property or with the identification of differential forms.

"In contrast to Chorin's operator splitting scheme, our approach guarantees divergence-free behavior in a functional manner. In comparison, the advection-projection method introduces numerical diffusion, as noted in prior works (Chang et al., 2002 [8]; Elcott et al., 2007 [5]). While we employ a first-order optimizer for the non-linear problem, which may result in numerical inaccuracies, our method is still a better alternative for the classic ones both theoretically (as a form of Lie advection for divergence free field) and empirically (in the energy presevation studies)."

You are referencing papers that are 20 years old to say that operator splitting introduces dissipation and therefore it wont work? What about all the other impulse-based formulations that were recently published? Are they all dissipative and inefficient as well? Take some time to reevaluate your perspective here.

We deeply appreciate the reviewer's enthusiastic and high evaluation of our work, as reflected in the rare 'Strong Accept' rating. Your thoughtful and detailed feedback has greatly encouraged us, and we are genuinely grateful for your recognition. Your comments on the paper's writing are especially valuable and will greatly assist us in presenting our work more effectively.

Q1: 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.

R1: We have include both figures in the revised version of the paper in Sec. 3.2 and 4.1.

Q2: 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.

R2: We will incorporate these refinements in the revised manuscript as follows:

1. Line 199: Add the description form of the divergence function.
2. Line 299: Include the reference image.
3. Line 234: Add a description of the vorticity.
4. Line 320: Include a reference image for the covariant derivative.
5. Line 327: Relate the inner product to the first order approximation.
6. Line 344: Provide an explanation for harmonic component modeling and refer to Appendix F.1 for further discussion.

We sincerely appreciate your efforts to help us improve the clarity and presentation of our work.

Q3: 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.

R3: We believe NFFS (Neural Functional Flow on Surface) is sufficient and we will include it in our revised manuscript.

Q4: Why introducing $f$ in equation (8) instead of using $\omega$ directly?

R4: Yes, I think we can directly use $\omega$.

Thank you for the reply. I appreciate the efforts to include the presentation changes. Following reviewer Y5ci, I also respectfully disagree with reviewer V8TK for the same reasons. This paper is not only a sound technical advance, but also a very good example of proper math writing in a machine learning paper, which is somewhat difficult to find among the numerous submissions. The insights about links between exterior calculus and riemannian geometry may also be very valuable for graduate students and researchers in numerous contexts. I will champion this paper unless a major flaw is identified afterwards.

We sincerely thank the reviewer for the enthusiastic and exceptionally positive evaluation of our work. Your insightful and detailed feedback has been deeply encouraging, and we are truly grateful for your recognition and support. Your comments are highly constructive and will help us further clarify the contributions of our work.

Q1: Clarfications on Performance vs. Storage vs. Accuracy.

R1: It is correct. The trade-off is effectively described in INSR-PDE, highlighting the advantages of compact neural representations. We will clarify this point in our contributions by explicitly mentioning the power of neural networks (as also referenced in Chen et al. [1]). We believe it is important to report this aspect and improvement though it is attributed to compact neural represnetation, as surface scenarios have not yet been simulated using the implicit neural representations (INR).

Q2: Clarfications on First to Simulate on Neural Implicit Surface

R2: That is correct. First, we will restate our contribution as: the first study to present simulation results of incompressible fluid flow on neural implicit surfaces with a guarantee of divergence-free behavior.

Additionally, we have included more results in Appendix E.3 to further illustrate the effectiveness of our divergence-free functional. These results also demonstrate its utility in improving the surface sampling method for handling incompressibility on the analytic surface. However, for non-analytic surfaces, designing a divergence-free functional without CPM is non-trivial. Consequently, the surface sampling method would fail without a proper divergence-free design by CPM similar as the failure in analytic surfaces as shown in the Appendix E.3.

Q3: Use of DEC Language

R3: Thanks for your advice. We follow the suggestion of Reviewer dCvp and add more illustrations to make it more intuitive for broader readers.

Q4: Handling Narrow Geometric Features in CPM

R4: Yes, narrow or thin features can introduce ambiguity. It will result in inefficiency, inaccuracy, and energy dissipation due to increased difficulties in convergence since the network will be misleaded by the non-unique closet point mapping. To mitigate this issue, as mentioned in Appendix F.1, we can follow the approaches of (King et al. 2023 [2], Marz and Macdonald 2012 [3]) and sample the neighborhood within a smaller tube radius (closer to the surface). In parctice, we can determine the sampling distance threshold to the surface as less than $\Delta x$ in the equation in Section 8, guided by the estimation of the reach distance (Aamari et al. 2019 [4]). This estimation can be locally constructed using a simple mesh extractor from the signed distance function (SDF), if no mesh is available, to detect thin regions roughly and adaptively adjust the sampling distance. This adaptive approach will improve efficiency and convergence and we will include more details about adaptations in Appendix F.1.

Q5: Missing Citations

R5: Thank you for pointing this out. We will include it in the related work section under the part Physical Simulation based on Neural Network.

Q6: Clarifying Performance Gains Over INSR

R6: Actually, we believe the improvement is primarily attributed to the Helmholtz decomposition. In the comparison between INSR and our method on the sphere jet, INSR is implemented using spherical coordinates with a fixed radius. We adopt two parameters ($\theta$, $\phi$) as input to the network, enabling the analytical verification of surface divergence in spherical coordinates without relying on CPM. Alternatively, this could also be imagined as maintaining a perfect CPM in 3D, where the value at each 3D point is taken as the projection onto the sphere. However, even under the idealized conditions, the results still exhibit significant error. If an inperfect CPM is included, additional errors in operator estimation would further worsen the results. Furthermore, in Appendix E.3, we adapt our divergence-free parameterization design to a surface-sampling Eigen-Net (using Spherical Harmonics to estimate the Laplacian-Beltrami operator analytically), and the results show improvement.

Nevertheless, we believe that CPM remains a critical component for handling arbitrary surfaces (especially non-analytical surfaces). It not only serves as a theoretical tool for constructing continuous differential forms to ensure divergence-free properties on arbitrary surfaces but also as a practical approach to enhances sampling efficiency by enabling uniform sampling in the ambient space, rather than directly on the iso-surface.

We thank the reviewer for the careful review and constructive advice. To address your questions and concerns:

Q1: The English in the current version of the paper needs to be improved.

R1: For any writing issues are present, we assure you that we will conduct a thorough proof reading and optimize the structure and descriptions throughout the paper.

Q2: Compared 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.

R2: Thank you for your valuable advice. We include more detailed preliminaries to assist readers who may not be familiar with differential geometry. For CP-EC and divergence-free design part, the main content introduces only the necessary concepts and notion, with the proofs provided in Appendix B. We will stress the point, allowing readers with a foundation in the topic to skip the preliminaries if desired.

Regarding the advection part, we believe it is essential to explain how Lie advection works, as this understanding is crucial for interpreting our loss design. We will refine the presentation to enhance clarity and ensure it aligns with the quality and precision of the referenced paper.

Q3: How do you do the interpolation of the pulled forms? In the CP-EC poster the authors recommended the Cubic Lagrangian. How does this interpolation affect the divergence-free property of the velocity field? Were there any numerical problems? Is it possible to do an ablation study on this point?

R3: Actually, interpolation for the pull forms is not required in our approach. In the classical CP-EC method, interpolation is necessary because the functional is represented on grid points. However, in our method, we utilize a continuous network representation, where interpolation can be considered included. The numerical issues encountered in our approach might be caused by inadequate fitting rather than interpolation errors.

Additionally, to elaborate further, if the classical method were used with interplation for CP-EC, divergence-free behavior would only be approximately preserved, with errors arising from the discretized operator computation.