Compositional Generative Multiphysics and Multi-component Simulation

For reference [1], this article employs the JFNK method to solve large systems of equations, which is a full-coupling strategy and a numerical technique for addressing multiphysics coupling problems. However, the full-coupling approach is not suitable for all scenarios, particularly when two physical processes require different numerical methods for their solutions. In addition, the cost of JFNK for solving complex problems is high.

For reference [2], as mentioned in the manuscript, this article establishes a surrogate model for the temperature field and then integrates it with other CFD solvers to accelerate multiphysics simulation. However the issue discussed is only a unidirectional coupling problem, while bidirectional coupling may introduce additional challenges. In contrast, our algorithm is not limited in this regard, and Experiment 2 encompasses a variety of scenarios that multiphysics simulation might encounter.

For reference [3,4,5,6], these articles are actually focused on the quantification of uncertainty in multiple numerical programs, which is entirely different from our work. The motivation is that numerical programs may have inaccurate internal parameters in new scenarios. To address this issue, the article calibrates the internal parameters of the program with new experimental data to ensure that numerical predictions match observed data.

As you mentioned, researcher Karen Willcox from UT-Austin is a highly influential scholar in her field. Her research focuses on model reduction, multifidelity methods, digital twins, and uncertainty quantification. The theme closest to this article is model reduction, which can also be understood as the use of surrogate models. In our manuscript, we use the surrogate model as a baseline for comparison and our method demonstrates superior performance. Researcher Youssef Marzouk from MIT is also a highly influential scholar in his field. His research focuses on uncertainty quantification, inverse problems, and Bayesian statistics. He has done extensive work using Bayesian methods in inverse problems and uncertainty analysis. In the field of multiphysics, he uses Bayesian methods to identify the most critical factors in coupled multi-physics systems. However, our research is actually concerned with forward simulation problems.

We hope the explanation above has answered all of the questions.

[1] Knoll, Dana A., and David E. Keyes. "Jacobian-free Newton–Krylov methods: a survey of approaches and applications." Journal of Computational Physics 193.2 (2004): 357-397.

[2] El Haber, George, et al. "Deep learning model to assist multiphysics conjugate problems." Physics of Fluids 34.1 (2022).

[3] Kennedy, Marc C., and Anthony O'Hagan. "Bayesian calibration of computer models." Journal of the Royal Statistical Society: Series B (Statistical Methodology) 63.3 (2001): 425-464.

[4] Friedman, Samuel, and Douglas Allaire. "Quantifying Model Discrepancy in Coupled Multi-Physics Systems." International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Vol. 50077. American Society of Mechanical Engineers, 2016.

[5] Willmann, Harald, et al. "Bayesian calibration of coupled computational mechanics models under uncertainty based on interface deformation." Advanced modeling and simulation in engineering sciences 9.1 (2022): 24.

[6] Subramanian, Abhinav, and Sankaran Mahadevan. "Error estimation in coupled multi-physics models." Journal of Computational Physics 395 (2019): 19-37.

We appreciate you pointing out the research field of Bayesian calibration, which could be beneficial for our future research. We have read through all the articles you have recommended, but upon thorough analysis, we find that these Bayesian-based methods primarily focus on analyzing well-established multi-physics simulation systems, including uncertainty analysis, error calibration, and decoupling approximations of coupled systems.

However, the motivation behind our method when applied to multiphysics simulation is that constructing coupled programs for complex problems can be exceedingly intricate and computationally inefficient. Our aim is to directly predict coupled solutions through decoupled programs, thereby eliminating the need for coupled program construction. For multi-component simulation, directly simulating the overall structure requires high computational cost and may encounter difficulties in convergence due to the increase in degrees of freedom. Our aim is to predict the large structure through small structure data. In the fields of engineering and scientific research, virtually all physicochemical processes involve the combined effects of multiple physical processes. Multi-component is common in large complex systems such as aircraft, buildings, and nuclear power plants, as well as in systems with highly intricate internal structures like chips and batteries. Rapid and accurate solutions to multiphysics and multi-component problems can lead to a better understanding of the physical behavior of systems, thereby enhancing their economic and reliability performance.

Thus, we believe our method is fundamentally different from the work described in these articles. We use decoupled training data to predict coupled solutions, and small structure data to predict large structures. This method shows great significance in science and engineering, representing a novel contribution.

The detailed analysis of the articles you mentioned is in the next response.

Dear Reviewer 6eU9,

Thank you for your time and effort in reviewing our work. We have carefully considered your detailed comments and questions, and we have tried to address all your concerns accordingly.

As the deadline for author-reviewer discussions is approaching, could you please go over our responses? If you find our responses satisfactory, we hope you could consider adjusting your initial rating. Please feel free to share any additional comments you may have.

Thank you!

Authors

We thank the reviewer for the constructive review, and are glad that the reviewer recognizes the novelty and generality of our method, the practicality of the dataset. In the following, we address the points the reviewer raised about, and answer the questions.

Re1: The multi-level “for loops” can cause a significant computational bottleneck, limiting the practical applicability of the method. While the author briefly acknowledged this limitation, they did not provide any metrics on the method’s computational efficiency, particularly in comparison to surrogates, which are a key consideration for physical simulations and their surrogates.

Answer: We greatly appreciate your feedback. Efficiency is indeed very important. Firstly, for the outmost loop that go over $k=1,2,...K$, we find that typically $K=2$ or $3$ is enough, and set $K=2$ for multiphysics simulation and $K=3$ for multi-component simulation. The second loop is for the denoising steps for diffusion models. To enhance efficiency, we have implemented DDIM for accelerated sampling in appendix H and conducted a comparative analysis of efficiency in appendix I. In the more complex Experiments 2 and 3, our method achieves 29 times and 41 times acceleration compared to numerical programs. When compared to the surrogate model, our algorithm takes approximately 2-3 times longer to run for multiphysics problems. For multi-component problems, our algorithm is more efficient, with a running time of about half that of the surrogate model. The results are shown in Table 20. Additionally, we would like to clarify that the use of multi-level "for loops" is primarily aimed at enhancing accuracy. In scenarios where efficiency is prioritized, the outmost loop can be omitted, and this adjustment does not significantly impact accuracy as shown in appendix F where we perform a hyperparameter study of $K$. Table 20 is also provided as follows, the time unit for each experiment is defined as the time required for a single neural network inference.

Re2: Although the authors noted the application of various compositional generative models in scientific domains in their related works, no compositional baselines were compared in the experiments.

Answer: Thank you for your comment. In fact, before starting on this work, we had considered using existing compositional generation methods. However, we found that these methods are not applicable to our problem. The most relevant studies to our issue are those by Du et al. [1] in the field of image generation and Wu et al. [2] who further extended the concept to inverse design problems in science. Their focus is on how a single object is influenced by multiple factors, such as generating images that meet various requirements in image generation or predicting the fluid fields and enhancing the lift-to-drag ratio under the influence of two wings in inverse design. However, our problem involves multiple objects, such as multiple physical processes and components, requiring the capture of interactions between these processes or components. Therefore, their methods are not suitable for our scenario.

[1]Du, Yilun, et al. "Reduce, reuse, recycle: Compositional generation with energy-based diffusion models and mcmc." ICML, 2023.

[2]Wu, Tailin, et al. "Compositional Generative Inverse Design." ICLR, 2024.

Dear Reviewer 1Pcu,

Thank you for your time and effort in reviewing our work. We have carefully considered your detailed comments and questions, and we have tried to address all your concerns accordingly.

As the deadline for author-reviewer discussions is approaching, could you please go over our responses? If you find our responses satisfactory, we hope you could consider adjusting your initial rating. Please feel free to share any additional comments you may have.

Thank you!

Authors

We thank the reviewer for the constructive review, and are glad that the reviewer recognizes the novelty of our method and the credibility of the experimental results. In the following, we address the points the reviewer raised about, and answer the questions.

Re1: In the comparative experiment, is the surrogate model too simple? Or can you compare your model with a more complex model? Or explain the current popularity of the surrogate model.

Answer: Firstly, we would like to clarify that in each experiment, for each network architecture, we have concurrently trained both diffusion models and surrogate models. Our primary objective is to compare the differences between our approach of multiphysics and multi-component simulation by composing diffusion models, and the conventional approach of learning surrogate models, rather than the differences between the network architectures themselves. In fact, both our approach and the conventional surrogate models approach can use suitable network architectures. Secondly, for the same network structure, both diffusion models and surrogate models employ the same network hyperparameters and training methods, thus their complexity is nearly identical. Furthermore, the FNO and U-Net network structures used in this manuscript are common baseline models in the neural PDE tasks, as demonstrated in studies by Georg Kohl et al. [1], Wang et al. [2], and Xiong et al. [3], among others. The Transolver used in Experiment 3, although newly proposed, is essentially a Transformer, an architecture that has been widely applied to a variety of tasks.

[1] Kohl, Georg, Liwei Chen, and Nils Thuerey. "Benchmarking autoregressive conditional diffusion models for turbulent flow simulation." ICML 2024 AI for Science Workshop. 2024.

[2] Wang, Haixin, et al. "Recent advances on machine learning for computational fluid dynamics: A survey." arXiv preprint arXiv:2408.12171 (2024).

[3] Xiong, Wei, et al. "Koopman neural operator as a mesh-free solver of non-linear partial differential equations." Journal of Computational Physics (2024): 113194.

Re2: I would like to know whether the model you designed has other innovations in structure and implementation compared with the diffusion model, in addition to the differences in parameter settings and application issues. I feel that the model is not deep enough, judging from the algorithms shown in the first six pages.

Answer: Our primary contribution lies in addressing multiphysics and multi-component simulation problems from a probabilistic generation perspective. In the fields of engineering and scientific research, virtually all physicochemical processes involve the combined effects of multiple physical processes. Multi-component is common in large complex systems such as aircraft, buildings, and nuclear power plants, as well as in systems with highly intricate internal structures like chips and batteries. Rapid and accurate solutions to multiphysics and multi-component problems can lead to a better understanding of the physical behavior of systems, thereby enhancing their economic and reliability performance. We have theoretically deduced the feasibility of our approach and validated its effectiveness with examples from engineering. We employ diffusion models to tackle these issues, making certain improvements to tailor them to our specific problems. In theory, other types of generative models, such as flow-based models, could also be utilized.

Re3: I also found that your experiments often combine your model with other NNs (FNO, etc.). Why don't you use your model to implement it independently? Because I am worried that NNs such as FNO will additionally correct the errors of your model (if they occur).

Answer: We would like to clarify that our contribution does not lie in the proposal of a new network architecture, but rather in the application of diffusion models from a probabilistic generation perspective to solve multiphysics and multi-component simulation problems. In addition, the diffusion model is a learning paradigm that can be implemented using various network architectures such as U-Net, FNO, and transformer. Our approach is theoretically versatile and can be applied to various network architectures, including U-Net and FNO, as well as various generative models such as diffusion models and flow-base models.

Re4: There are some unnecessary blank lines in the article that can be corrected.

Answer: Thank you for pointing out this issue. We have adjusted the layout to reduce the presence of blank lines.

We thank the reviewer for the constructive review, and are glad that the reviewer recognizes the novelty of our method. In the following, we address the points the reviewer raised about, and answer the questions.

Re1: The algorithm lacks clarity, and the model structure is not sufficiently detailed.

Answer: Our algorithm builds on top of diffusion models to make it suitable for multiphysics and multi-component simulations. It can combine with any network architecture as its denoising network, so long as the network architecture fits the data structure. The network architecture we have chosen includes widely used U-Net and FNO in their original form, and Geo-FNO and Transolver in the multi-component experiment. The parameter details of these network architectures are provided in the Appendix B, C, and D in the manuscript.

Re2: The iterative nature of MultiSimDiff, especially in multiphysics simulations, requires multiple diffusion steps for each field, which may lead to slow inference times. This constraint limits its practicality for scenarios requiring rapid predictions. While the authors recognize this issue and propose exploring faster sampling methods in future work, an initial investigation into such techniques within this paper could enhance its contribution.

Answer: Thank you for the suggestions. We have employed DDIM for accelerated sampling, the result is shown in appendx H. Compared to the original DDPM, it can achieve a 10-fold and 5-fold acceleration for multi-physics and multi-component problems, respectively, while maintaining accuracy. Additionally, we compared it with numerical programs in Table 20. In Experiment 2 and Experiment 3, we achieved accelerations of 29 and 41 times, respectively. The more complex the problem, the higher the efficiency of our algorithm compared to numerical programs. Table 20 is also provided as follows, the time unit for each experiment is defined as the time required for a single neural network inference.

Re3: For algorithm 1, it is identical to the alogrithm 2 in the DDPM paper [1]. And also the way identify $z_i$ is confusion when apply the third for loop.

Answer: Thank you for the question. Our primary contribution lies in addressing multiphysics and multi-component simulation problems from a probabilistic generation perspective instead of the improvement of diffusion model. We have theoretically deduced the feasibility of our approach and validated its effectiveness with examples from engineering. We employ diffusion models to tackle these issues, building on top of them to address our specific problems. In theory, other types of generative models, such as flow-based models, could also be utilized.

As for $z_i$, the $z_i$ in the third loop of Algorithm 1 represents the i-th physical fields with noise at the current diffusion step. The objective is to denoise it to obtain the desired coupled solution, which is similar to DDPM's denoising of images.

Re4: How time in involved in the alogrithm as you are doing time dependent system?

Answer: Both Experiment 1 and Experiment 2 deal with time-dependent systems. In these experiments, we treat time as an additional dimension, and generate the state trajectory across all time simultaneously. Consequently, for Experiment 1, which is a one-dimensional spatial-temporal problem, we employ a two-dimensional network; for Experiment 2, which is a two-dimensional spatial-temporal problem, we utilize a three-dimensional network.

We thank the reviewer for the constructive review, and are glad that the reviewer recognizes the novelty of our method. In the following, we address the points the reviewer raised about, and answer the questions.

Re1: claim (2): There are several datasets in the literature that can be described as simulations of multiphysics, multi-component, such as PDEBench [Takamoto et al., 2022]. The authors should better precise why the new datasets they propose is a contribution to the literature.

Answer: Thanks for your comments. Multiphysics simulation refers to the simultaneous consideration and modeling of multiple physical processes, such as heat conduction, fluid flow, and structural mechanics, each of which may involve one or multiple physical fields. Therefore, multiphysics does not simply mean multiple physical fields, i.e., there are many scenarios that involve several physical fields, but only one physical process. In the "PDEBench" [1], although a series of partial differential equation (PDE) cases are provided, including convection, diffusion reaction, and Navier-Stokes equations, there are no cases of multiphysics simulation, such as coupling the reaction-diffusion equation with the Navier-Stokes equations, which is very common in combustion dynamics. In this manuscript, Experiment 1 is to calculate the reaction-diffusion equation, which does exist in the literature [1]. But strictly speaking, this does not constitute a multiphysics problem because they all belong to the concentration field; it is merely for validating the proposed algorithm. We explain this in the updated manuscript. Experiment 2 is a simplified version of the nuclear thermal coupling problem in the nuclear engineering field, encompassing three physical processes, with the fluid field solved using the finite volume method and the other two fields solved using the finite element method. This problem covers most coupling types: strong and weak coupling between physical processes, regional and interface coupling, and unidirectional and bidirectional coupling, making it a very representative multiphysics simulation case. To our knowledge, such cases have not been found in literature [1] or other literature.

Multi-component simulation refers to the simulation of complex structures composed of multiple similar components, which is very common in civil, aerospace, and nuclear engineering fields. Multi-component simulation typically requires that in the inference time, it can due with larger structures than previously seen. For example, the reactor core typically consists of hundreds or thousands of fuel elements arranged in a square or hexagonal pattern. In fuel cells, the repeated array of ribs directly affects the cell's performance. The case domains in the PDEBench [1] are all on regular rectangular domains with relatively simple geometric structures and no interaction between multiple structural components. On the other hand, Experiment 3 in this manuscript solves the thermal and mechanical properties of multiple prismatic fuel elements, with mechanical and thermal interactions between adjacent fuel elements. Our predicted structure is more complex than in training, involving a large structure of 64 elements, each containing 804 grid points. Therefore, we consider this problem to be a typical case of multi-component problems with a certain level of complexity.

We add appendix K and provide Table 21 to outline the principal characteristics of these datasets and compare them with Reference [1]. The number of physics processes and the number of components are all 1 in Reference [1]. The table is also provided in next responce.

[1] Takamoto, Makoto, et al. "PDEBench: An extensive benchmark for scientific machine learning." Advances in Neural Information Processing Systems 35 (2022): 1596-1611.

Thus, we believe that our datasets in Exp2 and Exp3 constitute a contribution to the literature, which consists of multiphysics and multi-component simulations not present in previous benchmarks.

Re2: "a process we have mathematically proven", I don't think a "process" qualifies as being able to be "mathematically proven". You should precise what is "mathematically proven"; "This reverse diffusion process is also mathematically validated", same remark

Answer: Thank you for pointing out the inaccuracies in our previous statement. Indeed, a process should not be described as "be proven". What we want to express here is that we have mathematically derived the principles why our algorithm can obtain coupled solutions and large structure solutions. This has been corrected in the new manuscript.

Re3: "there do not exist utilized machine learning methods for multi-component simulation". I don't really understand the novelty since, technically, UNets [Ronneberger et al. 2015], FNO [Li et al. 2022], Transformer [Mccabe et al. 2023], diffusion [Kohl et al. 2024] models are already used on multi-component simulations.

Answer: In our responses to the above comments Re1, we have provided extensive explanations on the multi-component issue. In short, multi-component simulation refers to the simulation of complex structures composed of multiple similar components. Multi-component simulation typically requires that in the inference time, it can due with larger structures than previously seen. As was discussed in the response to the above Re1, previous benchmarks do not possess multi-component datasets, and the methods you mentioned were not specifically designed for multi-component simulations, yet some of them may be suitable baselines. Here, we provide a detailed discussion on the mentioned methods about their applicability to multi-component simulations:

• For the U-Net [1], although it has been applied to many learning physical simulation works before, it requires the data to form a regular grid. In contrast, multi-component problems typically have non-rectangular grids. Thus, U-Net is in general not applicable to multi-component problems.
• For the FNO [2], due to its FFT and IFFT being global operations in the full domain, it requires that the training and inference have the same domain structure. In contrast, multi-component simulation typically requires to simulate larger structures than seen in training, which FNO cannot handle.
• For the Transformer [3], it has the potential to address multi-component simulations, since it allows to handle larger input length during inference. The standard transformer [3,7] without graph structure embedding cannot handle multi-component simulation well, since they does not consider the complex graphical interactions between the components. On the other hand, graph transformers, such as SAN [5] which utilizes the eigenvectors as position embeddings, has the potential to address multi-component simulations, which we have added as a baseline in the revised manuscript.
• For the diffusion [4], the three tasks addressed in this article are: Incompressible Wake Flow, Transonic Cylinder Flow, and Isotropic Turbulence. The first two tasks simulate the flow of fluid around a cylinder in a rectangular domain, while the third task involves turbulence within a cuboid, with the computational domain being a 2D slice from a 3D dataset. These tasks are not related to multi-component problems. However, if the simulation involves a complex phenomenon such as fluid flowing around a bundle of rods which is a multi-component problem, our method might be applicable for solving it.

We have also modified the phrasing of this sentence in the Related Work to make it more accurate. We have also added the baselines of Graph Transformer SAN [5] and Graph Neural Network GIN [6] in Experiment 3 to compare the performance. Due to the uniformity of graph structures in all training data and the fact that SAN learns a global relationship, SAN fails to predict larger structures. We see that our method that is specifically designed for multi-component simulation, significantly outperforms the baselines. Part of Table 3 is also provided as follows (the unit is $1\times 10^{-2}$):

[1]Ronneberger, et al. "U-net: Convolutional networks for biomedical image segmentation." 2015.

[2]Wen, Gege, Z Li, et al. "U-FNO—An enhanced Fourier neural operator-based deep-learning model for multiphase flow." 2022.

[3]Alwahas, Areej, Kasper Johansen, and Matthew McCabe. "Crop Type Mapping Using Self-supervised Transformer with Energy-based Graph Optimization in Data-Poor Regions." 2023.

[4]Kohl, Georgi, et al. "Benchmarking autoregressive conditional diffusion models for turbulent flow simulation." 2024.

[5] Kreuzer, Devin, et al. "Rethinking graph transformers with spectral attention." Advances in Neural Information Processing Systems 34 (2021): 21618-21629.

[6] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In International Conference on Learning Representations, 2019.

[7] Vaswani, A. "Attention is all you need." Advances in Neural Information Processing Systems (2017).

Re7: As your mention, your method seems computationally intensive compared to a FNO for example, not just because of the denoising steps but also because of the loop over the physical fields. Do you have a rough estimation of how it compares in terms of execution time and in terms of FLOPs?

Answer: First, we would like to clarify that our algorithm is applicable to various network architectures, including FNO and U-Net. Therefore, the high computational cost you mentioned likely refers to the greater computational cost of U-Net compared to FNO. For a specific network architecture, we have trained both diffusion models and surrogate models with identical network parameter settings. This means that the time complexity for a single prediction by the neural network is almost the same for both the surrogate model and the diffusion model. However, the main difference between the two lies in the fact that the diffusion model requires multiple denoising steps to reach the final solution, while the surrogate model needs to iterate with other physical processes or components until the solutions on each physical process or component converge. We have added a comparative analysis of time complexity and computational time in appendix I. Compared to numerical programs, our method achieves 29 times and 41 times acceleration in experiments 2 and 3, respectively. When compared to the surrogate model, our algorithm takes approximately 2-3 times longer to run for multiphysics problems. For multi-component problems, our algorithm is more efficient, with a running time of about half that of the surrogate model.

Similarly, the number of floating-point operations (FLOPs) required for a single neural network prediction is almost identical for both the surrogate model and the diffusion model. However, due to the difficulty in calculating FLOPs in numerical procedures, we have only conducted a comparison of runtimes. Table 20 is also provided as follows, the time unit for each experiment is defined as the time required for a single neural network inference.

Re8: In algorithms 1 and 2, could you clarify what the "outer inputs" are in comparison to the "physical fields"?

Answer: Thank you for pointing out this issue. We are sorry that we didn't provide a clear definition before, but now we have updated it in the form of a footnote 1 in the manuscript. In this manuscript, "outer inputs" refers to the inputs of the physical system, physical fields are what we need to solve. For example, in experiment 1, they are the initial conditions; in experiment 2, they are the variations of boundary neutron flux density over time; and in experiment 3, they are the heat flux density.

We thank the reviewer for the constructive review, and are glad that the reviewer recognizes the novelty and effectiveness of our method. In the following, we address the points the reviewer raised about, and answer the questions.

Re1: Only accuracy is compared in the examples. Efficiency comparison is also important. Specifically, a fair comparison with standard numerical methods (such as FEM or FVM) is important.

Answer: Thank you for the suggestion. We have added the efficiency comparison in appendix I in the revised manuscript, the results are shown in Table 20 and also below. The time unit for each experiment is defined as the time required for a single neural network inference. The last column shows our method's speedup compared to the numerical program.

From the table, we see that because Experiment 1 is simple, the numerical program runs very quickly and does not achieve acceleration. However, Experiment 2 and Experiment 3 achieve accelerations of 29 and 41 times, respectively. The more complex the problem, the higher the efficiency of our algorithm compared to numerical programs.

Re2: The iterative nature of the diffusion process in MultiSimDiff can be computationally intensive, particularly for multiphysics simulations due to the high complexity.

Answer: We have employed DDIM for accelerated sampling, and the results are shown in appendix H. We also provide part of the Tables below. We see that compared to the original DDPM, DDIM sampling can achieve a 10-fold and 5-fold acceleration for multi-physics and multi-component problems, respectively, while maintaining accuracy. For DDPM, the diffusion step of all experiments is 250.

Re5: How does MultiSimDiff handle cases where the decoupled training data does not closely match the dynamics of coupled data? Would the model performance degrade significantly?

Answer: In fact, the training data used in experiments 1 and 2 does not closely resemble the coupled dynamics, but our algorithm can correctly predict the coupled solution. Details can be found in the above response to Re3 and the appendix G in the revised manuscript.

Re6: Could you clarify if there are specific scenarios where traditional numerical solvers might still outperform MultiSimDiff in terms of accuracy or computational cost?

Answer: Firstly, in terms of accuracy, since we currently regard the numerical program's solution as the ground truth, the accuracy can only be as close as possible to the numerical program. Regarding efficiency, for very simple problems, such as the solution of the reaction-diffusion equation in experiment 1, our algorithm does not hold an advantage. However, for slightly more complex cases, like experiments 2 and 3, our algorithm is capable of achieving significant acceleration. The more complex the problem, the higher the efficiency of our algorithm compared to numerical programs.

Thank you for your time and effort in reviewing our work. We have carefully considered your detailed comments and questions, and we have tried to address all your concerns accordingly. As the deadline for author-reviewer discussions is approaching, could you please go over our responses? If you find our responses satisfactory, we hope you could consider adjusting your initial rating. Please feel free to share any additional comments you may have.

Thank you!

Authors

Thank you for your time and effort in reviewing our work. We have carefully considered your detailed comments and questions, and we have tried to address all your concerns accordingly. As the deadline for author-reviewer discussions is approaching, could you please go over our responses? If you find our responses satisfactory, we hope you could consider adjusting your initial rating. Please feel free to share any additional comments you may have.

Thank you!

Authors