PEINR: A Physics-enhanced Implicit Neural Representation for High-Fidelity Flow Field Reconstruction

Thanks for your reviews. Here are our explanations about the weaknesses and problems.

1.Response to comments on transferability and generalization of PEINR

We appreciate your query regarding the generalization of our model. Since the physical phenomena in the flow field vary greatly from problem to problem, there is no truly infinite generalization. As demonstrated in our experiments (Fig. 4(a)-(f)), the three types of physical problems differ drastically in both macroscopic and microscopic aspects. At present, our model is not universally generalizable to all problems. However, with a short period of transfer learning, we can quickly adapt the model to converge across different numerical schemes for the same problem, as shown in Figure2(https://drive.google.com/file/d/1Ghv7-cWKf2DQUEt-5O3JUdZfXVkIf3v-/view).

2.Response to comments on the input of PEINR

Our method PEINR is based on implicit neural representation, which takes spatial and temporal coordinates as inputs, not the images, as described in sec 1 line41.

3.Response to comments on the motivation of using INR

We appreciate your inquiry into the rationale for our high-fidelity reconstruction method compared to simple downsampling. Our method, PEINR, aims to reconstruct high-fidelity flow fields from low-fidelity data while simultaneously enhancing grid resolution and numerical accuracy. This approach not only enables efficient storage of high-fidelity data but also alleviates the significant computational resources required for high-fidelity simulations.

While the reviewer suggests directly downsampling high-resolution data and using a low-resolution solver, this approach is problematic because flow fields, unlike images, have inherent physical meanings. Such downsampling would irreversibly lose critical physical information, and the truncation errors of a low-resolution solver would accumulate and propagate over time, leading to deviations from the true solution in long-term predictions.

In contrast, PEINR only stores low-fidelity data and reconstructs high-fidelity details on demand, capturing small-scale flow structures without explicitly solving them. Moreover, since PEINR takes spatiotemporal coordinates as input, it can generate local flow regions on demand rather than the entire global field, significantly reducing computational overhead.

4.Response to comments on the relation to broader scientific literature

Our method, PEINR, aims to reconstruct high-fidelity flow fields from low-fidelity data while simultaneously improving grid resolution and numerical accuracy, rather than solely performing reconstruction or super-resolution.

Nevertheless, we appreciate the opportunity to discuss these works. Our CFD data is obtained from discrete numerical solvers, which inherently introduce numerical dissipation and truncation errors, deviating from the governing Navier-Stokes equations. Forcing a PINN to simultaneously fit the data while satisfying residual equations may lead to non-convergence or unphysical solutions. Instead, our approach combines data-driven learning with physics constraints to ensure physical consistency.

Regarding super-resolution, neural operators leverage function space mappings and efficient spectral-domain computations to overcome the fixed-grid limitations of traditional methods. However, when high-frequency energy decays rapidly, neural operators may struggle to retain fine details compared to INR. Additionally, INR supports on-demand generation of localized regions, such as boundary layers or vortex structures, without requiring full-field computation, making it more adaptable and efficient.

5.Response to comments on missing citations

Thank you for your valuable suggestions. We will incorporate discussions of these references in the revised version.

6.Response to comments on inverse problems definition

In our work, the inverse problem refers to reconstructing high-fidelity physical fields from sparse data[1]. We will clarify this definition in the revised version before discussing our proposed method.

[1] Chen, Yang, et al. "HOIN: High-Order Implicit Neural Representations." arXiv preprint arXiv:2404.14674 (2024).

7.Response to comments on ablation study

The ablation study of neighbor coordinate, RBF and PCA for time encoding, and spectral block is shown in Table2. PEINR$^1$ removes both spectral block and the physical encoding, while PEINR$^2$ only removes the physical encoding. As shown in Table 2, PEINR outperforms PEINR$^2$ in the metric DD in all cases, which indicates that our physical encoding method can preserve more physical peculiarity. The superiority of PEINR$^2$ compared with PEINR$^1$ highlights that the added spectral block can also help the model to perform better.

8.Response to comments on Figure 2

Thanks for your suggestion, we have enlarged Figure 2 to make reading easier, as shown in Figure 1 of the link(https://drive.google.com/file/d/1Ghv7-cWKf2DQUEt-5O3JUdZfXVkIf3v-/view).

Thanks for your reviews. Here are our explanations about the weaknesses and problems.

1.Response to comments on the missing citation CoordNet(Line 246)

Thanks for your suggestions and we will correct it in our revised paper. ResuMLP is proposed by CoordNet, and combines residual connections with MLPs, aiming to alleviate the vanishing gradient problem in deep MLP training and enhance the model's representational capacity.

2.Response to comments on temporal interpolation

By utilizing an INR framework, a neural network learns the implicit expression of flow field data, mapping input spatiotemporal positions to corresponding attribute values. This enables continuous modeling and representation of the flow field, allowing for time interpolation without the need for retraining.

3.Response to comments on the RBF-enhanced NTK

We appreciate the reviewer’s question regarding the integration of the RBF kernel with the MLP in our NTK framework. Below is a detailed explanation of the implementation.

The RBF kernel achieves translation invariance by measuring similarity between time points (based on relative distances rather than absolute positions), which is crucial for temporal modeling. In our implementation, the original 1D temporal input is first mapped to a high-dimensional space via the RBF kernel to generate a kernel matrix. Kernel PCA is then applied to extract principal components, yielding low-dimensional temporal features that serve as one input to the MLP.

The NTK of a standard MLP inherently lacks translation invariance. However, the high-dimensional features generated by the RBF kernel enhance the network’s sensitivity to input variations. This modification alters the NTK’s eigenvalues (and spectral distribution), ultimately improving the network’s ability to capture high-frequency information and its generalization performance.

The NTK can be directly computed as: $\gamma\left(x, x^{\prime}\right)=\nabla_\theta f(x,\theta)^{\top} \nabla_\theta f\left(x^{\prime},\theta\right)$, where $f(x,\theta)$ represents the output of MLP and $\theta$ represents parameters of MLP. In our framework, the MLP’s input combines spatial coordinates $\Psi(\vec C)$ with RBF-enhanced temporal features $\Phi(t)$, enabling the model to simultaneously resolve high-frequency spatiotemporal details and long-term dynamics. Our MLP input form is [$\Phi(t)$,$\Psi(\vec C)$]. By incorporating the RBF kernel’s stationary properties, the RBF-enhanced NTK becomes: $\gamma_{\mathrm{RBF}-\mathrm{NTK}} =\gamma_{M L P}\left([\Phi(t), \Psi(\vec{C})],\left[\Phi\left(t^{\prime}\right), \Psi\left(\vec{C}^{\prime}\right)\right]\right) =\gamma_{\mathrm{MLP}}\left(\Psi(\vec{C}), \Psi(\vec{C})^{\prime}\right) \cdot K_{\mathrm{RBF}}\left(t, t^{\prime}\right)$. The RBF-enhanced NTK is more stable during training and enables PEINR to simultaneously model high-frequency details and long-term dynamics, significantly improving the accuracy of flow field reconstruction.

4.Response to comments on syntax errors

We sincerely appreciate the reviewer's careful reading and valuable feedback on grammar and phrasing. We have thoroughly revised the manuscript to address these issues.

5.Response to comments on training time

We appreciate the reviewer's attention to the experimental details. To clarify, the "Training Time" reported in Table 1 refers specifically to the time required per training epoch (not the total training time). We acknowledge that this distinction was not explicitly stated in the original manuscript and will add a clear note in the revised version to avoid any confusion.

6.Response to comments on training timesteps

In this paper, the original CFD data were saved at every 20 physical computation steps, so the 20 steps tested actually correspond to the evolution of 400 computational steps. During this phase, the key physical processes have fully evolved, and this experimental configuration has fully validated the long-term stability of the proposed method.

7.Response to comments on average results of our experiments

We repeated this process 3-5 times and calculated the error bars for all models to compare their robustness on high-fidelity flow field enhancement, as shown in Table1(https://drive.google.com/file/d/1Ghv7-cWKf2DQUEt-5O3JUdZfXVkIf3v-/view). It can be seen that the overall performance of the proposed PEINR is better than other state-of-the-art baseline models, indicating the effectiveness of our approach.

Thanks for your reviews. Here are our explanations about the weaknesses and problems.

1.Response to comments on missing citations

[1] de Vito et al. (2024) Implicit Neural Representation For Accurate CFD Flow Field Prediction.
[2] Du et al. (2024) Conditional neural field latent diffusion model for generating spatiotemporal turbulence.

We will add the citations in the revised manuscript to ensure completeness. The following are our descriptions: Recent work by de Vito et al.[1] in implicit neural representation for accurate CFD flow field prediction demonstrates the potential of INR architectures for steady-state CFD simulations. However, their method primarily focuses on single-resolution flow fields with stationary boundary conditions and does not address the critical challenges of reconstructing unsteady, multi-resolution flow fields with spatiotemporal coupling – a key limitation our PEINR framework resolves through physical encoding and TransSTF. We acknowledge the CoNFiLD model [2] for synthesizing turbulence via neural field latent diffusion, whereas PEINR uniquely prioritizes physics-constrained reconstruction of multi-resolution flow fields through physical encoding and spatiotemporal attention mechanisms.

2.Response to comments on spatial localization encoding clarification

The spatial localization encoding in PEINR directly aligns with CFD’s stencil-based computation paradigm, where numerical solutions inherently depend on local neighborhood interactions (e.g., finite volume discretization). By embedding this locality into our spatial localization encoding, PEINR explicitly leverages CFD’s template computation logic to resolve fine-scale flow structures. In spatial localization encoding, we expand the input spatial coordinates to include not only the original coordinates $\vec{C}$ but also those of nearby neighboring points $\vec{C}_{neighbors}$. For instance, in a 2D case, we consider the nearest 4 points (up, down, left, right), transforming the input from a single coordinate tensor of shape (batch_size,2) to an augmented tensor of shape (batch_size,5,2). This design explicitly encodes the local spatial correlations required for solving discretized Navier-Stokes equations, where each grid point’s solution depends on its neighbors’ physical quantities.

3.Response to comments on clarification on grid-independence

We sincerely appreciate the reviewer's insightful feedback. We acknowledge that there might be a misunderstanding regarding our method's assumption of grid dependence/independence, which we aim to clarify in this response. In conventional implicit neural representations (INRs) for fluid super-resolution, it is typically assumed that the physical attributes (e.g., velocity, pressure) at a given spatial coordinate remain consistent across different grid resolutions (grid independence assumption). However, as the reviewer rightly noted, this assumption often fails in real-world flow simulations where grid resolution significantly affects the discretized field values due to numerical diffusion, truncation errors, and turbulence modeling. In contrast, our method explicitly addresses this limitation by abandoning the grid independence assumption. Instead, we propose a novel grid-conditioned implicit neural representation that encodes both spatialtemporal coordinates and grid-resolution-dependent features. This allows our model to learn how physical attributes vary with grid resolution, thereby capturing the inherent grid dependence observed in practical CFD simulations. In Figure1, we illustrate that baseline INR methods fail to generalize across resolutions due to their grid independence assumption, while our method achieves consistent accuracy by accounting for grid dependence.

4.Response to comments on PSNR result of Riemann problem

We appreciate your observation. The observed PSNR difference stems from PEINR's design prioritizing physical accuracy over numerical optimization for discontinuous flows. Riemann problems involve strong discontinuities (shocks, contact surfaces) where PEINR's localized spatial encoding intentionally preserves sharp gradients, making it more sensitive to errors in these regions. While this approach leads to slightly lower PSNR (a mean-squared-error metric favoring smoothness), it avoids the physical distortions visible in CoordNet's results (Figure 4(c)(d)) where large errors near discontinuities appear. We will clarify this rationale in the revised manuscript.

5.Response to comments on our dataset availability

We confirm that HFR-Bench, our large-scale CFD dataset, will be publicly available to support further research. The dataset contains 33,600 unsteady 2D and 3D flow fields (totaling 5.4 TB), covering diverse grid resolutions and numerical precisions for high-fidelity flow reconstruction.

Thanks for your reviews. Here are our explanations about the weaknesses and problems.

1.Response to comments on the single 3D dataset

The SV case is a classical benchmark in CFD that incorporates both shock waves and vortical structures, which can effectively validate our method's capability in capturing multi-scale, strongly nonlinear physical phenomena. While turbulent boundary layers were not directly tested, they share fundamental flow characteristics and field structures with the SV case, thereby supporting the extensibility of our method. The present study focuses on the fundamental validation of the method using canonical flow problems rather than comprehensive engineering applications. In future work, we plan to develop more test cases to further verify the method's applicability across multiple scenarios.

2.Response to comments on the experimental setting

In this paper, the original CFD data were saved at every 20 physical computation steps, so the 20 steps tested actually correspond to the evolution of 400 computational steps. We trained 20 steps(460-480) and tested 20 steps (480-500). During this phase, the key physical processes have fully evolved, and this experimental configuration has fully validated the long-term stability of the proposed method.

3.Response to comments on the difference from other methods (grid-free methods and PINNs)

In CFD, while regular/irregular grids rely on explicit geometric discretization, grid-free methods represent a novel numerical computation technique that can discretize the solution domain using nodes or particles, thereby eliminating the need for complex mesh generation processes.

PINNs are an extended application of implicit neural representations in scientific computing. In this study, CFD data are solved based on discrete meshes, which inherently introduce numerical dissipation and truncation errors, deviating from the Navier-Stokes equations governing fluid flow. Forcing PINNs to simultaneously fit this data and satisfy the equation residuals may lead to network convergence issues or generate non-physical solutions[1].

[1] Farea et al. Understanding physics-informed neural networks: techniques, applications, trends, and challenges.

4.Response to comments on the quantization of the impact of mesh non-uniformity

To clarify, our method avoids grid-resolution independence, not grid-type uniformity. While mesh non-uniformity can affect traditional solvers, PEINR’s implicit neural representation inherently decouples performance from both resolution and local cell shapes, as it operates on continuous coordinate inputs rather than discrete grid topology. Quantifying uniform-mesh comparisons would deviate from our core contribution.

5.Response to comments on missing citations

We appreciate the references you suggested. But we can only find this paper: “MeshGraphNets” and we will include this citation in our revision.

6.Response to comments on 6D Coordinates We appreciate the reviewer’s insightful question.

In studies involving coordinate expansion, the implicit mapping is typically from spatiotemporal coordinates to physical quantities, so the dataset we provide does not consider physical parameters or boundary conditions. Directly increasing the input dimensions by adding physical parameters or boundary conditions to form a 6D input can lead to several issues, such as semantic ambiguity and optimization conflicts. To address physical parameters or boundary conditions,we plan to conduct comprehensive tests based on hypernetwork in the future.

7.Response to comments on sigma

We thank the reviewer for highlighting the potential of adaptive σ in Gaussian encoding. Our fixed $\sigma=0.1$ was empirically chosen to balance spectral coverage for cases in HFR-Bench, but we agree that dynamic $\sigma$ could better handle multi-scale temporal dynamics.

We will explore $\sigma(t)$ schedules (e.g., curriculum learning from high to low σ) or physics-aware $\sigma(x,t)$ (e.g., linking σ to local vorticity magnitude).

8.Response to comments on uncertainty quantification

We repeated this process 3-5 times and calculated the error bars for all models to compare their robustness on high-fidelity flow field enhancement, as shown in Table1. https://drive.google.com/file/d/1Ghv7-cWKf2DQUEt-5O3JUdZfXVkIf3v-/view

9.Response to comments on the practical limit for numerical precision enhancement $B_{max}$

$B_{max}$ depends on the training data. The numerical precision of data is higher, then $B_{max}$ is higher.

10.Response to comments on transferability of PEINR

Since the physical phenomena in the flow field vary greatly from problem to problem, there is no truly infinite generalization. Our model is not generalizable to all problems now, but with transfer learning, we can quickly adapt the model to converge across different numerical schemes for the same problem, as shown in Fig2 https://drive.google.com/file/d/1Ghv7-cWKf2DQUEt-5O3JUdZfXVkIf3v-/view.