PINP: Physics-Informed Neural Predictor with latent estimation of fluid flows

Weakness 1: Overclaimed title.

Reply: Excellent Remark! We have taken your suggestion into account. Finally, We have revised the title to better reflect the paper's contributions: "PINP: Physics-Informed Neural Predictor with Latent Field Estimator of Fluid Flows".

Weakness 2a: Accuracy at low resolution.

Reply: In our experiments, we used 3D scenes with a resolution of 64×64×64, which is not high. Under the same discretization method, our approach consistently outperformed all baselines.

We acknowledge that discretization introduces errors, which is why we designed a correction network to mitigate them. Physics-informed neural networks (PINNs) using automatic differentiation can achieve higher precision but are more suited to MLP-based designs, which can be less effective for comprehensive predictive tasks, particularly for complex or long-term predictions due to limited representation ability.

Weakness 2b: Necessity of predicting pressure fields and its ablation study.

Reply: The predicted pressure field serves not only as an auxiliary feature for predictions but also provides interpretability. Its similarity to actual distributions aids in understanding the dynamics, especially around obstacles or boundaries. In our paper, this constraint is written as:

$\boldsymbol{e_1}(\boldsymbol{x},t_k)=\Delta \boldsymbol{u}(\boldsymbol{x},t_k) + \left(-\boldsymbol{u}(\boldsymbol{x},t_k) \cdot \nabla \boldsymbol{u}(\boldsymbol{x},t_k) - \nabla p(\boldsymbol{x},t_k) + \text{Re}^{-1}\nabla^2 \boldsymbol{u}(\boldsymbol{x},t_k)\right) \Delta t.$

The constraint we introduce can actually be viewed as a temporal constraint on the velocity field ($\Delta \boldsymbol{u}( \boldsymbol{x},t_k)$ denotes the difference in velocity between consecutive frames). The inclusion of this temporal constraint helps stabilize the velocity field over time, which is crucial for long-term predictions.

Following your suggestion, we designed an additional ablation study by removing $\boldsymbol{e_1}$, referred to as no Velocity-Pressure Constraint. The results are summarized below in Table C. We have also revised the paper accordingly (Table 4, page 9, Section 5 and Appendix D.1, page 19).

Table C. Comparison of Evaluation Metrics Between Different Constraint Configurations and the Original Method.

In addtion, the result presented in Figure 9(a) reflects an average outcome across all 2D fluid test cases. The 40-step extrapolation limit made it challenging to illustrate the long-term impact of different components. To better illustrate the effectiveness of the Physical Constraint, we visualized the results on the Smoke2D dataset with 100-step extrapolation, which better highlights such an effectiveness. We have updated the paper accordingly (Figure 9(a), page 10, Section 5).

Weakness 2c: Necessity of the correction network.

Reply: Our predictions rely on the following equation:

$\hat{c}'(\boldsymbol{x},t_{k+1}) = c(\boldsymbol{x},t_k) + \left(-\boldsymbol{u}(\boldsymbol{x},t_k) \cdot \nabla c(\boldsymbol{x},t_k) + \text{Pe}^{-1} \nabla^2 c(\boldsymbol{x},t_k)\right) \Delta t.$

The use of second-order differences to approximate $\nabla$ introduces unavoidable errors. The correction network overcomes these limitations by minimizing these errors. Without it, the model struggles with long-term stability. The quantitative comparisons highlight the correction network's significance (Table 4, page 9, Section 5), with the results also shown in Table D.

Table D: Comparison of our model’s results with those obtained after removing the Correction Network on 2D smoke data.

Weakness 2d: How likely is the Temporal Constraint function to work?

Reply: Great question! In Appendix C.2, we have provided an analysis of the temporal loss variation during training. To further clarify the role of the Temporal Constraint, we have updated the presentation in Figure S5 (right) (page 18, Appendix C.2) of the revised paper to improve intuitiveness.

Here is a summary of the results:

• At the beginning of training, the temporal loss is non-zero.
• As training progresses, the probability of the temporal loss approaching zero increases. However, the temporal loss does not become consistently zero.

Hence, we can conclude that the Temporal Constraint works and plays an important role in improving the model's prediction accuracy.

Weakness 3: About the efficiency.

Reply: We agree that the inclusion of the temporal constraint and additional loss functions requires higher computational resources, particularly GPU memory. However, the impact is manageable and comparable to other advanced baselines. For example, in the 3D scenario, the GPU memory consumption of PINP is on par with Helmfluid as shown in Table E.

Table E: Comparison of GPU Memory Consumption Between Our Model and HelmFluid on 3D Smoke Data.

In summary, we appreciate your constructive comments and suggestions. Please let us know if you have any other questions. We look forward to your feedback!

Thank you for your insightful comments and suggestions, which are very helpful in improving the clarity of our paper!

Q1: The prediction blur issue in NowcastNet.

Reply: Excellent remark! NowcastNet can be divided into two components: the deterministic prediction model (Evolution Network) and the probabilistic generative model (GANs). The clarity of the predictions stems from the testing phase, where the "nearest" mode wrapping and the probabilistic generative model are applied. The experimental results for the Evolution Network alone and the complete NowcastNet are summarized below in Table A.

Table A: Comparison of NowCastNet results under different settings with Our Model

It is crucial to note that there are two main approaches for nowcasting precipitation prediction:

• Deterministic prediction models.
• Enhancing deterministic models with probabilistic generative models.

Since our model is deterministic (similar to the Evolution Network), we selected the Evolution Network's better results in our comparison. While incorporating a probabilistic model might improve visual results, it could negatively impact evaluation metrics. We believe presenting NowcastNet's results would not fairly represent the model's contributions.

The above additional explanations have been included in the main text (Line 429-431, page 8, Section 4.2), and the NowcastNet results are provided in the Appendix A.5 (page 15).

Q2: The issue of periodic oscillation in baseline results.

Reply: The periodic "shaking" phenomenon in some baseline model predictions is a form of periodic oscillation. Initially, this puzzled us as we did not find any bug in the implementation code. Upon further investigation, we discovered that the oscillation period (4 steps) matches the training rollout step size. Specifically, our training uses 4 frames to predict one frame unrolling over four steps, while the testing (inference) extends the rollout process to 40 steps. This periodic oscillation may arise because some models (except ours) struggle to learn motion dynamics within short sequence training.

To further investigate this phenomenon, we conducted three validation experiments as follows:

• Case 1: Training with four frames predicting one frame for four steps.
• Case 2: The same as Case 1 but with an adjusted batch size.
• Case 3: Training with four frames predicting one frame for six steps.

Our findings include:

• Experiments on Case 1 and Case 2 exhibited oscillations with a period of 4 throughout training (even at epoch 200).
• Experiment on Case 3 initially had oscillations with a period of 6, but these gradually diminished as training progressed.

This suggests that some baseline models struggle to capture long-term motion features when trained on short-term sequences. Under the configuration of Case 3, the oscillations in LSM were significantly reduced, yet our model still outperformed it, as shown in Table B.

Table B: Comparison of Results Between Our Method and LSM Under Experimental Setting Case 3.

The details of the above analysis and results are included in our paper (page 7, Section 4.1 and page 30, Appendix F).

Q3: Spatial generalization.

Reply: In our tests, the spatial generalization was demonstrated by adding/removing obstacles or altering their shapes—variations not present in the training data. The model's ability to generalize spatially arises from:

• The Signed Distance Field (SDF), which effectively captures spatial information.
• The explicit embedding of Navier-Stokes equations into the model and loss function, which enhances the learning of physical dynamics.

For instance, the model learns that higher pressure in front of obstacles constrains velocity, resulting in spatial generalization in fluid flows.

Dear Reviewer 3QEU,

Again, thanks for your constructive comments. We would like to follow up on our rebuttal to ensure that all concerns have been adequately addressed. If there are any further questions or points that need discussion, we will be happy to address them. Your feedback is invaluable in helping us improve our work, and we eagerly await your response.

Thank you very much for your time and consideration.

Best regards,

The Authors

Q2: Concerns About Over-Complicated Designs (Temporal Consistent Loss and Correction Networks).

Reply: Regarding the impact of the Correction Network and Temporal Constraint, we have clarified their contributions in Table B, where both modules show clear benefits for prediction accuracy. Below, we explain the motivations for these components in detail.

Motivation for the Temporal Constraint

The prediction task can be expressed as:

$

c(\boldsymbol{x},t_{k+1})=\int_{t_k}^{t_{k+1}} \dot{c}(\boldsymbol{x},t) dt + c(\boldsymbol{x},t_k) \tag{a}

$

Directly solving this integral is challenging. By leveraging the Lagrange Mean Value Theorem and assuming material concentration $c(\boldsymbol{x}, t)$ is continuous, there exists a moment $t' \in [t_k, t_{k+1}]$ such that:

$

{\int_{t_k}^{t_{k+1}} \dot{c}(\boldsymbol{x},t) dt = c(\boldsymbol{x},t') \Delta t, \ t' \in[t_k, t_{k+1}].\tag{b}}

$

Thus, the prediction can be reformulated as:

$

\begin{split} c(\boldsymbol{x},t_{k+1}) &= c(\boldsymbol{x},t_k) + c(\boldsymbol{x},t') \Delta t\ &= c(\boldsymbol{x},t_k)+\left(-\boldsymbol{u}(\boldsymbol{x},t')\cdot \nabla c(\boldsymbol{x},t') +\text{Pe}^{-1} \nabla^2 c(\boldsymbol{x},t')\right) \Delta t . \end{split} \tag{c}

$

While $\boldsymbol{u}(\boldsymbol{x}, t')$ can be modeled using a neural network, handling $c(\boldsymbol{x}, t')$ requires further consideration. We explored two approaches:

• Approximate $c(\boldsymbol{x}, t')$ with $c(\boldsymbol{x}, t_k)$.
• Use a neural network to map past concentrations to $c(\boldsymbol{x}, t')$.

The first approach performed well for short-term predictions but deteriorated rapidly in long-term extrapolation (Figure 9$c$, page 10, Section 5) due to accumulated errors in velocity fields (Figure S8, page21, Section D.3).

Therefore, we adopted the second approach, introducing Temporal Constraints to ensure physical meaning of $c(\boldsymbol{x}, t')$ and $\boldsymbol{u}(\boldsymbol{x}, t')$. Because in Equation $c),$c(\boldsymbol{x}, t')$is coupled with$\boldsymbol{u}(\boldsymbol{x}, t')$. Without these constraints,$c(\boldsymbol{x}, t')$would lose its physical meaning, which would in turn cause$\boldsymbol{u}(\boldsymbol{x}, t') to lose its physical validity (Figure S6, page 20, Section D.1), ultimately negatively impacting long-term predictions (Figure 9\(a, page 10, Section 5).

Motivation for the Correction Network

Equation $c$ approximates gradient operators with second-order differences, which introduces inherent errors. Without a Correction Network, these errors persist in the final prediction step, limiting the model's performance, particularly in long-term forecasting.

For instance, predictions results in manageable errors during short-term testing. However, when tested on long-term, error accumulation becomes severe. As illustrated in Figure 9(b) (page 10, Section 5), while both models perform similarly within 40 frames, the model with the Correction Network significantly outperforms in longer sequences. Figure S7 (page 21, Section D.2) shows visual comparisons, highlighting the Correction Network's importance.

Given the challenges of long-term sequence prediction, we conclude that the Correction Network is essential for mitigating accumulated errors.

In summary, we believe that the introduction of the Correction Network and Temporal Constraint is essential. The Temporal Constraint is a natural requirement of the method we propose, while the Correction Network is designed to overcome the model bottleneck caused by errors introduced by central difference methods in the experiments. The Temporal Constraint helps preserve physical validity, and together with the Correction Network, it improves the performance of long-term predictions.

Thank you very much for your great questions! Please let us know if you have any other questions. Looking forward to your response!

Q1: The Role of Pressure Fields in Assisting Concentration Prediction.

Reply: Thank you for this insightful comment. We believe that the pressure field $( p )$ aids concentration prediction by effectively constraining the velocity field $\boldsymbol{u}$, which, in turn, improves the prediction of concentration. We formalize this constraint as the Velocity-Pressure Constraint, defined by the following equation:

$\boldsymbol{e_1}(\boldsymbol{x},t_k)=\Delta \boldsymbol{u}(\boldsymbol{x},t_k) + \left(-\boldsymbol{u}(\boldsymbol{x},t_k) \cdot \nabla \boldsymbol{u}(\boldsymbol{x},t_k) - \nabla p(\boldsymbol{x},t_k) + \text{Re}^{-1}\nabla^2 \boldsymbol{u}(\boldsymbol{x},t_k)\right) \Delta t.$

In this equation, by introducing $H \times W + 1$ additional unknowns ( $p(\boldsymbol{x},t_k) \in \mathbb{R}^{H \times W}$ and $\text{Re} \in \mathbb{R}$ ), we impose $2 \times H \times W$ constraints ( $\boldsymbol{e_1} \in \mathbb{R}^{2 \times H \times W}$ ). This improves the velocity field's accuracy, which positively impacts concentration predictions.

Referring to [1], we evaluated the correlation between predicted and true latent physical quantities, with and without the Velocity-Pressure Constraint. The results, summarized in Table A, show a clear improvement in the correlation of velocity components when the constraint is applied:

Table A: Correlation Between Predicted and True Latent Physical Quantities

Figure S6 (page 20, Section D.1) provides a visual comparison of velocity fields with and without the constraint.

For concentration prediction, Table B compares evaluation metrics across different settings, including configurations without the Velocity-Pressure Constraint, Temporal Constraint, or Correction Network.

Table B: Comparison of Evaluation Metrics Across Different Settings

The results demonstrate that introducing the Velocity-Pressure Constraint significantly improves concentration prediction accuracy.

In summary, we believe that the pressure field $p$ not only enhances interpretability but also contributes to improving concentration prediction.

Reference:

[1] Kochkov, et al. Machine learning–accelerated computational fluid dynamics. PNAS, 2021, 118.21: e2101784118.

Thank you for your constructive comments and suggestions!

Q1. The gradient discretization used here is a second-order central difference approximation.

Reply: Thank you for your insightful comment! Your feedback helps us clarify the approach used in our method.

• Automatic Differentiation and Central Difference. Physics-Informed Neural Networks (PINNs) typically use automatic differentiation within neural networks to achieve high accuracy in approximations. However, this differentiation method relies on the MLP (multi-layer perceptron) network structure, which is typically less effective for comprehensive predictive tasks, particularly for complex or long-term predictions, due to its limited representation ability.

• Second-order Central Difference and Higher-order Central Difference. In our experiments, we primarily use the second-order central difference approximation. In non-boundary regions, higher-order central differences did not show a significant advantage in our model's prediction accuracy. However, in boundary regions, using more points in higher-order differences often led to a larger influence from zero-value boundaries, which adversely affected the prediction accuracy. This impact was especially notable in long-term predictions, where error accumulation became more significant. Hence, we employ the second-order central difference considering the above tradeoff.

Q2. Two Different Sets of Baselines.

Reply: Thank you for this comment. In nowcasting, we treat it as the transport of rain clouds by atmospheric motion, but compared to simulated data, it presents unique challenges. These challenges include the lack of strict adherence to the Navier-Stokes (NS) equations and the presence of significant noise. Most current neural operator methods are tested with noise-free or low-noise simulated data.

In our method, we discretize the Partial Differential Equations (PDEs) and embed them into our model and loss function. Although the precipitation process does not strictly follow the NS equations, we believe the dynamics described by the NS equations still captures some essential motion features that benefit the prediction. By using nowcasting precipitation, we test our model’s ability to handle scenarios with noisy data and partial adherence to physical equations.

Weakness 1. About Interpretability.

Reply: Thanks for your concern regarding interpretability. However, we believe that providing unsupervised predictions of the velocity and pressure fields based solely on observed concentration data is physically meaningful. This aids in understanding the future dynamics.

Weakness 2. The pertinence of benchmarking nowcasting.

Reply: Excellent Remark! As previously mentioned, the SEVIR test is designed to assess the predictive capability of our model under conditions of noisy data and in scenarios where the governing equations do not strictly satisfied. In addition, we tested the most representative neural operator, namely, FNO, for nowcasting. Unfortunately, its performance is unsatisfactory given the SEVIR dataset. Hence, we decide not to include such results in our paper.

Weakness 6. Need for detailed derivation of Equation 3.

Reply: Excellent remark! The proof relies on the Lagrange Mean Value Theorem, and we appreciate the suggestion to clarify this further. We have included additional explanations in the paper (lines 208-211, page 4, Section 3.1). Please see below.

"However, considering the continuity of material concentration $c(\boldsymbol{x},t)$ over time, based on the Lagrange Mean Value Theorem, there must exist a moment $t'$ between time $t_k$ and $t_{k+1}$ such that:"

$

\int_{t_k}^{t_{k+1}} \dot{c}(\boldsymbol{x},t) dt = c(\boldsymbol{x},t') \Delta t, \ t' \in[t_k, t_{k+1}].

$

Regarding the time notation $\Delta t$, we agree that our original description was imprecise. We have revised the formulations in the entire paper thoroughly. Specifically:

• $k \in \{1, 2, \dots, N\}$ denotes the $k$-th frame.
• $t_k$ represents the time of the $k$-th frame.
• $\Delta t$ is the time interval between $t_k$ and $t_{k+1}$.

Weakness 7. Justification for using U-Net over transformers.

Reply: The Transformer model typically consumes more GPU memory than the U-Net model. Due to page limitations, we could not provide detailed descriptions of the network architecture in the main text. Hence, we placed the detailed network architecture in Figure S3 (Appendix A.4, page 15). We hope for your understanding.

Weakness 8. Lack of details on potential interference between components.

Reply: The model is trained in an end-to-end manner. We directly use the loss function we designed during training. Importantly, there is no interference between components in the actual training process. The process is straightforward and stable.

Weakness 9. Treatment of noise in real-world data and explicitness of underlying PDEs.

Reply: Excellent comment! We would like to clarify that we herein use the Navier-Stokes (NS) equations as an approximation. While the atmosphere’s dynamics (e.g., raincloud transport) do not strictly follow NS equations, these equations effectively capture key motion patterns. This inductive bias is considered helpful to improve the model's robustness against noise.

Weakness 10. Result in Figure 9(a) is unclear.

Reply: Thank you for pointing out the issue with Figure 9(a). This result reflects an average outcome across all 2D fluid test cases. The 40-step extrapolation limit made it challenging to illustrate the long-term impact of different components.

To address this, we visualized the results on the Smoke2D dataset with 100-step extrapolation, which better highlights these effects. We have updated the results in the revised paper accordingly (Figure 9(a), page 10, Section 5). The results are also presented in Table A below.

Table A: Comparison of Evaluation Metrics Between Different Constraint Configurations and the Original Method.

Weakness 11. Insufficient details about the settings for 2D fluid data.

Reply: The symbols in the paper were deliberately chosen to avoid implying that these quantities are known. In our setup, the 2D data comprises grayscale image sequences (videos) of concentration, and the specific physical scales are unknown. We have added further clarifications and additional details in Figure S4 (page 16, Appendix B.1).

Weakness 12. Absence of code and dataset links.

Reply: Open-sourcing the code and data is our priority. We will make both resources publicly available after the peer review stage.

Thank you for your constructive comments and suggestions!

Before addressing the specific questions, we would like to clarify the primary problem we are motivated to solve as follows:

We consider partially observed data environments, e.g., the rising and diffusion of smoke in the air. In such scenarios, while smoke concentration can be easily observed (by video cameras), the velocity and pressure fields are not directly observable. Our goal is to predict future smoke concentration while simultaneously estimating velocity and pressure fields as interpretable evidence in an unsupervised manner. To align with potential real-world applications, our 2D simulated data consists of grayscale image sequences where even the spatial scale remains unknown.

Weakness 1. Lack of a clear overview of the method; the approach is an extension of FDM which lacks novelty.

Reply: Thanks for your comments! We acknowledge that our original overview of the proposed method was overly brief. This section has been thoroughly revised, and the updated version is now presented in lines 153–188 (pages 3–4, Section 3) in the revised paper.

Unlike numerical methods such as the Finite Difference Method (FDM), our approach leverages the temporal continuity of concentration and applies the Lagrange Mean Value Theorem for temporal discretization. This embedding strategy, coupled with our loss function design, enables the model to provide interpretable estimation of velocity and pressure fields, without the need of training data, while maintaining stability in long-term predictions.

Moreover, we emphasize that our contribution extends beyond achieving high prediction accuracy over longer time horizons. A key novelty lies in our ability to infer hidden physical quantities (e.g., velocity and pressure) in an unsupervised manner. These quantities closely resemble actual physical fields and offer interpretability, helping to enhance the understanding of future motion processes. This forms fundamental differences compared with FDM.

Weakness 2. Inaccuracy in claims about velocity field observability.

Reply: Great remark! Our initial phrasing was indeed overly absolute, and we have revised it in lines 40–41 (page 1, Section 1) of the paper. Please also see below.

"Successful approaches usually rely on supervised learning for these quantities, but such data is not easy to obtain in practical scenarios.""

Nevertheless, while velocity fields can be observed using methods like Particle Image Velocimetry (PIV) and Particle Tracking Velocimetry (PTV), these methods face notable challenges, such as particle occlusion, high experimental costs, limited spatial resolution, etc.

Weakness 3. Ambiguity in terms like "past observable data" .

Reply: Excellent comment! Our task indeed involves partially observed data environments, where only the concentration field is directly observable. We infer the underlying mapping based on the past observations of concentration data. To better reflect this, we have revised the phrasing in line 69 (page 2, Section 1) of the paper. Please also see below.

"Neural networks are employed to establish a mapping function from past measurement (observed data) to both intermediate observed quantities and latent physical variables, enabling predictions through the discretized NS equations."

Weakness 4. Inappropriate statement about the difficulty of obtaining initial conditions.

Reply: Obtaining explicit initial conditions is often impractical in real-world scenarios. For instance, when observing the rising of smoke, it is far more common to have access to a video rather than explicit initial conditions. This is why our 2D simulation data is structured as grayscale image sequences, with the actual spatial size remaining unknown.

Weakness 5. Misinterpretation of Table 1 categories and the capabilities of FNO.

Reply: Thanks for your careful observation! While methods like FNO can map concentration to velocity fields, this mapping is supervised and requires velocity field data during training. Our approach, in contrast, predicts velocity fields using only concentration data, without requiring any ground-truth velocity fields for supervision.

Regarding Table 1, we have made modifications and added descriptive explanations (lines 147–149, page 3, Section 2).

Comment: Quantitative Comparison of Latent Physical Quantities.

Reply: Excellent comment! Referring to [1], we calculate the Correlation to quantitatively evaluate the accuracy of our predictions of the latent physical quantities compared to the true latent physical quantities. For example, we consider $x_i$ as the predicted values and $y_i$ as the true values, with $\overline{x}$ being the mean of all predicted values and $\overline{y}$ being the mean of all true values. The Correlation metric is defined as follows:

$

\mathrm{Correlation} = \frac{\sum_{i=1}^n\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right)}{\sqrt{\sum_{i=1}^n\left(x_i-\bar{x}\right) ^2} \sqrt{\sum_{i=1}^n\left(y_i-\bar{y}\right)^2}}

$

The results of the Correlation metrics are shown in Table A.

Table A: The Correlation between the predicted latent physical quantities and the true values.

We can see that high Correlation values (>0.9) in Table A further demonstrates the accuracy of the predicted hidden physical quantities. The above results have been updated in the latest version of our paper (Figure 6$c$, page 8, Section 4.1 and Section B.2, page 17).

Thank you very much for your great suggestion! Please let us know if you have any other questions.

Reference:

[1] Kochkov, et al. Machine learning–accelerated computational fluid dynamics. PNAS, 2021, 118.21: e2101784118.

Comment 2: Clarification on the evaluation metrics.

Reply: We would like to clarify the rational use of correlation as the evaluation metric. As discussed in [1], correlation is a commonly used metric in evulating the similarity between two quantities. In particular, when the correlation exceeds 0.9, the variables is strongly correlated.

We appreciate your suggestion on the use of MSE to evaluate the predicted velocity and pressure fields. We would like to highlight that due to the unknown physical scale (since only the grayscale video sequences of the concentration field are provided as the training data), our predicted velocity fields and the true values have a scaling relationship, expressed as follows:

$\hat{\boldsymbol{u}} = \alpha \boldsymbol{u}$

where $\alpha$ is a scaling constant related to the actual physical scale. When only concentration videos are available, we cannot determine the exact physical scale of the region represented in the video (e.g., the scale may lie in any range such as from 10 cm to 1 m in the smoke dataset). This uncertainty introduces a constant factor difference between the predicted and true values.

Thus, directly computing the MSE might be inappropriate. To address this, we considered two methods for calibration:

• Case 1: Using the true initial velocity values (assumed known a priori) to compute $\alpha$.
• Case 2: Using just one true data point to estimate $\alpha$.

We then apply the computed $\alpha$ to correct the predicted results (note that HelmFluid also requires calibration). The calibrated MSE values are shown in Table B below.

Table B: Comparison of calibrated MSE for latent physical quantities between our model and HelmFluid.

The results in Tables A and B highlight the superiority of our approach in predicting the latent physical quantities. Moreover, even with a constant factor difference, a high correlation (>0.9) is sufficient to demonstrate the consistency between our predictions and the true data in terms of distribution, regardless of scaling. This can serve as supplementary information for concentration prediction and provide interpretable evidence for the future evolution of fluid dynamics.

In conclusion, with concentration data as the sole training labels, our model outperforms all baselines for all tested examples.

Additionally, we have also revised Table 1 (page 3, Section2), with the modifications highlighted in blue to enhance the clarity of the paper (please see the updated .pdf file). Thank you very much for your insightful comments, and we look forward to your feedback!

Reference:

[1] Akoglu, Haldun. "User's guide to correlation coefficients." Turkish Journal of Emergency Medicine 18.3 (2018): 91-93.

Thank you for your constructive comments and suggestions!

Q1. Concerns on SEVIR: (1) underperformed MSE metric, (2) adaptability of NS equations in nowcasting, (3) comparison with PySteps.

Reply: Insightful comments! We would like to firstly clarify that, although the Navier-Stokes (NS) equations are not strictly satisfied in SEVIR, the corresponding test was taken to evaluate our model's predictive performance and robustness to noise under conditions where the equations are not fully obeyed. Importantly, no specific adaptations, such as adjustments to the loss function weights, were designed for this task. As noted, the nowcasting scenario may not strictly adhere to the NS equations. However, we believe that incorporating the NS equations helps to capture the fundamental motion characteristics of nowcasting.

To address the first two points raised, we conducted the following experiments:

• Lowered the physical loss weight (weight = 0.1) while keeping other settings unchanged.
• Removed the physical loss entirely.

The results are summarized below in Table A.

Table A: Comparison of Evaluation Metrics on the SEVIR dataset under Different Physical Loss Configurations.

The results in Table A demonstrate that while nowcasting processes do not strictly adhere to the NS equations, these equations effectively capture certain underlying motion characteristics. When the physical loss weight is reduced, loosening the constraints, the Mean Squared Error (MSE) decreases. However, this relaxation comes at the cost of reduced ability to model essential motion features, as reflected in the decline of the Critical Success Index (CSI). The corresponding analysis and results have been included in Appendix D.4 (page 20).

Regarding comparisons with pySTEPS, we sincerely appreciate the reviewers' suggestion. However, after careful consideration, we believe this is unnecessary since NowcastNet (our comparable baseline model) has already demonstrated improvements over pySTEPS.

Q2. Simplicity of the Fluid 2D Dataset.

Reply: When calculating the Reynolds number, we defined the characteristic length as the length of the widest obstacle, resulting in the Reynolds number around 450. For the 2D fluid training set, the obstacle positions were varied. However, in the test set, we increased complexity by altering obstacle shapes and numbers in addition to their positions. This introduces greater variability and poses higher demands on the neural network's ability to learn motion laws and generalize effectively.

Q3. Flickering in Baseline Predictions.

Reply: The flickering phenomenon observed in some baseline model predictions is a form of periodic oscillation. Initially, this puzzled us as we did not find any bug in the implementation code. Upon further investigation, we discovered that the oscillation period (4 steps) matches the training rollout step size. Specifically, our training uses 4 frames to predict one frame unrolling over four steps, while the testing (inference) extends the rollout process to 40 steps. This periodic oscillation may arise because some models (except ours) struggle to learn motion dynamics within short sequence training.

To further investigate this phenomenon, we conducted three validation experiments as follows:

• Case 1: Training with four frames predicting one frame for four steps.
• Case 2: The same as Case 1 but with an adjusted batch size.
• Case 3: Training with four frames predicting one frame for six steps.

Our findings include:

• Experiments on Case 1 and Case 2 exhibited oscillations with a period of 4 throughout training (even at epoch 200).
• Experiment on Case 3 initially had oscillations with a period of 6, but these gradually diminished as training progressed.

This suggests that some baseline models struggle to capture long-term motion features when trained on short-term sequences. Under the configuration of Case 3, the oscillations in LSM were significantly reduced, yet our model still outperformed it, as shown in Table B below.

Table B: Comparison of Results Between Our Method and LSM Under Experimental Setting Case 3.

The details of the above analysis and results are included in our paper (page 7, Section 4.1 and page 29, Appendix F).

Q4. Velocity Visualization.

Reply: Excellent suggestion! While we believe the field-based visualization provides more comprehensive information compared with the arrow-based visualization, we understand its limitation in intuitiveness. Therefore, followed your suggestion, we have included the arrow-based visualization in Figure S10 (page 22, Appendix E) for better illustration.

Q5. Writing and Grammar.

Reply: We appreciate this feedback. We have thoroughly revised the paper to improve the language and formatting, ensuring clearer and more effective presentation. Please see our revisions marked in red color in the revised paper.

Weakness: Modest Improvements on Smoke3D.

Reply: We acknowledge the modest improvements observed on specific datasets like Smoke3D. However, we achieved consistently the best results across all simulation datasets. Additionally, we would like to emphasize that our contributions go beyond the prediction accuracy. Our model is capable of predicting hidden physical quantities (e.g., velocity and pressure) in an unsupervised manner. These predicted quantities align closely with real physical fields and serve as interpretable evidence which deepen the understanding of future motion processes. This enhances the value of our approach. Hope this clarifies your concern.

In summary, we appreciate your constructive comments and suggestions. Please let us know if you have any other questions. We look forward to your feedback!

Thank you for the extra experiments and detailed answers.

Q1. Concerns on SEVIR: Firstly, I would like to express my gratitude to the authors for explaining the applicability of the N-S equation as a loss function. I think this helped the model learn better physical characteristics to a certain extent. However, this paper does not include visualization of the physical quantities learned from the SEVIR dataset. I would be very interested in looking at the visualization of the learned velocity fields using pyplot.quiver and comparing it with the velocity fields learned by NowcastNet.

Secondly, I appreciate the authors' explanation for the MSE metric, but I still believe this indicates that the authors have not treated the baseline fairly. As shown in Fig. 8, the predicted intensity of PINP is significantly higher than that of other models. This does not clarify whether the higher CSI compared to other models is due to higher forecast intensity by introducing various losses or the more accurate position provided by velocity prediction.

Thirdly, the authors stated that only the Evolution Network part was used for NowcastNet. This network applies an advection scheme directly on the input observation. Thus, the results should not show a decrease in intensity, which contradicts the results provided in Figures S17 and S16. Additionally, this does not demonstrate that the Evolution Network part has already shown improvements over pySTEPS. As mentioned in the previous point, this severely affects the fairness of the CSI metric comparison. Therefore, I insist on suggesting that the authors check the correctness of the implementation of the Evolution Network or provide comparisons with pySTEPS.

Q2. About Fluid 2D dataset: I appreciate the authors' provision of the parameters utilized within the Fluid 2D Dataset. However, the motion characteristics of visualization suggest no turbulence presence in the dataset. I highly recommend that the authors consider employing a dataset with a higher Reynolds number that encompasses turbulence or leverage the Bounded N-S dataset from the baseline HelmFluid to ascertain the model's efficacy in forecasting turbulent flows. Otherwise, the practical application value of this model in real fluid prediction scenarios is somewhat limited.

Q3. Flickering in Baseline Predictions. Thank you to the authors for explaining the prediction results' flickering and providing additional training outcomes. I have no further questions about this.

Q4. Velocity Visualization. I appreciate the authors' visualization of the velocity field. The current results are clear and well-presented.

Q5. Writing and Grammar. Thank you for revising the paper and updating the expressions. Since the entire article is now more readable, I have increased the presentation score to 3.

Weakness: Modest Improvements on Smoke3D. I also appreciate the authors' explanation regarding the issues on the Smoke3D dataset. However, even though the model provides additional variable predictions with physical interpretability, I still believe that the improvement in the metrics is not significant enough.

In summary, the authors have partially addressed my concerns about this paper. However, (1) the issue with the baseline and metrics on the SEVIR dataset still exists, and (2) the problem that the Fluid 2D Dataset only contains laminar flow and is overly simplistic remains unresolved. Therefore, I will not change my score for now.

Thanks for your additional comments and suggestions!

Q1a: Velocity Field Visualization and Comparison with NowcastNet.

Reply: Excellent suggestion! We have included the figure comparing the velocity fields of our method with those of NowcastNet and pySTEPS in Figure S11 (Section E, page 23) in the revised paper.

Q1b: The comparison with pySTEPS

Reply: Great suggestion! We have run pySTEPS, obtained the prediction results, and compared this method with NowcastNet (under different settings) as well as our own method. The comparison results are shown in Table A below, and the above results have already been updated in our revised paper (Table S1, page 16, Section A.5).

Table A: Comparison of pySTEPS with NowcastNet (under different settings) as well as our own method.

Based on the results shown in Table A, our method outperforms pySTEPS.

Q1c: The issue of weakened prediction intensity in NowcastNet.

Reply: Excellent remark! This weakening is caused by the "bilinear" mode used during "wrap". If the "nearest" mode is applied, the weakening issue does not occur.

As shown in Table A, the reason for choosing "bilinear" mode is that it performs relatively better on the evaluation metrics. The new comparison results are added and presented in our revised paper (Table S1, page 16, Section A.5).

Q2: Evaluation on more complex fluid datasets.

Reply: We understand the reviewer’s concern. We generated data with incident velocities 4 times and 6 times the original velocity (corresponding to Reynolds numbers of 1800 and 2700) and tested our trained model (inference). The results are shown in Table B below. Our model still achieves the best performance.

Table B: Results at higher Reynolds numbers.

Moreover, for your suggestion regarding testing the Bounded N-S dataset used in Helmfluid, we feel a little bit confused and believe it might not be suitable for evaluation of our model on turbulence prediction, as its Reynolds number is only 300 (Table 8, Appendix B.1 in HelmFluid). Anyway, thank you for your suggestions!

Concluding Remark: In summary, we appreciate your constructive comments and suggestions. Please let us know if you have any other questions. We look forward to your feedback!