PEINR: A Physics-enhanced Implicit Neural Representation for High-Fidelity Flow Field Reconstruction
Authors: Liming Shen
Liang Deng
Chongke Bi
Yu Wang
Xinhai Chen
Yueqing Wang
Jie Liu
Venue: ICML 2025 poster (2025)
Keywords: Implicit Neural Representation
Time-varying data visualization
Standard benchmark datasets
High-fidelity flow field reconstruction
Physical encoding
Primary area: applications->chemistry_physics_and_earth_sciences

Abstract: Implicit neural representation (INR) has now been thrust into the limelight with its flexibility in high-fidelity flow field reconstruction tasks. However, the lack of standard benchmarking datasets and the grid independence assumption for INR-based methods hinder progress and adoption in real-world simulation scenarios. Moreover, naive adoptions of existing INR frameworks suffer from limited accuracy in capturing fine-scale structures and spatiotemporal dynamics. Tacking these issues, we first introduce HFR-Beach, a 5.4 TB public large-scale CFD dataset with 33,600 unsteady 2D and 3D vector fields for reconstructing high-fidelity flow fields. We further present PEINR, a physics-enhanced INR framework, to enrich the flow fields by concurrently enhancing numerical-precision and grid-resolution. Specifically, PEINR is mainly composed of physical encoding and transformer-based spatiotemporal fuser (TransSTF). Physical encoding decouples temporal and spatial components, employing Gaussian coordinate encoding and localized encoding techniques to capture the nonlinear characteristics of spatiotemporal dynamics and the stencil discretization of spatial dimensions, respectively. TransSTF fuses both spatial and temporal information via transformer for capturing long-range temporal dependencies. Qualitative and quantitative experiments and demonstrate that PEINR outperforms state-of-the-art INR-based methods in reconstruction quality.

PEINR: A Physics-enhanced Implicit Neural Representation for High-Fidelity Flow Field Reconstruction
Liming Shen 12
Liang Deng²
Chongke Bi
Yu Wang'
Xinhai Chen 12
Yueqing Wang? Jie Liu
Abstract
1. Introduction
Implicit neural representation (INR) excels in high-fidelity flow field reconstruction through flexible enhancement of numerical precision and grid resolution. However, its broader adoption faces two barriers: the absence of standardized benchmarks for flow reconstruction tasks, and the impractical grid independence assumption in real-world simulations. Current INR frame- works also struggle to resolve fine-scale structures and spatiotemporal dynamics, particularly under severe temporal-spatial data imbalance, where temporal sensitivity degrades significantly. Tack- ing these issues, we first introduce HFR-Bench, 5.4 TB public large-scall scale CFD dataset with 33,600 unsteady 21) and 3D vector fields for re- constructing high-fidelity flow fields. We further present PEINR, a physics-enhancer INR frame- work, which is mainly composed of physical encoding and transformer-based spatiotemporal fuser (TransSTF). Physical encoding decouples temporal and spatial components through Gaus- sian temporal encoding, which can enhance high- dimensional features and nonlinear characteristics in temporal information, and localized spatial en- coding, which can implement stencil-based dis- cretization in the spatial dimension. TransSTF fuses both spatial and temporal information via transformer for capturing long-range temporal de- pendencies. Qualitative and quantitative experi- ments demonstrate that PEINR outperforms state- of-the-art based methods in reconstruction quality. Code and dataset are released here.
High- High-fidelity flow fields contain richer flow information and finer-scale flow structures, which play crucial role in understanding the flow behaviors in various physical and natural phenomena. In computational fluid dynamics (CFD), increasing the grid-resolution and employing higher- order numerical schemes can typically generate high-fidelity flow fields, but the simulations require several days to com- plete and incur substantial computational costs. Recently, inspired by their success in computer vision, deep-lcarning- based super-resolution (SR) methods have increasingly been used to reconstruct high-fidelity flow fields from uw-fidelity counterparts without repeatedly solving complex partial dif- ferential equations (Fukami et al., 2023).
Rewokrios
Low- Low-Fideling Fidelity
PEINR
FigeeFydeliiy
Nameical Accumay
Figure 1. The red circles explain the error in the assumption of grid independence, showing that as the grid resolution (in the row direc- tion) and numerical accuracy (in the column direction) increase, the attrihute values at the same location differ and the highufrey part becomes more pronounced in the corresponding frequency domain representations, which means that the structure of the flow field more complex and refined.
II 'Laboratory of Digitizing Software for Frontier Equipment, National University of Defense Technology, Changsha, China e National Key Laboratory of Parallel and Distributed Comput- ing, National University of Defense Technology, Changsha, China i "Computational Aerodynamies Institute, China Aerodynamics Re- search and Development Center, Mianyang, China "College of Intelligence and Computing, Tianjin University, Tianjin, China. Correspondence to: Liang Deng <dengliangl @ enu...edu.cn>>.
Existing methods of high-fidelity flow field reconstruction based on convolutional neural networks (CNNs) (Fukami et al., 2021; Yousif et al., 2021; Xu et al., 2023; Hu et al., 2024; Shen et al., 2024b;a) are clearly confined to equally- spaced Cartesian meshes because the nodes of the computa- tional grid are reinterpreted as pixels. Nevertheless, irregular meshes are widely used in the industry because they can
Proceedings of the 4nnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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PEINR: A Physics-cnhaneed Implicit Neural Representation for High-Fidelity Flow Field Reconstruction
adequately delineate complex geometries and easily deal with localized regions that require different resolution, e-g- cells are small near walls to capture the boundary layer where flow transitions are sharp whereas they are large in the freestream where gradients are smooth. A grid can also be regarded as a graph and graph neural networks (GNNs) are reasonable candidates. However, in order to propagate information over long distances between nodes, many graph convolutional layers need be stacked. Moreover, although pooling is a cheap operation in CNNs, it is rather challeng- ing in GNNs (Grattarola et al., 2022; Kashefi et al., 2021).
grid-resolutinn gap and the highest uumerical-precisiin can reach up to 64 times and 7th-order, respectively.
Compared to high- figh--iielity simulation, the coarse discretiza- tion of the low-fidelity simulation introduces inaccuracies and truncation errors. As shown in Figure 2(a), we cal- culate the correction errors between low and high-fidelity flow fields. Effectively representing the error field is the target we strive to reach with the application of the INR net- work. We propose can simultaneously enhancing the numerical- numerical-precision and grid grid-resolution of flow fields. PEINR consists of physi- cal encoding and transformer- based spatiotemporal fuser (TransSTF).I In physical encoding, as shown in Figure 2(b), spatial coordinates C are expanded from neighbors, taking into account the nature of stencil discretization. Temporal information is expanded by Gaussian kernel encoding to ad- dressing the issue of disproportionate spatiotemporal dimen- sions. As shown in Figure 2(c), TransSTF, temporal-aware encoder based on the multi-head attention mechanism, aims to fuse both spatial and temporal information and capture long-range temporal dependencies. We also leverage spec- tral block (Patro al., 2023), to capture the high frequency components of high fidelity flow fields. We perform quali- tative and quantitative analyses, to experimentally demon- strate that our approach is well-suited for high-fidelity flow field reconstruction tasks.
eehyance-e INR (PEINR) which
The implicit neural representation (INR) frameworks (Han & Wang. 2023; Pan et al., 2023; Tang& & Wang. 2024; Jiao et al., 2024) typically learn to represent a flow field as a continuous function which can map the spatiotemporal coor- dinates their corresponding values, thus are highly adapt- able for different mesh types and having emerged as new mesh-agnostic paradigm for flow reconstruction. However, these INR-based methods still face several limitations in high-fidelity flow field reconstruction: (1) In the absence of standard benchmarks datasets, existing INR-based methods assume grid independence, where attribute values at the same location in the flow field remain unchanged across different mesh resolutions. However, in the case of real simulation data, this assumption does not hold. As shown in Figure 1, the red circles highlight the same locations in flow fields with varying mesh resolutions and numerical accuracies, where the corresponding density values clearly differ. (2) Due to the I/O bottleneck (Fajardo et al., 2018), only selective time steps can be saved. There is significant disparity between temporal and spatial complexity. Exist- ing methods overlook this gap by simply coupling space and time, thus failing to fully simulate the complex spa- tiotemporal dynamics. (3) The spectral bias issue of INR method (Xu, 2018; Rahaman et al., 2019) concentrate more on the low-frequency information, which can have deficien- cies in capturing fine-scale structures of flow fields. As shown in Figure 1, the corresponding frequency domain representations illustrate that high-fidelin flow fields cap- turing small-scall flow features more effectively can exhibit a richer presence of high-frequency information.
The key contributions of our work are summarized as fol- lows: (1) We release HFR-Bench, a truly large-scale CFD dataset including both uniform Cartesian and non-uniform meshes with 33,600 unsteady 2D and 3D vector fields, total of 5,4 TB of data. For each simulation, data are provided at both low and high resolution, making the dataset suitable for the specific task of reconstructing high-fideliyy flow fields. (2) We propose a novel physics- enhanced INR model for concurrently handling numerical- precision and grid-resolution enhancement for both uniform and non-uniform meshes. The physical encoding can allevi- ate the disparity between temporal and spatial complexities, capture the nonlinear characteristics of spatiotemporal dy- namics and the stencil discretization of spatial dimensions. (3) We utilize the spectral block in TransSTF to alleviate the spectral bias on flow field learning. The result of ablation study can further illustrate that our method can alleviate the negative impact.
amounting
In this paper, we first introduce scale dataset containing 24,000 unsteady flow fields of three canonical two-dimensional (2D) flow problems, 1,600 un- (3D) flow problem
truly large-
steady flow fields of with uniform Cartesian mesh, and 8,000 2D unsteady flow fields with non-uniform mesh, amounting to a total of 5.4 TB of data. For each problem, we discretize the domain into four different grid resolutinn configurations, and perform a simulation at each grid-resolution using four different menerrccl-precission settings. Low- and high-fidelity sim- ulations in each pair start from identical initial conditions and share the same physical parameters. The maximum
2. Related Work
This section discusses the related works of implicit neural representation, and super-resolution for flow field.
Implicit neural representation. Implicit neural represen- tation (INR) is powerful framework that seeks to train a neural network to parameterize continuous, implicit func-
2

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PEINR: A Physics-cnhaneed Implicit Neural Representation for High-Fidelii Flow Field Reconstruction
Treaky k)
Localized Sporial Enceding
ResuML
ResuMLP
Tinska last
Timesep f
Timesap fg
ResiMLP
Tincaiop bo
FM
6f
Spectral
Block
Multi-Head
Spectral Block
Guasiia Temporal Encoding
Attension
p
ResuMLP
-D-
L
(a) Inverse Problems Inputs
(b) Physical Encoding
(c) TransSTF Block
(d) Inverse Problems Solutions
Figure 2. Overview of method PEINR. (a) We first calculate the difference field between the counterpart. (b) In physical encoding, we leverage Gaussian coordinate encoding and localized encoding to handle temporal information and spatial coordinates. (c) The TransSTF block fuses temporal and spatial information and consists of ResuMLP. multi-heat attention and spectral block. (d) The generated high-fidelity flow fileds are obtained by combining diffenrence fields and low-fidelity counterparts.
flow field and low- fwwffidlitty
tion. This function maps a given spatiotemporal position, which consists of both spatial and temporal coordinates, to its corresponding value.
by simply replacing the red, green, and blue components with physical variables (Fukami et al., 2023). Among these works, CNN-based super-resolution models (Zhenglei et al., 2017; Fukami et al., 2019; 2021; Liu et al., 2020; Obiols- Sales et al., 2021; Gao et al., 2021) have been actively studied for range of flows. To further improve the model performance, many works complicate their model by incor- porating GAN (You et al., 2018; Zhiwen et al., 2019; Yousif et al., 2021; 2022; Yu et al., 2022; Han & Wang, 2019; 2020; Wurster et al., 2023; Hyojin et al., 2021), transformer (Wang et al., 2022; Xu et al., 2023; Hu et al., 2024: Shen et al., 2024b;a), for capturing fine-scale flow features. However, these methods are limited flow field data on Cartesian grids.
Recently, incorporating Fourier features (Mildenhall et al., 2021), periodic activations (Sitzmann et al., 2020), or multi- resolution Hash tables (Müller et al., 2022) has significantly improved the performance of INR for scene reconstruc- tion (Jiang et al., 2020; Chabra et al., 2020) and shape modeling (Genova et al., 2020; Atzmon & Lipman, 2020). Recent work by de Vito et al. (de Vito et al., 2024) in im- plicit neural representation for accurate CFD flow field pre- diction demonstrates the potential of INR architectures for steady-state CFD simulations. However, their method pri- marily focuses single-resolution flow fields with station- ary 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.
Neural operators (Kovachki et al., 2023) leverage function space mappings and efficient -oomral-domain computations to overcome the fixed-grid limitations of traditional meth- ods. However, when high- geghffreueency 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 vortex struc- tures, without requiring full-field computation, making it more adaptable and efficient.
PINNs (Karniadakis et al., 2021; Dong & Polak, 2024; Hos- seini & Shiri, 2024) are an extended application of implicit neural representations in scientific computing. Nevertheless, our CFD data are solved based on discrete meshes, which inherently introduce numerical dissipation and truncation errors, deviating from the Navier-Stokes equations govern- ing fluid flow. Foreing PINNs to simultaneously fit this data and satisfy the equation residuals may lead network con- vergence issues or generate non-physical solutions (Farea et al., 2024).
Existing INR metods for flow field (Han & Wang, 2023; Pan et al., 2023; Tang & Wang, 2024; Jiao et al., 2024) only fo- cuse on grid-resolution enhancement, and PEINR is the first attempt to simultaneously enhance the grid-resolution and numerical-precde by leveraging transformer and physical- enhanced INR model.
Super-resolution for flow field. In the past several years, various deep learning-bas baasid methods have been applied to tackle single image super-resolution (SISR) tasks (I.u et al., 2022; Zhang et al., 2022: Chen al., 2023). Inspired by their success in computer vision, SISR methods have in- creasingly been used to reconstruct high-fidelit flow fields
3. Method
We propose a framework with an INR-hased model for high- fidelity flow field reconstruction. The framework consists of 4 components: inverse problem input encoding, physi-
3

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PEINR: A Physics-enhanced Implicit Neural Representation for High- Flow Field Reconstruction
mapping
mapping
NTK Fourter Spectrum
20-
20-
04-
40
0.4
60-
121
-0.4
0.2
-22 00
60
Irpet Spuce
Prepurny
(8)
(d)
Figure (a) The NTK without mapping exhibits irregular patterns, indicating dependency on absolute time positions and leading to inconsistent predictions. (b) Applying the Gaussian kernel mapping results in structured diagonal pattern, ensuring translation invariance and improved generalization. (c) The NTK spatial decay plot shows how kernel values diminish across input space, with smaller emphasizing local interactions and larger capturing broader dependencies. (d) The NTK Fourier spectrum demonstrates that larger values allow high-freuenncy variations, while smaller a favor smooth, low-frequency behaviors.
cal encoding, TransSTF, and inverse problem solutions, as shown in Figure 2. Our method allows approximating the accurate variables (density p. the components u, v and w of vector variable velocity V) of high-fidelity flow field by learning its time information and the dependency of spa- tial information. We detail all framework components and learning schemes in the following subsections.
field but still contains residual errors.
As shown in Figure (a), to quantify these discrepancies, we compute the error field F., Fe -F which captures the missing fine-grained details. We obtain 8F, through the BI method, which allows to efficiently handle the discrep- ancies between the low-fidelity and high-fidelity flow fields, thus laying the foundation for subsequent error learning and precise reconstruction. The goal of the PEINR network is to effectively learn and represent the error field 8Ft, enabling the accurate reconstruction of high-fidelity flow fields from low-fidelity counterparts. As shown in Figure 2 (d), ob- tain the inverse problems solutions, generated by INR model along with the low-fidelity flow field to reconstruct the high-fidelity flow field.
3.1. Inverse Problem Input Encoding and Solutions
The inverse problem refers to reconstructing high-fidelity physical fields from sparse data (Chen et al., 2024). The CFD simulation flow field is closely related to the numerical accuracy and grid resolution, and these two factors influence each other. Finer grids can better capture flow details and re- duce discretization errors, which can lead to discrepancies in flow field property values at the same location. To enhance the applicability of INR, chose use INR to learn the error field 8 between high-fidelity and low-fidelity flow fields.
the error field
3.2. Physical Encoding
To accurately capture the spatial and temporal dependen- cies in flow field representations, we introduce specialized encoding techniques for both spatial and temporal informa- tion.
The process is as follows: to establish the connection be- tween the low-fidelity flow field Ft at timestep l, and high-fidelity counterpart F.H, we need address the in- accuracies and truncation errors introduced by the coarse discretization in low-fidelity simulations. First, upscale the low-fidelity flow field to match the spatial resolution of the high-fidelity field using BI (bicubic interpolation) method. Although traditional upscaling methods or trans- ferring existing mature super-resolution models could chosen, we opted for the traditional BI method due to the diversity of flow field grid types. BI method estimates the value each point by considering the surrounding 4 4 grid of known values and applying cubic interpolation along both axes, resulting in a smoother and more accurate up- scaled flow field compared to simpler methods like bilinear interpolation. The resulting upsealed flow field, denoted as B provides a better approximation of the high-fidelity
3.2.1. SPATIAL DISCRETIZATION
Spatial discretization in computations typically involves stencil-based methods, whether using finite volume, finite difference, finite element methods. These approaches rely on information from grid cells and their neighboring cells to construct the discretized equations. In most con- ventional INR methods, the network learns to represent a flow field as a continuous function with the input of depen- dent coordinates Ö and temporal information t, ignoring the importance of spatial dependency.
To address this, after randomly sampling sufficient number of points, we expand the input spatial coordinates to include not only the original coordinates but also those of nearby

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PEINR: A Physics-cnhaneed Implicit Neural Representation for High-Fidelity Flow Field Reconstruction
neighboring points:
solely dependent on relative differences:
VOO) = C, C
[[,
K (ti, 12) = K -t))
(3)
neighhors
This property allows the INR model to maintain consistency across different time steps, improving long-term reconstruc- tion accuracy.
This localized encoding aligns with stencil-based computa- tion paradigm of CFD, where numerical solutions inherently depend on local neighborhood interactions (e.g., finite vol- ume discretization). By embedding this locality into our spatial localization encoding, PEINR explicitly leverages the template computation logic to resolve fine-scale flow structures. For instance, in a 2D case, we consider the near- est 4 points (up, down, left, right), transforming the input from single coordinate tensor of shape (hatchsize, 2) to an augmented tensor of shape (batchsize, 5, 2). This design explicitly encodes the local spatial correlations required for solving discretized Navier-Stokes equations, where the so- lution of each grid point depends on physical quantities of its neighbors.
To analyze the impact of the RBF kernel on model stability and generalization, we apply Neural Tangent Kernel (NTK) theory, which describes the training evolution of infinitely wide neural networks and is essential for convergence anal- ysis. Traditional NTKs derived from MLPs lack translation invariance, making them unsuitable for temporal modeling. By incorporating the RBF kernel, we convert the NTK into stationary kernel that depends on relative time differences, improving stability, especially in solving time-dependent PDEs. As shown in Figure 3, the RBF-enhanced NTK ex- hibits structured patterns that better capture temporal depen- dencies, promoting smoothness and consistency over time, which enhances generalization and robustness in dynamic enviromments. The spectral properties of the NTK reveal that adjusting the kernel bandwidth a allows for flexible adaptation to various time scales, balancing local and global temporal interactions.
Spatial discretization allows the model to capture spatial derivatives more effectively, reflecting the underlying prop- erties of partial differential equations (PDEs) that govern flow field computations. By considering spatial derivatives, the model can better account for the local gradients and variations in the flow field, enhancing its ability to represent fine-scale structures and accurately approximate the dynam- ics dictated by the governing PDEs, as shown in Figure 2 (b).
Step 2: Extracting Temporal Components with Kernel PCA. Kernel PCA is applied to extract significant temporal features from the kernel matrix by first centering the ma- trix to ensure zero-mean data. This is achieved using the equation:
3.2.2. NONLINEAR TEMPORAL ENCODING
k= - 1rK - KIN + 1yK1N
(4)
To effectively model the nonlinear temporal dynamies in flow fields, we employ two-step approach that combines the Gaussian Radial Basis Function (RBF) kernel and Kernel Principal Component Analysis (Kernel PCA). This method transforms the oneddimensional temporal input into a high- dimensional, nonlinear feature space and then reduces the dimensionality while preserving essential temporal features.
where IN is an N x N matrix with all elements equal to 1/N.
Next, eigenvalue decomposition of the centered kernel ma- trix K yields eigenvalues and eigenvectors, from which the top m eigenvectors are selected form a -iwddiensiooall representation. Temporal inputs are projected into this sub- space to obtain their encoded representations:
Step 1: Temporal Encoding Using RBF Kernel. The RBF kernel maps the one-dimensional temporal input t into nigh-dimensional feature space where nonlinear temporal patterns can he better represented. The RBF kernel function is defined as:
(5) where (tt) is the nonlinear feature representation of t in the reduced space, Vi the matrix of the top m eigenvectors, and K (t,) is the similarity vector of time point t, with all other points (i.e.. the i-th column of the kernel matrix).
T (t+) = V K (t:)
t - 2 20²
K (t,t') exp
The combination of the RBF kernel and Kernel PCA cap- tures complex temporal relationships, retaining key features through dimensionality reduction, ensuring smooth tran- sitions in representation, and enhancing the INR model's efficiency and stability in learning nonlinear temporal dy- namies.
where and t' are two time instances. is a hyperparameter controlling the kernel's width, determining how quickly the similarity between i and l' decays with their distance.
Given a set of temporal data points {t,, f2....,,tyy we construct the RBF kernel matrix K of dimensions N x N, encoding pairwise temporal relationships. This transforma- tion ensures translation invariance by making the kernel
The encoded temporal features (t)) are concatenated with the localized encoding spatial coordinates vi (C) and fed
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PEINR: A Physics-cnhaneed Implicit Neural Representation for High-Fidelity Flow Field Reconstruction
into the INR model. The INR framework approximates the error field FF (x. ys t, by learning a continuous mapping:
numerical-preecison between FL and I data can be de- scrihed by the upscaling factor and the improvement factor B:
8 F ( 1,) (L)(()))CC))) O),
(6)
== ==STRF/STaa
B=SPHF/SPP
(8)
where ST denotes the number of grid points and SP de- notes the number of points in the stencil.
where NV is the neural network parameterized by 8. This ap- proach enables the model to leverage the enriched temporal features for improved reconstruction accuracy.
Baselines. We compare our method with three classic meth- ods: (1) Bicubic interpolation (BI) is common traditional image processing algorithm based on interpolation. (2) NIF (23' JMLR) (Pan et al., 2023) consists of two modi- fied MLPs: ShapeNet, which isolates and represents spatial complexity, and ParameterNet, which represents other in- puts, such time. (3) CoordNet (23TVVC) (Han & Wang, 2023) is coordinate-basee and leverages encoder-decoder based INR to learn mapping from coordinates to values.
3.3. TransSTF
As shown in Figure 2 (c), TransSTF combines spatial and temporal information to generate complex embeddings us- ing higher-order features. It consists of multi-head attention (MHA) layers, ResuMLPs, and spectral blocks. The MHA layer with query (Q), key (K), and value (V) inputs aims to refine feature maps, where Q, K, and V come from mixture of spatial-temporal data and ResuMLP-exxtratted features. The output matrix of MHA (M) is shown as fol- lows:
Implementation Details. All experiments are conducted on a single NVIDIA A100 80GB GPU. The batch size (BS), memory (MM) and experimental variable and time per epoch are listed in Table 1.
M(Q, K, V)= = Concat (att 1...., att W
Our method and all the baseline methods are trained with the MSE loss in 2000 epochs and every method including ours can converge within 1000 epochs. Learning ratio is set to le - and decrease after 20 epochs if there is no loss degradation and we adopt the Adam optimizer for optimization. In spatial discretization, for 2D cases, we consider the nearest 4 points, and for 3D cases, the nearest 9 points. In temporal nonlinear encoding, we set the as 10 with time steps normalized to [0.1]. During design, we set the number of residual layers of a ResuMLP to 10 and the max number of neutrons in a ResuMLP is 64. Experiments are conducted within a range of + 10% around the optimal hyperparameters, using the inference results as the standard to confirm that the hyperparameters are indeed optimal. For the other models compared in the table, we utilized their original code and conducted experiments by only varying the input length.
(Q.WKKWW. ddu
where att; = softmax
V.WY
ResuMLP. The ResuMLP (Han & Wang, 2022) improves gradient propagation by adding depth and complexity to the network. It includes residual blocks and a SIREN (Sitz- mann et al., 2020) activation function (sin(wx)), with w set to 30, as recommended (Sitzmann et al., 2020). The output scaled to the [-1, 1] range, suitable for sinusoidal activations.
Spectral Block. The spectral block (Patro et al., 2023) cap- tures different frequency components and reduces spectral bias. It includes spectral gating network with FFT for converting to spectral space, weighted gating to adjust fre- queney component weights, and an inverse FFT (IFFT) to return the signal to physical space, enhancing the capture of fighffreueency features like edges.
Experiments are conducted on a single channel for construct- ing in flow fields, and two (three) channels for constructing velocity fields in 2D (3D). For uniform Cartesian meshes, flow fields from timesteps 460 to 480 (out of 500) are used for training, with the final 20 steps as test samples. Results for uniform meshes (FFS, RM. RT, SV) are shown for step 500. For non-uniform meshes, training samples are taken from steps 400 to 500 at intervals of 5, with the remaining non-multiples of 5 used for testing. Results for non-uniform meshes are presented for step 494.
4. Experiment
Dataset. We generate truly large-scale dataset HFR- Bench, containing 2D and 3D unsteady flow fields including both uniform Cartesian and non-uniform meshes in different high-order weighted essentially non-oseillatory (WENO) schemes, amounting to a total of 5.4 TB of data. In uniform Cartesian meshes, we choose the Rayleigh- ity (RT), Riemann (RM), and Forward Facing Step (FFS) problems for the 2D flow fields, and the shock-longitudinal vortex interaction (SV) case in 3D. In non-unifome struc- tured meshes, we simulate the problem of a flow past a cylinder (Cylinder). The discrepancy in grid-resolution and
instabil-
4.1. Quantitative Comparison
We take MSE (Mean Squared Error) as the loss function and report PSNR (Peak Signal-to-Noise Ratio), SSIM (Struc- tural Similarity Index), CORR (Correlation Coefficient) and
6

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PEINR: A Physics-enhaneed Implicit Neural Representation for High- Flow Field Reconstruction
Grourd Truth
NIF
CooediNet
Ours
Ground Truh
(4)
(b)
CaordNet
Our
(c)
(
Trath
Figure Comparison results of the variable p flow field of uniform Cartesian meshes. given by NIF, CoordNet and method in the 2D (upscaling a 16) and 3D datasets (upscaling = 64). Results of FES datasets in (a) WENO3 (88 = 1), (b) from WENO3 to WENO7 with 3 2.6. Results of RM datasets (c) WEN03 (33 1), (d) from WENO3 WENOS with 8 1.4. Results of RT datasets in (e) WENO3 (8 1), (f) from WENO3 to WENOS with , 1.4. (g) (h) Results of SV datasets from WENO3 to WENOS with 3 1.4 from different viewpoints.
DD (Dissipation difference) results as the evaluation metrics. The dissipation operator used assess the model perfor- mance terms of capturing the discontinuities and rotating features with lower values indicating better performance with lower values indicating better performance.
rics Table 2 (a) across all extrapolated timesteps. The results clearly demonstrate that PEINR exhibits absolute advantage in generalization performance.
Ablation Study To evaluate the effectiveness of importance sampling and spectral block of PEINR, we compare with two baseline methods PEINR and PEINR PEINR' re- moves both spectral block and the physical encoding, while PEINR only removes the physical encoding. As shown in Table 2, PEINR outperforms PEINR in the metric DD in all cases, which indicates that our physical encoding method can preserve physical peculiarity. The superiority of PEINR compared with PEINR" highlights that the added spectral block can also help the model to perform better.
The quantitative results are shown in Table 2, which quan- titatively compares PEINR with different grid-resolution and numerical-pie preecisin against the results generated by baselines using data-level metrics (higher values are bet- except for DD). In Table 2, all experiments channel. condueted to reconstruct the variable of flow fields. PEINR can generally outperform the baselines, ex- cept for the slightly inferior PSNR with respect to Coord- Net (Han & Wang, 2023) in the RM dataset. The observed PSNR difference stems from PEINR's 's design prioritizing physical accuracy over numerical optimization for discon- tinuous flows. Riemann problems involve strong discon- tinuities (shocks, contact surfaces) where PEINR's local- ized 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), avoids the physical dis- tortions visible in CoordNet's results (Figure 4(c)()) where large errors near discontinuities appear.
one-
4.2. Qualitative Comparison
The qualitative evaluation provides several contour plots and streamline rendering results. Figure 4 displays syn- thesized flow fields density p of FFS datasets (a) (b), RM datasets in (c) (d), RT datasets in (c) (f) and 3D SV datasets in (g) (h) with different upscaling factors and improvement factors To facilitate analysis, we normal- ized all results to the interval [-1, for ease of observation. The areas of interest to domain experts marked with red squares in the first column, and we zoom in on these areas in the following columns. BI can not capture complex fluids
As shown in 5, also present the four evaluation met-

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PEINR: A Physics-enhanced Implicit Neural Representation for High-Fidelity Flow Field Reconstruction
Table The details of each data set. GR, NP, BS and MM denote grid resolution, numerical precision, batch size and memory, respectively.
Input
Output
Train
Inference Time(s)
dataset
Variable
GR
GR
NP
BS
Time(s)
MM(GH)
WENO3
P.M,2
480
WENO3
x 1920
WENOS
8000
10
22.68
16.71
WENO3
RM
P,
x00
WENO3
x00
WENOS
8000
11
10.53
13.59
36x12
48
WENO3
FFS
P.u.v
180x48
WENO3
x20
WENO7
8000
11
17.29
15.4
WENO3
Cylinder
0,v
51x381
WENO3
101 761
WENOS
8000
19
5.11
9.59
SV
P,U,2,,10
96x32
32
WENO3
384x128 128
WENO5
36000
23
68.45
75.31
Table 2. PSNR, SS"Mx10", CORRx10* and DDx10 values with different upscaling factors and improvement factors 8. The best
values are highlighted bold.
Dataset
(a) RT a=16 3=1
(d) RM a=16 3=1
(g) FFS
a=16 3=1
Metric
PSNR
SSIM
CORR
DD
PSNR
SSIM
CORR
PSNR
SSIM
CORR
DD
NIF
17.53
87.70
94.04
22.46
24.83
92.85
98.33
26.45
35.09
86.44
95.78
2991.72
CoordNet
23.33
93.17
98.40
10.25
45.34
99.67
99.98
21.80
42.71
94.88
98.61
98.46
PEINR
24.82
94.51
98.86
9.31
34.16
98.70
99.80
4.93
50
96.47
98.96
72.92
PEINR
26.91
95.85
99.28
8.17
37.10
98.96
99.90
3.28
70.03
99.80
99.98
15.93
PEINR
26.96
95.91
99.31
7.76
44.22
99.76
99.98
2.51
74.15
99.84
99.98
4.61
Dataset
(b) RT
1=16
#=1.4
(e) RM 1=16
$3=1.4
(h) FFS
n=16 /=2.66
Metric
PSNR
SSIM
CORR
DD
PSNR
SSIM
CORR
DD
PSNR
SSIM
CORR
DDx10*
NIF
15.15
83,81
89.98
85,02
23,93
90.002
97.95
17.46
48.43
95.64
98.81
193.82
CoordNet
20.73
89.20
97.16
71.38
37.98
98.46
99.91
8.78
41.16
93.92
98.41
246.5
PEINR
21.45
89.65
97.61
26.32
33.27
97.57
99.75
7.20
37.14
96.50
99.10
314.26
PEINR
22.40
91.01
97.88
24.58
37.55
98.66
99.90
3.71
51.10
99.40
99.92
58.97
PEINR
22.46
91.07
97.93
22.09
37.81
98.62
99.91
3.18
70.91
99.46
99.96
25.38
Dataset
(c) Cylinder q=4 0=11
(f) Cylinder
a=4 Ø31.4
(i) SV
a=
$=1.4
Metric
PSNR
SSIM
CORR
DD
PSNR
SSIM
CORR
DD
PSNR
SSIM
CORR
DDx 10
NIF
16.28
80.12
82.88
131.58
19.08
78.20
84.80
286.9
27.47
89.67
99.21
20.75
CoordNet
21.80
84.32
92.20
12.76
19.31
85.09
90.30
172.7
27.34
89.57
99.19
21.96
PEINR
39.15
98.52
99.85
11.53
35.7
98.88
99.86
49.6
42.21
99.12
99.97
9.91
PEINR
39.47
98.52
99.85
7.63
38.9
98.89
99.81
11.25
43.55
99.29
99.98
11.22
PEINR
40.04
99.10
99.89
2.35
40.34
99.28
99.88
1.27
45.26
99.50
99.98
4.21
due to its simple interpolation, and the low-fidelity and high- fidelity data are very different in near-real circumstances, as shown in Figure 4 (g) (h). NIF tends blur the turbulence structures, resulting in poor reconstructions of the small- scale ground-truth features, regardless of or 3. PEINR can efficiently handle the simultaneous enhancement of the numerical- numerical-precision and grid-resolution.
throughout the simulation. Figure 7 displays the compar- ison results of Cylinder dataset in non-uniform structured meshes, and it can illustrate that our method achieves better results both uniform Cartesian grid data and non-uniform structured data compared to other baselines.
5. Conclusion
In Figure 6, we compare streamline rendering results of the synthesized vector fields generated by baselines. Figure (a) and (b) display 48 streamlines generated from different seeds in the SV data set. While NIF and CoordNet can capture more details compared with BI but the tail of vor- tex obtainted are diverging and they fail to produce more complex flow details. In contrast, our approach not only maintains a high level of detail but also ensures that the complex behaviors of the flow, including vortex dynamics and the underlying fluid structure, are consistently preserved
We introduce a large-scale comprehensive flow simulation dataset covering 5 canonical flow problems 2D and 3D in both uniform Cartesian and non-uniform meshes. For each problem, results were obtained using four grid-resolution and four numerical-precisiin settings. In total, the dataset contains 33,600 vector fields, resulting in approximately 5.4 TB of data. Using this dataset, we creatively propose the coordinate- based INR method with the attention mechanism and physical encoding, which achieves satisfactory results
8

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PEINR: A Physics-cnhanced Implicit Neural Representation for High-Fidelin Flow Field Reconstruction
HI
0.022
3.00
0.50
1C89-1
0.97
PEAR)
0.80
REAR
0.65
0.003
0.000
420
407
dV
Figure 5. Comparison of model generalization performance on unseen timesteps beyond the training range. The PEINR method consistently achieves higher PSNR. SSIM and CORR values while maintaining lower DD compared to other methods, demonstrating its superior generalization capability in long-term flow field predictions.
a p=1
a 1==4 ß= 1.4
BI
NIF
Ours CoordNet
Figure 6. Comparison streamline rendering results of three- channel experiments of SV dataset. Results given by BI, NIF and CoordNet are the inferred results (i.e., the networks do not see these vector fields during training). (a) and (b) display streamlines from different seeds.
Ground Truth
in the field of high-fidelity flow fields reconstruction, so that domain experts can observe high-fidelity flow fields with less storage cost. However, the PEINR model is, thus far, tailored and appraised for particular flow scenarios. In the future, would like to explore meta- meta-learning approaches to enhance the performance out-of-disse distribbiiin samples by considering the relationship among different flow fields.
(b)
Figure 7. Comparison results of variable a field with 25 uniformly distributed contours (upscaling a == 4) given by BI, NIF, CoordNet and our method in Cylinder dataset of non-uniform structured meshes. Results in (a) WENO3 with 3 = and (b) from WENO3 to WENOS (3 = 1.4).
Impact Statement
References
This work introduces HFR-Bench, the first large-scale benchmark dataset for high-fidelity flow field reconstruction. By integrating physics-based encoding with transformer ar- chitectures, our proposed PEINR framework significantly improves the accuracy and robustness of implicit neural representations. These contributions pave the way for more generalizable and physically consistent solutions in fluid simulation tasks.
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Acknowledgment
This work was supported by the the National Key Research and Development Program of China 1704), and the National Key Research and Development Program of China (2021 YFB0300101).
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PEINR: A Physics-cnhaneed Implicit Neural Representation for High-Fidelity Flow Field Reconstruction
A. Dataset Details
This section provides a detailed introduction to the large-scale high-fidelity flow field dataset, HFR-Bench, which was developed in-house. The dataset covers various grid types, including Cartesian grids, structured grids, and unstructured grids, to accommodate different fluid dynamics simulation needs. In terms of numerical simulation methods, HFR-Bench includes unsteady compressible Direct Numerical Simulations (DNS) and unsteady incompressible Navier-Stokes (RANS) methods, ensuring the capture of complex dynamic behaviors in the flow fields. Moreover, the dataset employs high-precision numerical techniques, with Cartesian and structured grids using the commonly applied Weighted Essentially Non-Oscillatory (WENO) scheme, while the unstructured grids are modeled using the Discontinuous Galerkin (DG) method to enhance the accuracy and reliability of the simulations. For each flow configuration, there are four grid resolutions and four levels of numerical precision, with high-fidelity and low-fidelity flow fields aligned based on real simulation time. Each flow field contains corresponding grid information and raw variables. With its diverse simulation settings, -ench provides rich data support for fluid dynamics research, advancing the development and validation of related algorithms and models.
Syynolds-Averaged
A.1. Cartesian Grid Flow Field Data
For the 2D flow fields with uniform Cartesian mesh, choose the Riemann (RM), Sayleigh-Taylon instability (RT), and forward facing step (FFS) problems, which contain both discontinuities and complex flow features (Jing et al., 2003). The physical laws underlying our 2D flow fields are represented by the compressible Fuler equations for the conservative variables, which can be expressed as
u, + f(u)x + gu), 0,
(9)
where
(p. pv,
pv, E
[u) = pu? + P. pun, u(E + p))
(10)
gu)) = pue. pv² + o(E +p))²
Here, p is the density, (u, v) is the velocity, E is the total energy, p is the pressure, and fu) and g(u) the convective fluxes. Four different high-order weighted essentially non-oscillatory (WENO) schemes are employed to solve these equations. For the conservation laws, the derivative f(u), at (x,, yj) is approximated along the line y; by convective flux difference:
1 Ar f(-1)))
f(u)zle=z,
(11)
where, for the kth-order WENO scheme, the numerical flux fi+ depends on 2k-1 values f(umj), = i-k-I,..,,i+ k-k/ We set k to 3, 5, 7, or 9 to give third-order (WENO3), fifth-order (WENO5), seventh-order (WENO7), and ninth-order (WENO9) Runge-Kutta Kutta method, and the Lax-Friedrichh building blocks
versions. Time discretization achieved via the third-order Runge
and local characteristic decomposition are employed.
Riemann (RM) Problem. The two-dimensional RM problem containing multiple dimensions in the flow field is a common example of testing the efficiency of numerical methods. It solves the two-dimensional Euler equations in the unit square region [0, 1]x[0, [0, 11, with initial conditions as follows:
(1.5,0,0, 0.5,
for 0 0
(0.5323, 1.206, 0.3),
for 0.8 0,
u. (p.u...p)=
(12)
(0.138, 1.206, 1.206, 0.029),
for 0.8 1 for 0 x < 0.8, 0.8 $ <1
(0.532, 1.206, 0,0.))
The computational grid sizes used are 200 x 200, 400 x 400, 800 800, and 1600 x 1600, with the simulation time set to = 0.8. The time step for storage is set to 0.0016. Figure 8(a) illustrate the flow field results generated by the WENO method under different grid resolutions. At the same grid resolution, the flow field generated by WENO methods with different accuracies exhibit distinct multi-scale flow field structures. Similarly, for the same accuracy the WENO method, flow field cloud plots at different grid resolutions also display varying sulti-scale flow field structures. Both numerical precision and grid resolution significantly enhance the multi-scale resolution.
13

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PEINR: A Physics-cnhanced Implicit Neural Representation for High-Fidelity Flow Field Reconstruction
Rayleigh-Taylor Instability (RT) Problem. In this problem, two fluids with different densities interact under gravity, with the denser fluid moving downward and the lighter fluid moving upward, which leads to the development of instability. The computational domain is defined as [0, 0.25] x [0. IJ with initial conditions:
(2. 0, 0.02 0 2y 1), 0 S y < 0.5
(p. u, v,
(13)
(1. 11.00,0.05 (,000-00005/y/////ocoo0tor)a)a 1.5),
0.5 - S
where the heat parameter is specified as 5/3. The boundary conditions for the left and right sides are taken as reflective, while the top and bottom boundaries are assigned with constant values:
(1,0,0,25) y= 1
(p, (,,uv..))(,.u.1))
(14)
0,,0, y = 0
The computation ends & 1.95, with flow field storage timestep of 0.0039. The grid resolutions used are 120 480, 240 x 960, 480 x 1920, and 960 x 3840. Figure 8(b) shows the flow field results generated using different accuracy WENO methods at the same grid resolution. As observed, with increases in numerical precision and grid resolution, the vortex structures in the flow field become clearer and more intricate.
This problem is commonly used in high-order numerical schemes and has been well-received numerical testing, particularly for its ability to capture the instabilities in the fluid flow due to the presence of a source at the Mach stem and the resulting vortex formations, which are of great interest in fluid dynamics. The computational domain is defined [o, 0.6] [0, 0.2] U [0,3] [0.2, I], which corresponds length of and a height of I, where a step with length of 2.4 and height of 0.2 is placed at position 0.6 units away from the left inlet of the wind tunnel. The initial flow field in the computational region is given by:
Forward Facing Step (FFS) Problems.
small wind tunnel with
(p,u, u, p, = 31,4,300,,,1,
(15)
The simulation ends at t 4, with flow storage timestep of 0.008. The grid resolutions used are 180 x 48 U 36 x 12, 360 x U6 U72 x 24, 720 x 192 U 144 48, and 1440 x 384 U 288 x 96. Figure (a) present the flow field results generated using different accuracy WENO methods at the same grid resolution. As can be observed, as the numerical precision and grid resolution increase, the vortex structures become clearer.
Shock-longitudinal Vortex Interaction (SV) Problem. For our 3D SV flow field, the underlying physical laws are represented by the following 3D unsteady Navier-Stokee equations for the conservative variables:
1 - (fo(u)z
U, f(u)z + gu), h(u);
Re
(16)
+ (u), + he (u)s),
where Re is the Reynolds number f. g..(u,, (u) are the viscous fluxes. A stationary shock is initially located at the == 0 plane. Upstream of the shock (x i 0),
1,
U2 0.05
u, = ug O, p 1,
(17)
where Mg., ur, and ug are the axial, radial, and azimuthal velocity components. The downstream mean solution (x 0) is
(7 + 1)M
U₂ = 707 -05---0 M)M +2) ( 1)M?
p (7 - 1)M? +
(18)
u, = ug = 0, p= M7 (7-
7+1
An isentropic vortex is superimposed on the mean flow upstream of the shock. The axis of the vortex along the r-axis (y = z= 0). The perturbations in the azimuthal velocity and temperature associated with the vortex are expressed as
ug = er
c 0.5(1-+2)
2π
T' = (7 - 1) (1-22)
(19)
2
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PEINR: A Physics-enhanced Implicit Neural Representation for High-Fidellis Flow Field Reconstruction
WENO3
WENO5
WENO3
WENO5
WENO7
200 X 200
120 X 480
3
480 X 1920
800 X 050
Figure 8. (a) x-direction velocity component u of dataset RM contours with 25 equally spaced isosurfaces in diffenrent grids and computation accuracy. (b) p flow field in different grids and computation accuracy.
where r= v22 + 22 is the radius to the vortex axis, to is the vortex core radius, and e is dimensionless circulation at r 1. In our simulations, the value of Re, 7, 1, and æ are set to 10S, 1.4, 2, and 7, respectively. The grid sizes are 96x x 32, 192x64 64, 384x 128x 128, and 768 x56xx2266 Moreover, the final simulation time 11.0 and the time step 0.11 s. Therefore, 1,600 4xx4x4x 100) 3D snapshots are separately collected. Figure 9(b) display the component u and v contours for SV for WENO3 and WENO5 using the 96 32 32 and 384 128 x 128 grids, respectively. We can clearly observe significant difference between the high-fidelity results and their low-fidelity counterparts. The grid-resolution (GR) and sumerical-precisti (NP) of four datasets are listed in Table 1.
A.2. Flow Field Data with Non-uniform meshes
For the non-uniform meshes, we select the circular jet standard test case. The maximum grid spacing difference and the highest precision can reach 64 times and 9 orders of magnitude, respectively. The control method is a wwo-dimensional incompressible RANS equation group. The Reynolds number re is 200, and the calculation time is stopped at f. = 600.0, with a time step dt = 0.11. Figure 10 shows the flow field contours under different precision WENO grid models for the same grid model. As can seen from figures, as the precision and grid refinement increase, the flow field's turbulence becomes clearer.
B. Experimental Details
Our model is implemented with Pytorch 1.10 (Paszke et al., 2019), and all experiments are conducted on a single NVIDIA A100 80GB GPU. The batch size (BS), memory (MM) and experimental variable and time per epoch are listed in Table I. Our method and all the baseline methods are trained with the MSE loss in 2000 epochs. Learning ratio is set to le - and decrease after 20 epochs if there is no loss degradation on the traning set and the max epochs is 2000. (Every
15

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PEINR: A Physics-cnhanced Implicit Neural Representation for High-Fidelity Flow Field Reconstruction
WENO3
WENO7
WENO3
WENO5 x 128 128
36 x 12
200 18
144 144-X-4 48
720 192
(a)
(b)
Figure 9. (a) x-direction velocity component и of dataset FFS contours with 25 equally spaced isosurfaces in diffenrent grids and computation accuracy and bottom rows displays the difference between low-resolution and high-resolution within same computation scheme. (b) The two rows displays x-direction velocity component and y-direction velocity component in the flow field of SV in different grids and computation accuracy.
method including ours can converge within 1000 epochs). During the trainin process, we adopt the AdamW optimizerfor optimization. During design, we can set the number of residual layers of Resu MLP to 10 and the max number of neutrons Resu MLP is 64. To promote the learning of bases and ensure consistency of time series across different dimensions, we normalized the time series during training and performed inverse normalization when outputting the results. For the other models compared in the table, we utilized their original code and conducted experiments by only varying the input length.
PSNR (Peak Signal-to-Noise Ratio), SSIM (Structural Similarity Index), CORR (Correlation Coefficient) and DD (Dissipa- tion difference) results used evaluate the quality of the synthesized HF flow fields. These three metrics are defined below, where the high-fidelity flow field and synthesized flow field are F and F$ respectively.
PSNR. This represents the ratio of the maximum possible power of the signal and the destructive noise power that affects its representation accuracy, PSNR computed using the mean squared error by the following formula:
(2 1)²
PSNR(F" FH _FS) = 10 x logı0 MSE(FH,FS
(20)
WENO3
WENO5
WENOS
101 x 761
51 381
Figure x-direction velocity component of datasct cylinder contours with equally spaced isosurfaces in diffenrent grids and computation accuracy.
16

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PEINR: A Physics-cnhanced Implicit Neural Representation for High-Fidelity Flow Field Reconstruction
SSIM. This measure of the similarity of two images, mimicking human perception by focusing primarily edge and texture similarity. It is calculated by the following formula:
(2 pos + c1) p2 + C2)
SSIM(F" F))
(21)
(pLT + p +c) (ops + ops + 02)
where рeв and Hys are the average values of Fd and ps respectively. of and of are the variance of f and Fs. respectively. rra pe is the covariance of Fol and F$.
CORR. The correlation coefficient reflects the degree of correlation between variables, and is defined as:
E (F ppss) (F" 1pH
CORR (F"l, Fb)
(22)
V (FS (Mps) V Ei= (F 1pp
Dissipation difference. The dissipation operator is used to assess the model performance in terms of capturing the discontinuities and rotating features. The dissipation of each velocity component (u, e, and w) evaluated. The dissipation operator is defined by
x(F) = OF
OF Oy
OF
+
(23)
dx
Oz
The dissipation experiments aim measure the difference in flow gradient between the ground truth data (GT) and the reconstructed data given by baselines and our method. This is represented by x x09) (Q+)) with lower values indicating better performance.
17


Reviews:
[{"review_id": "xN2rVvdPdb", "reviewer": "Reviewer_zZ8o", "summary": "The paper introduces PEINR (Physics-enhanced Implicit Neural Representation), a framework designed for high-fidelity flow field reconstruction. The authors highlight the limitations of current implicit neural representation (INR) methods in handling complex spatiotemporal dynamics and accurately capturing fine-scale flow structures, especially when applied to computational fluid dynamics (CFD). To overcome these challenges, PEINR integrates physical encoding techniques and a transformer-based spatiotemporal fuser (TransSTF). The physical encoding decouples spatial and temporal components, improving resolution and accuracy. TransSTF captures long-range temporal dependencies using a multi-head attention mechanism, enhancing the model's ability to reconstruct high-fidelity flow fields. The authors introduce a large-scale dataset, HFR-Bench, which includes over 33,600 unsteady 2D and 3D flow fields across various grid-resolutions and numerical precisions. Experimental results demonstrate that PEINR significantly outperforms existing INR-based methods in both qualitative and quantitative evaluations, achieving superior reconstruction quality in flow simulations.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your reviews. Here are our explanations about the weaknesses and problems.\n\n1.Response to comments on missing citations\n\n[1] de Vito et al. (2024) Implicit Neural Representation For Accurate CFD Flow Field Prediction.  \n[2] Du et al. (2024) Conditional neural field latent diffusion model for generating spatiotemporal turbulence.\n\nWe will add the citations in the revised manuscript to ensure completeness. The following are our descriptions:\nRecent 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. \nWe 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.\n\n2.Response to comments on spatial localization encoding clarification\n\nThe 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.\nIn 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. \n\n3.Response to comments on clarification on grid-independence\n\nWe 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. \nIn 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.\nIn 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.\nIn 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.\n\n4.Response to comments on PSNR result of Riemann problem\n\nWe appreciate your observation. \nThe 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. \nWe will clarify this rationale in the revised manuscript.\n\n5.Response to comments on our dataset availability\n\nWe 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.", "claims_and_evidence": "The paper is self-evident with its claims.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "Experimental designs and analyses are valid.", "relation_to_broader_scientific_literature": "The method and dataset is valuable for physics and simulation applications, especially CFD / fluid dynamics.", "essential_references_not_discussed": "de Vito et al. (2024) Implicit Neural Representation For Accurate CFD Flow Field Prediction.\nDu el al. (2024) Conditional neural field latent diffusion model for generating spatiotemporal turbulence.", "other_strengths_and_weaknesses": "Strengths:\n\n- The introduction of HFR-Bench, a large-scale dataset containing 5.4 TB of flow field data, provides a valuable resource for benchmarking and advancing research in flow field reconstruction.\n\n- Innovative Physical Encoding Techniques: The use of Gaussian coordinate encoding for temporal information and localized encoding for spatial coordinates enhances the model's ability to capture fine-scale structures and spatiotemporal dynamics.\n\n- Architecture: The introduction of the TransSTF block, leveraging multi-head attention, allows PEINR to model long-range temporal dependencies effectively, which is crucial for capturing the full complexity of fluid dynamics.\n\n\nWeaknesses:\n\n- Some important method details are not clear enough for me to fully understand the method, e.g. the Localized spatial encoding.\n\n- The method assumes grid independence for flow field attributes across different mesh resolutions, which, as acknowledged in the paper, may not hold true in all real-world simulations. This assumption could limit the model's application in certain practical scenarios where grid dependence is significant.", "other_comments_or_suggestions": "I would expect an explanation on the Riemann problem why PEINR performs weaker than CoordNet in terms of PSNR.", "questions_for_authors": "- Please clarify the localized spatial encoding and how the information of the neighbors are encoded.\n\n- As claimed as one of the major contributions, will the dataset be publicly available?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wRies59XP2", "reviewer": "Reviewer_4FEk", "summary": "This paper introduces PEINR, a physics-enhanced implicit neural representation framework for high-fidelity flow field reconstruction. The authors address three key limitations of existing INR methods: 1) invalid grid independence assumption, 2) temporal-spatial complexity disparity, and 3) spectral bias. The work contributes: 1) HFR-Bench: A 5.4TB CFD dataset (33,600 fields) with uniform/non-uniform meshes; 2) PEINR framework combining physical encoding and transformer-based fusion; 3) Novel temporal encoding via Gaussian RBF kernels and spatial stencil-aware encoding.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your reviews. Here are our explanations about the weaknesses and problems.\n\n1.Response to comments on the single 3D dataset\n\nThe 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.\nThe 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.\n\n2.Response to comments on the experimental setting\n\nIn 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. \nWe 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.\n\n3.Response to comments on the difference from other methods (grid-free methods and PINNs)\n\nIn 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. \n\nPINNs 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]. \n\n[1] Farea et al. Understanding physics-informed neural networks: techniques, applications, trends, and challenges.\n\n4.Response to comments on the quantization of the impact of mesh non-uniformity\n\nTo 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. \n\n5.Response to comments on missing citations\n\nWe appreciate the references you suggested. But we can only find this paper: “MeshGraphNets” and we will include this citation in our revision. \n\n6.Response to comments on 6D Coordinates\nWe appreciate the reviewer’s insightful question.\n\nIn 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.\n\n7.Response to comments on sigma\n\nWe 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.  \n\nWe 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). \n\n8.Response to comments on uncertainty quantification\n\nWe 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.\nhttps://drive.google.com/file/d/1Ghv7-cWKf2DQUEt-5O3JUdZfXVkIf3v-/view\n\n9.Response to comments on the practical limit for numerical precision enhancement $B_{max}$\n\n$B_{max}$ depends on the training data. The numerical precision of data is higher, then $B_{max}$ is higher.\n \n10.Response to comments on transferability of PEINR\n\nSince 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.", "claims_and_evidence": "1 Grid independence assumption violation: Fig.1 shows the attribute values at the same location change as the grid resolution (in the row direction) and numerical accuracy (in the column direction) increase.\n\n2 Nonlinear Temporal Encoding effectiveness: The analysis of NTK in Fig.3 demonstrates improved temporal consistency.\n\n3 Spectral bias mitigation: Ablation study (PEINR1 vs. PEINR2 in Table.2) shows spectral blocks reduce dissipation difference (DD).", "methods_and_evaluation_criteria": "Methods: The proposed method, for the first time, addresses both grid-resolution enhancement and numerical-precision enhancement simultaneously through a single model, breaking through the limitations of traditional methods that focus solely on a single objective.\n\t\nBenchmark: The 5.4 TB of data proposed cover five classic flow problems (2D/3D, structured/unstructured grids), and for the first time, provide a strict pairing of grid resolution ($\\alpha$=1\\~64) and numerical precision ($\\beta$=1\\~7 orders), filling the gap in the CFD community's lack of standardized benchmarks.\n\t\nMetrics: All metrics are explained in the supplementary material and are quite relevant to this issue.", "theoretical_claims": "1. NTK theory adaptation for temporal encoding shows translation invariance (Eq 2-3)\n\n2. The combination of the RBF kernel and Kernel PCA (Eq4-5)\n\n3. Spectral blocks enable high-frequency capture (Table 2)", "experimental_designs_or_analyses": "1. The introduction of HFR-Bench is a comprehensive benchmarking and experiments isolate the effects of grid resolution(\\alpha) and numerical precision(\\beta) by parsing low/high-fidelity fields with identical initial conditions and physical parameters. The testing on diverse flow regimes and the ablation study are sufficient. There are still some concerns in experiments.\n\nOne of my concerns is that there is only one 3d case(SV problem) that is tested, and what about the scalability to larger 3D domains(e.g., turbulent boundary layer)? I may not know much about it, but hopefully, there's a reasonable explanation. Another question is about the training-test temporal split. Why only test on the final 20 steps for uniform meshes ( the second column of L312 on page 6)? I think this may not fully stress long-term stability, and if a more aggressive split (e.g., training on first 50% steps and testing on latter 50%) would better assess robustness? After searching relevant literature, I found that in addition to regular and irregular grids, there are also grid-free methods(e.g., PINNs). The omission of this part may make the value of the proposed method discounted to a certain extent. I hope the authors can discuss this issue.\n\n2. The evaluation and the ablation study are convincing, and the proposed theories are well supported. While PEINR avoids the flawed grid-independence assumption, the paper does not quantify how mesh non-uniformity impacts performance, and there is no comparison to uniform-mesh performance under similar α/β.", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "The following references should be discussed in the paper:\n1. MeshGraphNets (Learning Mesh-Based Simulation with Graph Networks, 2020) for irregular meshes", "other_strengths_and_weaknesses": "Strengths\n- Comprehensive dataset with rigorous numerical basis\n- Novel integration of spectral blocks with transformer\n- Strong empirical results\n\nWeaknesses\n- Limited 3D validation (only 1 test case)\n- Could there be a comparison with grid-free methods or some discussion on those grid-free methods?\n- No comparison to uniform-mesh performance under similar α/β, I think it would be essential experiments about the insight of theories of this paper.", "other_comments_or_suggestions": "NA", "questions_for_authors": "1\tHow does the localized encoding scale to 3D+time with 6D coordinates?\nIn a 3D + time scenario, the input coordinate dimension is increased to 4D (x, y, z, t). If physical parameters or boundary conditions are further considered, it may rise to 6D. The paper does not verify such high-dimensional extendibility, nor does it analyze the trend of parameter increase in the encoding layer.\n\n2\tCould adaptive σ in Gaussian encoding improve temporal modeling?\nIn this paper, the parameter \\(\\sigma\\) in the Gaussian RBF kernel is set to a fixed value of 0.1 (Section 4), while the NTK analysis in Fig.3c-d indicates that \\(\\sigma\\) affects the spectral properties (small \\(\\sigma\\) captures high frequencies, while large \\(\\sigma\\) smooths out the spectrum). The paper does not explore dynamically adjusting \\(\\sigma\\) to adapt to different time scales (such as transient versus steady-state flow), which might optimize multi-scale dynamics modeling.\n\n3\tHave you explored uncertainty quantification for reconstruction errors?\nIn the field of CFD (Computational Fluid Dynamics), there is a high sensitivity to the credibility of errors (for example, in aviation safety assessments). However, the experiments only report mean metrics (such as PSNR and SSIM) without providing confidence intervals or analyses of error distributions. I would like the authors to discuss this question. But this does not detract from the value of the work itself.\n\n4\tWhat's the practical limit for numerical precision enhancement (β_max)?\n\n5\tHow transferable is PEINR across different fluid regimes (e.g., turbulent vs laminar)?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DAXyU4hbfy", "reviewer": "Reviewer_1eeX", "summary": "This paper introduces HFR-Bench, a large-scale CFD dataset, with 33,600 unsteady 2D and 3D vector fields across various grid resolutions and numerical precisions, providing a benchmark for flow field reconstruction. It also proposes PEINR, a novel physics-enhanced INR model that improves flow field reconstruction by enhancing numerical precision and grid resolution using physical encoding and a transformer-based spatiotemporal fuser (TransSTF). Experiments show that PEINR outperforms existing methods like NIF and CoordNet.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your reviews. Here are our explanations about the weaknesses and problems.\n\n1.Response to comments on the missing citation CoordNet(Line 246)\n\nThanks 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.\n\n2.Response to comments on temporal interpolation\n\nBy 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.\n\n3.Response to comments on the RBF-enhanced NTK\n\nWe 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.\n\nThe 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.\n\nThe 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. \n\nThe NTK can be directly computed as: \n$\\gamma\\left(x, x^{\\prime}\\right)=\\nabla_\\theta f(x,\\theta)^{\\top} \\nabla_\\theta f\\left(x^{\\prime},\\theta\\right)$, where $f(x,\\theta)$ represents the output of MLP and $\\theta$ represents parameters of MLP.\nIn 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. \nOur MLP input form is [$\\Phi(t)$,$\\Psi(\\vec C)$]. By incorporating the RBF kernel’s stationary properties, the RBF-enhanced NTK becomes:\n$\\gamma_{\\mathrm{RBF}-\\mathrm{NTK}} \n=\\gamma_{M L P}\\left([\\Phi(t), \\Psi(\\vec{C})],\\left[\\Phi\\left(t^{\\prime}\\right), \\Psi\\left(\\vec{C}^{\\prime}\\right)\\right]\\right)\n=\\gamma_{\\mathrm{MLP}}\\left(\\Psi(\\vec{C}), \\Psi(\\vec{C})^{\\prime}\\right) \\cdot K_{\\mathrm{RBF}}\\left(t, t^{\\prime}\\right)$.\nThe 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.\n\n\n4.Response to comments on syntax errors\n\nWe sincerely appreciate the reviewer's careful reading and valuable feedback on grammar and phrasing. We have thoroughly revised the manuscript to address these issues.\n\n5.Response to comments on training time\n\nWe 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.\n\n6.Response to comments on training timesteps\n\nIn 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.\n\n7.Response to comments on average results of our experiments\n\nWe 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).\nIt 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.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "The main contribution is a novel large-scale CFD dataset, providing a benchmark for flow field reconstruction.", "essential_references_not_discussed": "Yes. \nLine 246, missing citation because the ResuMLP architecture is proposed by CoordNet.", "other_strengths_and_weaknesses": "Strengths:\n1.\tThis paper introduces a novel large-scale CFD dataset, providing a benchmark for flow field reconstruction.\n2.\tThe experiments are extensive and demonstrate the advantage of PEINR. \nWeaknesses: \n1.\tPEINR seems unable to perform temporal interpolation when using the proposed temporal encoding, otherwise it requires retraining.\n2.\tThe RBF-enhanced NTK is confusion. How is it implemented in detail? For example, Line 235 “By incorporating the RBF kernel”, how does MLP combine with RBF kernel?\n3.\tLine 246, missing citation because the ResuMLP architecture is proposed by CoordNet.", "other_comments_or_suggestions": "Line 64: “can … enhancing” -> “can … enhance”\nLine 89: “concentrate” -> “concentrates”\nFigure 3 (c) (d), the \\sigma value are not labeled on the curves.\nLine 328, “9 points” -> “8 points” ?\nLine 369, “lower values indicating better performance” is redundant.", "questions_for_authors": "1.\tWhy are training times in Table 1 shorter than the inference times? \n2.\tHow does PEINR perform with long time series? There are 500 timesteps in the experimental data, but only a few timesteps are used.\n3.\tAre the quantitative results in Table 2 averages or values from a single timestep (as described in the Implementation Details)? The average results are more convincing.\n4.\tSee weaknesses for other questions.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2GFZoHo50y", "reviewer": "Reviewer_ijmU", "summary": "The paper introduces two contributions:\n\n1. Multi-Fidelity Large-Scale CFD Dataset: A comprehensive dataset generated using the WENO solver, comprising five different configurations of 2D and 3D flows that include shock phenomena. Each configuration is provided at two resolutions: a low-resolution version (approximately 100×100) and a corresponding high-resolution version (approximately 1000×1000).\n2. Physics-Enhanced Implicit Neural Representation (PEINR) Framework: A novel neural architecture designed to learn the discrepancy between low-fidelity and high-fidelity simulation data. The PEINR incorporates several specialized components:\n- Spatial and temporal encoding using RBF and PCA\n- Transformer blocks\n- Spectral blocks \n\nThe framework aims to bridge the gap between computationally efficient low-resolution simulations and more accurate high-resolution simulations by learning the difference between these fidelity.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your reviews. Here are our explanations about the weaknesses and problems.\n\n1.Response to comments on transferability and generalization of PEINR\n\nWe 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. \nAt 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).\n\n2.Response to comments on the input of PEINR\n\nOur 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.\n\n3.Response to comments on the motivation of using INR\n\nWe appreciate your inquiry into the rationale for our high-fidelity reconstruction method compared to simple downsampling. \nOur 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. \n\nWhile 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. \n\nIn 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.\n\n4.Response to comments on the relation to broader scientific literature\n\nOur 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. \n\nNevertheless, 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. \n\nRegarding 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.\n\n5.Response to comments on missing citations\n\nThank you for your valuable suggestions. We will incorporate discussions of these references in the revised version. \n\n6.Response to comments on inverse problems definition\n\nIn 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.\n\n[1] Chen, Yang, et al. \"HOIN: High-Order Implicit Neural Representations.\" arXiv preprint arXiv:2404.14674 (2024).\n\n7.Response to comments on ablation study\n\nThe 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\nPEINR$^2$ only removes the physical encoding. As shown in\nTable 2, PEINR outperforms PEINR$^2$ in the metric DD in\nall cases, which indicates that our physical encoding method\ncan preserve more physical peculiarity. The superiority of\nPEINR$^2$ compared with PEINR$^1$ highlights that the added\nspectral block can also help the model to perform better.\n\n8.Response to comments on Figure 2\n\nThanks 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).", "claims_and_evidence": "The dataset presented appears to be a valuable contribution to the community. However, I have several concerns regarding the PEINR model that require clarification:\n\n### Model Architecture and Training Approach\n- Is a separate model trained for each flow case (as in traditional INR approaches), or is a single model trained to handle a family of cases (similar to super-resolution/operator learning paradigms)?\n- What precisely serves as the model input? Are these spatial coordinates or low-resolution images?\n\n### Unclear Motivation\nThe fundamental motivation for using an INR to represent the difference between low- and high-fidelity simulations requires further explanation:\n- If the goal is efficient storage of high-resolution data, why not directly downsample the high-resolution dataset and use a low-resolution solver instead?\n- If the aim is super-resolution from low-fidelity to high-fidelity simulations, how does the model generalize across different simulation cases?\n\n### Model Components\nThe model architecture incorporates several components that appear somewhat arbitrarily selected. Additional ablation studies would significantly strengthen the paper by demonstrating the necessity and contribution of each component.", "methods_and_evaluation_criteria": "The paper proposes a new dataset. But I am not sure what is the motivation to do recontruction of low- and high- fidelity simulation.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "N/A", "relation_to_broader_scientific_literature": "If the goal is to reconstruct a fluid field, it is very similar to physics informed neural nework (PINNs)[1] which should be discussed.\nIf the goal is super-resolution. It should discuss standard models in super-resolution as well as neural operators [2].\n[1] Karniadakis, George Em, et al. \"Physics-informed machine learning.\" Nature Reviews Physics 3.6 (2021): 422-440.\n[2] Kovachki, Nikola, et al. \"Neural operator: Learning maps between function spaces with applications to pdes.\" Journal of Machine Learning Research 24.89 (2023): 1-97.", "essential_references_not_discussed": "Previous works on augmenting coarse grid solution:\nPathak, Jaideep, et al. \"Using machine learning to augment coarse-grid computational fluid dynamics simulations.\" arXiv preprint arXiv:2010.00072 (2020). \nLi, Zongyi, et al. \"Physics-informed neural operator for learning partial differential equations.\" ACM/JMS Journal of Data Science 1.3 (2024): 1-27.", "other_strengths_and_weaknesses": "Overall the method section could be improved. The inverse problem has no definition. Please first define the problem setting and then discuss the method.\n\nAre all components such as neighbor coordinate, RBF and PCA for time encoding, and spectral block necessary? Each component in PEINR may need an ablation study. If some components of the architecture is not so important, I suggest remove them into the appendix and use the space to define the problem.\n\nFig 2 is too small and it is hard to understand.", "other_comments_or_suggestions": "I like the dataset, and I am happy to raise my score if my questions on the method can be addressed.", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: Reviewers all appreciated the method, and found it sound and well supported by the experiments, though there are questions around the motivation and generality of the approach. Overall, the rebuttal was thorough and addressed some of the concerns, leading to a clear consensus towards acceptance. The dataset was found to be a valuable contribution. As a result, I believe that this work would be of interest to the community and I recommend for acceptance. I would ask the authors to incorporate the discussions into the updated manuscript and revisit the motivation at the light of the discussions.
Decision: Accept (poster)