RealPDEBench: A Benchmark for Complex Physical Systems with Real-World Data
Authors: Peiyan Hu
Haodong Feng
Hongyuan Liu
Tongtong Yan
Wenhao Deng
Tianrun Gao
Rong Zheng
Haoren Zheng
Chenglei Yu
Chuanrui Wang
Kaiwen Li
Zhi-Ming Ma
Dezhi Zhou
Xingcai Lu
Dixia Fan
Tailin Wu
Venue: ICLR 2026 Oral (2026)
Keywords: complex physical system
PDE
benchmark
real-world data
prediction
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We propose the first benchmark for complex physical systems with paired real-world data and simulated data, and explore how to bridge simulated and real-world data.

Abstract: Predicting the evolution of complex physical systems remains a central problem in science and engineering. Despite rapid progress in scientific Machine Learning (ML) models, a critical bottleneck is the lack of expensive real-world data, resulting in most current models being trained and validated on simulated data. Beyond limiting the development and evaluation of scientific ML, this gap also hinders research into essential tasks such as sim-to-real transfer. We introduce RealPDEBench, the first benchmark for scientific ML that integrates real-world measurements with paired numerical simulations. RealPDEBench consists of five datasets, three tasks, nine metrics, and ten baselines. We first present five real-world measured datasets with paired simulated datasets across different complex physical systems. We further define three tasks, which allow comparisons between real-world and simulated data, and facilitate the development of methods to bridge the two. Moreover, we design nine evaluation metrics, spanning data-oriented and physics-oriented metrics, and finally benchmark ten representative baselines, including state-of-the-art models, pretrained PDE foundation models, and a traditional method. Experiments reveal significant discrepancies between simulated and real-world data, while showing that pretraining with simulated data consistently improves both accuracy and convergence. In this work, we hope to provide insights from real-world data, advancing scientific ML toward bridging the sim-to-real gap and real-world deployment. Our benchmark, datasets, and instructions are available at https://realpdebench.github.io/.

Published as a conference paper at ICLR 2026
REALPDEBENCH: A BENCHMARK FOR COMPLEX PHYSICAL SYSTEMS WITH HEAL-WORLD DATA
Peiyan Hu* i Haodong Feng*! Hongyuan Liu"! Tongtong Yan Wenhao Deng' Tianrun Gao 1,4 Rong Zheng 155 Haoren Zhengi 1.2 Chenglei Yu! Chuanrui Wang' Kaiwen Lit 1,2 Zhi-Ming Ma Dezhi Zhou² Xingcai Lu Dixia Fan' Tailin Wu (hupeiyan, fengha liuhengyuan, wutailing wutailinjewestlakee @westlake edu on. "School of Engineering, Westlake University; Glooba College, Shanghai Jiao Tong University; "Academy of Mathematics and Systems Science, Chinese Academy of Sciences;
4 'Department of Geotechnical Engineering, Tongji University: SScooo of Physics, Peking University;
Key Laboratory for Power Machinery and Engineering of M. O. E., Shanghai Jiao Tong University
Real-world data
Simulated data
Cylinder
Controlled Cylinder
Fail
Combustion
Figure 1: Five scenarios in RealPDEBench with the corresponding real-world and simulated data. demonstrates the differences between real-world and simulated data, such as the different modalities, measurement noise, and numerical errors. These discrepancies motivate and call for our proposed benchmark, RealPDEBenchh to systematically collect data and conduct experimental analysis.
ABSTRACT
Predicting the evolution of complex physical systems remains central problem in science and engineering. Despite rapid progress in scientific Machine Learn- ing (ML) models, critical bottleneck is the lack of expensive real-world data, resulting in most current models being trained and validated on simulated data. Beyond limiting the development and evaluation of scientific ML, this gap also hinders research into essential tasks such as sim-to-real transfer. We introduce RealPDEBench, the first benchmark for scientific MI that integrates real-world measurements with paired numerical simulations. RealPDEBench consists of five datasets, three tasks, nine metrics, and ten baselines. We first present five real- world measured datasets with paired simulated datasets across different complex physical systems. We further define three tasks, which allow comparisons be- tween real-world and simulated data, and facilitate the development of methods to bridge the two. Moreover, we design nine evaluation metrics, spanning data- oriented and physics-oriented metrics, and finally benchmark ten representative baselines, including state-of-the-an models, pretrained PDE foundation models, and traditional method. Experiments reveal significant discrepancies between simulated and real-world data, while showing that pretraining with simulated data consistently improves both accuracy and convergence. In this work, hope provide insights from real-world data, advancing scientific ML toward bridging the sim-to-real gap and real-world deployment. Our benchmark, datasets, and instructions are available at htttt::////. http://aaaldddddddeee github. io/
"Equal contribution. Work done intern Westlake University. 'Corresponding author.

---

Published as a conference paper at ICLR 2026
INTRODUCTION
Predicting the future evolution of complex physical systems central and enduring problem in both science and engincering, Such systems play fundamental roles in a wide range of domains, including fluid dynamics (Griebel et al. 1998] Ferziger & Peric 2002). combustion (Poinsot Vey-
nante 2005| Glarborg et al. 2018). and plasma physics (Chen 2015). These systems are charac- terized by complex spatiotemporal dynamics. Their high-dimensionall strongly nonlinear, unsteady, and tightly coupled nature presents a formidable barrier to accurate prediction.
To more accurately predict complex physical systems, a series of traditional numerical methods have been developed (Neumaier 1987 Bratanow 1978 Dormand & Prince 1980; Griffith & Patankar 2020). These methods approximate complex physical phenomena by formulating them as symbolic Partial Differential Equations (PDEs), which are then discretized and solved numerically to obtain approximate solutions. More recently, numerous scientific Machine Learning (ML) models have emerged. These models train neural networks by optimizing loss functions defined either from data (Praff et al. 2020 Li et al., 202T) or from governing PDEs (Raissi et al. 2 Cuomo et al. 2022) thereby learning the dynamics of complex physical systems.
Although scientific MI. has gained widespread attention due to its efficiency, it still has some is- sues. One of the most significant ones is that most current models are only learned and validated on simulated data from traditional numerical solvers, lacking experiments in real-world scenarios (Thyaggainngam et al. 2022). However, due factors such as numerical errors, measurement noise, and unmeasured modalities (variables of the physical system). there is huge gap between real- world measured data and numerically simulated data (Roache, 1998] Oberkampf & Trucano| [2002] Kravchenko & Moin 2000 Veynante & Vervisch, 2002) Hochgreb 2019). As a result, we can- not truly evaluate how current scientific ML models perform in the real world against numerical methods. The absence of real-world datasets has fundamentally restricted both the development and evaluation of scientific ML. Moreover, this limitation further hinders progress on many important tasks, such learning from noisy real-world data, sim-to-real transfer, and understanding how limi- tations of measurement techniques influence model performance. Consequently, real-world datasets crucial for the advancement of scientific ML, yet they have remained largely scarce (Thiya-) galingam et al. 2022). This scarcity primarily stems from the high cost of acquiring real-world data, as it requires the construction of experimental setups and the rich experience of measurement.
To bridge this gap, we propose the first scientific ML benchmark in real-world complex physical sys- tems, RealPDEBench, which contains paired real-world data and simulated data. Our contributions are fourfold: data, tasks, metrics, and baselines. First, from the data perspective, we provide more than 700 trajectories, each exceeding 2000 frames under distinct operating conditions, that cover five scenarios in the domains of fluid dynamics and combustion. For each scenario, multiple system parameters are included, and for every parameter setting, both real- world data and simulated data of equal temporal duration are provided. In particular, the Combustion dataset highlights the im- portance of real-world data, as its inherent complexity, multi-physics, and multi-scale nature make accurate simulation challenging. Second, from the task perspective, evaluated on real-world data. Specifically, the models are (i) trained on simulated data, (ii) trained on real-world data, and (iii) pretrained simulated data and then finetuned on real-world data. Through experiments across the three tasks, we are able to compare the respective strengths and limitations of real-world and simulated data, Our work provides foundation for further exploring how to combine the advantages of both and achieve improved models for future work.
consider three settings, all
Third, we provide comprehensive set of evaluation metrics, comprising nine in total. Specifi- cally, these metrics can be categorized into data-oriented and physics-oriented metrics, along with a dedicated metric designed for investigating the impact of pretraining on simulated data. Fourth, for the baselines, we consider nine data-driven scientific ML models and one traditional method. They include not only the latest state-of-the-art (SOTA) architectures but also the pretrained foun- dation PDE model. We systematically compare their modeling capabilities on real-world data. All the aforementioned datasets, tasks, metrics, and baselines are integrated into a unified and highly modular code framework, which enables rapid adaptation of prediction and sim-to-real tasks to new datasets or models. Our experimental findings reveal a gap between real-world and simulated data. On the other hand, also observe that reasonably leveraging simulated data can improve prediction performance on rorl-world data, which offers guidance for advancing Scientific ML models toward real-world applications.
2

---

Published as a conference paper at ICLR 2026
2 RELATED WORK
Several benchmarks have been proposed (Chung et al.
2022 Guppa & Brandstetter 2022 Toshey
ctt al. 2023 Koehler et al. 2024 Liu al. 2024bb Nathaniel et al. 2024) evaluate the scientific
ML models on PDEs (Li cl al. 2021] Raissi et al. 2019; Lu et al. 2022 Li et al. 2024), fluid
scenarios (Chen al. 2025). and inverse problems (Wu et al. 2024b Zheng al. 2025). ). Each
benchmark has its own focus: PDEBench (Takamoto et al. (2022) provides a vast amount of data
and benchmarks across different PDEs, the Well (Ohana et al. (024) focuses on the data quality
including high resolution and large data volume. [uo et al. (2023) and Tali et al. (2024) provide large amount of data in fluid mechanics which is governed by the complex Navier-Stoker equation. In the domain of science, some works provide real observational data (Kang et al. 2023 2025
Lyu et al.) 2024 Zhang et al. 2025) (Casey Wintergerste 2000), but they are not designed for machine learning, with limited data and sparse operating conditions. A concurrent work, REALM (Mao et al. 2025), , proposes a benchmarking framework designed to test neural surrogates on multi- physics reactive flows realistic regimes. In comparison, our proposed benchmark RealPDEBench contains more than 700 real-world physics experiments and numerical simulations in multiple phys- ical systems, and we collect the paired real-world measurements and numerical data. These data are leveraged to evaluate the performance of scientific ML in real-world scenarios and their ability to transfer learning from simulation to reality.
Scientific ML. has been extensively studied, and related works can be found in the following overviews (Lavin et al. 202T Brunton & Kutz 2022 Karniadakis et al., 2021 Cuomo et al. 2022 Wang et al. 2024a). As for the baselines considered in our work, we focus on approximating the evo- lution of physical system states using data-driven deep learning models. These works can divided into two categories: small models trained on a single dataset to solve specific problems (Li et al.)
2021 Wang et al. 2024b Hu et al., 2024). and pre-trained or foundation models trained on large- scale datasets (Hao et al. [2023] Ye et al. 2024 Herde al. 2024 Feng et al. (025bb). Methods such as neural operators (Li et al. 2021) are initially used to solve specific problems in a single physical system. This type of method widely used in various directions, such as fluids (Chen et al. 2025
Feng et al. (225a)), combustion (Weng et al., 2025). electromagnetism (Zhou et al.) 2025). 5). power
systems (Huang & Wang. 2022), , etc. With the increase of available data and the development of models, more and more works are dedicated to developing universal foundation models for multiple physical systems (Totounferoush et al., (025). Our proposed RealPDEBench can be used by all the scientific ML models mentioned above, verifying their performance on real-world measurements, and providing a unique reference and data support for developing new models.
REALPDEBENCH: BRIDGING REAL-WORLD AND SIMULATED DATA
In this section, we present four key components of RealPDEBench. First, we define the tasks con- sidered in RealPDEBench, and discuss the roles of real-world and simulated data. Next, we provide a brief description of each dataset in RealPDEBench, along with the data collection, data format, and extensibility. We then introduce the employed evaluation metrics, which can be categorized into data-oriented and orysics-oriented types. Finally, we describe the baselines considered in the experiments.
3.1 TASK DEFINITION: ROLES OF REAL-WORLD AND SIMULATED DATA
Prediction task. All tasks considered in RealPDEBench fall into the category of prediction tasks, namely, predicting the future evolution of complex physical systems. More formally, we aim to learn mapping between the input and output spaces, F A x r + U, where the input is given by the Cartesian product of initial discretized states of system states a A and system parameters y I, and the output corresponds to the subsequent discretized temporal evolution of the system.
Roles of real-world and simulated data. Real-world and simulated data each possess distinct advantages and limitations, which in turn motivate us to leverage the advantages of both. Real- world data avoid numerical errors and simplified physics but is costly, noisy, and often limited in observability. For example, the incoming flow cannot be strictly guaranteed to be uniform, and camera noise leads to measurement errors. Simulated data are relatively cheaper and offer broader modalities (variables of physical systems) with dense parameter coverage, yet suffer from numerical
3

---

Published as a conference paper at ICLR 2026
errors often caused by modeling like Large Eddy Simulation (LES) and discretization like second- order convergence (Weymouth & Yue 2011). Since the ultimate goal of scientific ML is to model real systems, evaluation is all conducted on real-world data, while the training paradigm admits multiple possible designs.
Three categories of prediction tasks. Based the characteristics of the two types of data, design three categories of tasks: real-world training, simulated training, and simulated pretraining with real-world finetuning (abbreviated real-world finetuning in Sec. 4. For each dataset, there are N real-world samples and N simulated samples. n real-world samples are used for training, (N n2/2 for validation, and (N n for testing. For all tasks, the validation and test set is fixed to be these N - n real-world samples, ensuring consistent evaluation protocol. For training, the three task settings are defined as follows. In real-world training, models are trained directly on the real-world samples. In simulated training, models are trained on all N simulated samples. In simulated pretraining with real-world finetuning, models are first pretrained on the N simulated samples and subsequently finetuned using the real-world samples. This setup reflects common practical scenarios, where real-world data are scarce, and simulated data are abundant.
3.2 OVERVIEW OF DATASETS AND PDES
In order to comprchensively evaluate the ability of scientific ML in the real world, an ideal bench- mark is expected include series of key physical challenges, such transition turbulence (Bloor] 1964), prediction of control response (He et al.] (000). nonlinear multi-physics coupling AAthni al. 2025), three-dimennionaal effects (Spalart & Venkatakrishnan (0166), and physical processes involving chemical reactions (Cathonnet] 2003). Following these principles, we carefully select and construct five representative scenarios (Huang al. 2023 Kumar et al.) 2016: Lin al.) 2024 (Shukla et al. [024] Liu et al.) (2024a) for RealPDEBench. Specifically, the scenarios include classical transition to turbulence at wake (Cylinder), controlled system (Controlled Cylinder), fluid-structure interaction (FSI), three-dimensional effects of fluid dynamics (Foil), and reactive flows (Combustion). There are total of 736 trajectories of real-world and simulated data. The governing equations of the above systems range from the Navier-Stoke equations for basic cylin- der flows to coupled FSI equations, and reactive Navier-Stokes equations with species transport for combustion systems. Developing surrogate models that approximate these real-world challenges with high fidelity across different parameter regimes remains a significant challenge (Conti et al., 2024]. We argue that achieving this capability is a necessary precondition to applying such models for real-world problems. While these problems have been studied in prior work on simulated data Takamoto et al. 2022) Ohana et al. 2024 Luo al. 2023). comprehensive real- real-world benchmark dataset spanning complex physical systems is, to the best of our knowledge, not available.
In the following parts, we provide brief introduction and important features of each dataset. More details are provided in Appendix B
Cylinder represents a fundamental benchmark problem in fluid dynamics, featuring wake dynamics behind circular cylinders across various Reynolds numbers (Re) from laminar to turbulent regimes. This dataset captures the classical Kármán vortex street formation and provides both simulation and real-world measurements for studying unsteady flow phenomena.
Controlled Cylinder extends the basic cylinder flow by introducing active control through external forcing, spanning different Reynolds numbers and control sequences (periodic sinusoidal control at different frequencies), which highlights the challenges of learning control-forced fluid dynamics.
Fluid Interaction (FSI) captures phenomena where the circular cylinder undergoes struc- tural vibrations due to fluid forces, representing critical coupling dynamies encountered in real-world applications such as bridges under wind loading and offshore platforms in ocean currents. Parame- ters include different Re, mass ratios, and damping coefficients, covering lock-in in phenomena (Zhang et al. 2015) and galloping instabilities (Sun et al. 2020) across various configurations.
Foil contains cross-sections data extracted from 3D simulations and experiments. The 3D effect in- troduces increased fluid complexity, generating enhanced small-scale vortex structures. Our dataset covers diverse angles-of attack (aoa) and Reynolds numbers, which is particularly valuable for foil design and fluid dynamics optimization in marine engineering applications.
4

---

Published as a conference paper at ICLR 2026
(a)
(b)
(d)
Figure 2: Photos of real-world data collection. (a) Water tunnel after laser irradiation. (b) Particle imaging photos taken by the camera. (c) Motion control equipment. (d) Swirl combustor equipment.
Combustion presents experimental and numerical cross-sectional data of 3D swirl-ttabilized NHy/CH//air flames. Combustion involves complex multi-field coupling problems, such as the pressure field, temperature field, sound field, velocity field, concentration field, etc. Accurately pre- dicting the evolution of concentration, temperature, and other fields caused by combustion is of great significance to efficiently control the thermoacoustic instability for aircraft and space engines.
Collection of real-world data. Real-world data of fluid systems are measured in the circulating water tunnels using the Particle Image Velocimetry (PIV) technique (Abdulmouti] 2021). The circu- lating water tunnel is employed to generate incoming flow that interacts with internal structures, thereby inducing complex fluid-structure interactions and associated physical phenomena. Mean- while, in order to measure the changes in the flow field, sprinkle fluorescent particles (hollow- glass microspheres, 10um) into the water and irradiate with a continuous laser (forming a 2mm thick laser layer on the shooting surface), thereby observing the velocity of the fluid through the velocities of the fluorescent particles, as shown in Figure 2a These velocities are recorded using high-speed cameras (Figure 2b). Afterwards, we use PIVLab to process the photos and obtain the flow velocity states at each time step. In addition, the structure is installed on a rail driven by a stepper motor. Through the motor, we can control the movement of the structure, as shown in Figure 2c Next, we measure the well-wolld data of combustion using the flame chemiluminescence (CL) imaging technique (Alviso et al., 2017). Specifically, we construct swirl combustor and use mass flow controllers to control the injection ratio of air, NH3, and CH4 (Figure 2d). These components are ignited the combustor, and then we measure the light intensity using an OH* CL camera. More details are provided in Appendix B.3
Collection of simulated data. Furthermore, we use Computational Fluid Dynamics (CFD) to gen- erate the simulated data, which have the same parameters as real-world experiments. Specifically, we generate simulated data for four fluid systems using the Finite Volume Method (FVM) and Im- mersed Boundary Method (IBM) (Weymouth & Yue 2010). For 2D experiments, we apply Lilypad (Weymouth & Yue (011) as the solver, and for 3D simulations, we apply Waterlily (Weymouth & Font (025) on GPUs, which is efficient 3D CFD solver. Next, for the Combustion dataset, we conduct three-dimensional, implicit, unsteady LES to calculate thermoacoustic instabilities in a wwirl-stabilized flame. The Eddy Dissipation Concept (EDC) (Lourier et al. 2017 is employed to model the turbllence-chemisty interaction.
Data format and extensibility. All datasets are stored in HDF5 (Folk et al. 201 format. Each file contains one trajectory sampled at equal time intervals on uniform spatial grid, represented as a NumPy y (van der Walt et al. 2011 1) array with modalities of shape (T, X, Y). It also records the corresponding system parameters. such as Re, oscillation frequency, and equivalence ratio. Our codebase is implemented on the PyTorch (Paszke et al., 2019) platform. All datasets are wrapped under common module RealDataset, and baselines are implemented under module Model, which enables straightforward extensibility to new datasets and baselines (Appendix(C) ())
3.3 OVERVIEW OF METRICS
We propose comprehensive set of nine evaluation metrics for the three tasks, which can be cate- gorized into data-oriented and physics-oriented metrics. Below, we denote { as ground truth samples, and { as model's predictions. We additionally denote temporal coordinates for each sample (t) (1)=1,.... T, and spatial coordinates (a..)=-,,....-
5

---

Published as a conference paper at ICLR 2026
3.3.1 DATA-ORIENTED METRICS
RMSE, MAE, Rel L2. We first employ several standard metrics in ML, including Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Relative L2 Error (Rel L2).
Coefficient of determination (R²). In addition, we consider the coefficient of determination (R²), which measures how well the observed outcomes are replicated by the model, defined by the pro- portion of variance in the ground truth explained by the predictions, as
Ex(y - Un)²
n² = Σ. (yx z)²
where means the average and y Σx yk/K.
Update Ratio. Finally, the Update Ratio measures the relative efficiency of simulated pretraining with real-world finetuning versus real-world training from scratch. Let RMSE, denote the best RMSE achieved with real- weal-world training. Define N1 and N2 as the number of finetuning and training updates required to reach R.ME,, respectively. The metric then given by Ni N1/N2.
3.3.2 PHYSICS--RIIENTED METRICS
From the physical perspective, adopt three types of evaluation metrics.
Fourier Space Error (IRMSE). First, we consider the Fourier Space Error (fRMSE), which is frequency-domace metric for evaluating prediction accuracy in spectral space (Takamoto et al. 2022). It is computed by applying a 3D Fast Fourier Transform (FFT) to both prediction and target fields, taking squared differences of Fourier coefficients, grouping them by frequency magnitude, and averaging within each group. To provide a more fine-grained analysis, we report the Fourier error different frequency bands (low, middle, and high) by partitioning the spectrum.
Frequency Error (FE), Second, since lots physical systems exhibit periodicity, accurately cap- turing their temporal cycles is of critical importance. To further evaluate temporal dynamics, we compute the Frequency Error (FE). This is achieved by first summing the predicted and ground-truth fields over spatial dimensions obtain temporal signals, applying 1D FFT, and then measuring the MAE between the two spectra (Feng al. 2024):
FE
Σ F
Σ (( - Σ she (1,
KT
Kinetic Energy Error (KE). The Kinetic Energy Error (KE) a metric applied to the velocity field Wang et al. 2020), which measured as
KE = c a).
c (w')² (w))²
(u')
T Iu(())
1 -
2
where u and two channels of y. denote the velocity field in the and y-directions, respectively.
Mean Velocity Profile Error (MVPE). Finally. in many numerical studies of fluid literature (Kravchenko Moin, 2000; Ma al.) (000); Wissink & Rodi] 2008 Neunaber et al.) (025). the mean velocity profile (MVP) Is a generally employed summary statistic to measure discrepancies between simulated data and real-world data. For example, there is significant decrease of u near the Nymte wake region behind the cylinder. We select different probes positioned at [ Uprobe. })) } to calculate the differences between the time-average velocity field of real-world data, simulated data, and model predictions, forming the MVPE:
MVPE
Σ
[[(Sepoobe.k.) Uprobe -ū (ITpeobe.k k. Uprobe
K. K.Nonobe
kJj
We note that this metric evaluates the long-term performance, adopt it for autoregressive evaluation (Sec. 4.5) and the comparison between simulated and real-world data (Appendix B.4).
3.4
BASELINE MODELS
DMD Kutz al. 2016) is reduced order model that extracts spatiotemporal coherent structures from mene-series flow field data through a data-driver matrix decomposition approach. U-Net (Ron-) neberger et al. 2015) is auto-encoding architecture that propagates information efficiently

---

Published as a conference paper at ICLR 2026
different scales. CNO (Raonic al. 2023 is CNN modification designed to maintain structural consistency across continuous- discrete mappings and achieve artifact-free operator approximation. DeepONet (Lu et al., 202T) is a deep operator network using branch-trunk architecture to ap- proximate nonlinear operators by learning neural operator based the Fourier transform that solves PDEs by learning operator mappings in the frequency domain, which has the advantages of resolution invariance. WDNO (Hu et al.) 2025) applies diffusion-basee generation within the wavelet space to model trajectories, effectively capturing abrupt changes. MWT (Gupta et al. 2021) compresses the associated operator's kernel using fine-grained wavelets and learns the projection of the kernel onto fixed multiwavelet poly- nomial bases through explicit embedding of inverse multiwavelet filters. GK-Transforna (Cao) 2021 is an attention-basen neural operator that modifies the Transformer's self-attention mecha- nism by removing softmax normalization and incorporating Galerkin-type projections to improve approximation capacity and efficiency in learning PDEs solution operators. Transolver (Wu et al.) 2024a) introduces attention-bast baseo learning of physical states, enabling the model to achieve intrin- capacity while improving the modeling of physical correlations. DPOT (Hao al., 2024) is auto-regressive denoising operator transformer for large-scale PDE pre- training. We consider its pretrained small model (30M) and the pretrained large model (509M) in experiments, and finetune them on our datasets. Please refer to Appendix B for more details.
cunction-toofunctinn mappings. FNO (Li et al. (021) is
sic eeooeery-indeeeneee
4 EXPERIMENTS
4.1 EXPERIMENT SETUP
In this section, we conduct experiments on the tasks and datasets introduced in Sec. 3.1 and Sec. The train, validation, and test data are split at the parameter level. We report the RMSE, Relative L2 Error, IRMSE, and Update Ratio of all datasets and baselines under the three training categories, as summarized in Table The results for all other metrics are in Appendix [..] To better accommodate the differing characteristics of real-world and simulated data, we tailor simulated training with two strategies, adding noise and randomly masking unmeasured modalities (see Sec. During evaluation, to additionally assess the models' ability for long-term prediction, we include an option for autoregressive evaluation, with details in Sec. 4.5
3.2
4.2
Please note that our results mainly compare baselines on real-world data, setting that differs from the majority of their prior applications. Below, we will discuss some common findings from experi- ments. The code and data are available here
4.2 GAP OF REAL-WORLD AND SIMULATION
We first aim to demonstrate the gap between simulated data and real-world data, which is also one of the core motivations for proposing RealPDEBench. We demonstrate these gaps from three aspects: different modalities, the errors of simulated training and real-world training, and the measurement and numerical errors. First, the modalities of simulated and real-world data are different. Due to the limitations of measurement techniques, simulated data usually contain more modalities than real- world data (as shown in Figure I.. To better leverage these additional modalities, we randomly mask unmeasured modalities with a certain probability when training with simulated data (details in Appendix D) Second, we train all baselines separately on simulated and real-world data, and test them on the same real-world test dataset (as mentioned in Sec. 3. In Ta there are significant differences between the models trained on the two types of data, and the real-world trainings have from 9.39% to 78.91 improvements on Rel L2. Models trained on simulated data are difficult to generalize directly to real-world data, even if their physical parameters are consistent. Third, both real-world data and simulated data have errors. Real-world data generally have significant noise due to measurement technology limitations, as shown in Figure [ while simulated data have numerical errors, which may be caused by simplified physical processes or ideal conditions. Therefore, to improve the generalization of models trained on simulated data, we add additional noise to the simulated data to approximate the distribution of real-world data. In addition, Figure 3a shows that simulated training yields much higher Frequency Errors than real-world training, highlighting that simulated data cannot perfectly capture periodicity in real-world systems.
7

---

Published as a conference paper at ICLR 2026
Results of RMSE, Rel L2, IRMSE, and Update Ratio. Different datasets have different colors. Because DMD lacks the training process, we place its inference results in the last column and leave the rest blank. The smaller the error result, the darker the color. The bolded is the best.
Table I:
Simulated Training
Real-world Training
Real-world world Finetuming
Dataset
Baseline
Params
RMSE Rel L2
FRMSE
RMSE Rel L2 IRMSB
RMSE
Rel L2 IRMSE
Ratio
U-Net
23,0M
0.0758
0.2165
0.0122
0.0700
0.0701
0.0103
0.0632
0,0728
0.0097
0.3636
CNO
8.0M
0.0729
0.1849
0.0113
0,0424
0.0897
0.0069
0.0403
0.1013
0.0066
1.0000
DeepONet
3.6M
0.0863
0.3592
0.0132
0.0713
0.1528
0.0108
0.0661
0.1503
0.0107
0.5758
FNO
50.4M
0.0739
0.2575
0.0115
0.0585
0.0855
0.0087
0.0545
0.0780
0.0087
0.1111
WDNO
104.3
0.0699
0.2053
0.0105
0.0689
0.1666
0.0091
0.0513
0.0969
0.0073
0.1250
Cylinder
MWT
2.9M
0.0733
0.2240
0.0107
0.0540
0.0832
0.0081
0.0584
0.0850
0.0088
0.9253
GK- frrTTrnsformer
84.4M
0.0876
0.2941
0.0139
0.1196
0.0898
0.0166
0.0995
0.0908
0.0146
0,4400
Transolver
4.3M
0.1121
0.3224
0.0174
0.1093
0.1887
0.0160
0.0965
0.1723
0.0144
1.0030
SPOT-S.FT
30.8
0.0513
0.1474
0.0076
0.0586
0.0983
0.0086
0.0440
0.0766
0.0067
0.1250
L.
649.8 M
0.0486
0.1446
0.0070
0.0390
0.0812
0.0056
0.0394
0.0733
0.0059
1,0000
MI. Average
0.0752
0.2356
0.0115
0.0692
0.1106
0.0101
0.0613
0.0997
0.0093
0.5666
DMD
0.0862
0.3590
0.0114
U-Net
23.0M
0.0195
0.1360
0.0029
0.0080
0.0555
0.0010
0.0079
0.0543
0.0009
0.7632
CNO
8.0M
0.0167
0.1209
0.0022
0.0081
0.0583
0.0009
0.0080
0.0574
0.0009
0.8400
DeepONet
3.6M
0.0540
0.4256
0.0134
0.0309
0.2399
0.0058
0.029
0.2284
0.0052
0.5200
FNO
50,4M
0.0285
0.2007
0.0059
0.0097
0.0723
0.0012
0.0094
0.0702
0.0011
0.5217
Controlled Cylinder
WDNO
359.8 M
0.0240
0.1829
0.0037
0.0115
0.0927
0.0014
0.0101
0.0752
0.0013
0.3617
MWT
2.9M
0.0230
0.1594
0.0037
0.0102
0.0747
0.0011
0.0102
0.0746
0.0010
0.5487
GK OK-Trersfommer
50.8
0.0261
0.1816
0.0050
0.0103
0,0767
0.00122
0.0101
0.0748
0.0012
0.5200
Transolver
4.3M
0.0294
0.1894
0.0069
0.0171
0,1200
0.0014
0.0168
0.1176
0.0022
0.4285
D-OT-S-FT
30
0,0233
0.1722
0.0039
0.0084
0,0598
0,0010
0.0085
0.0615
0,0010
1.0000
DPOT-L-FT
649.8 M
0.0248
0,1784
0.0043
0.0084
0,0603
0.0010
0.0085
0.0611
0.0010
1.0000
ML Average
0,0268
0.1947
0.0052
0.0123
0.0910
0.0017
0.0119
0.0875
0.0016
0.6504
DMD
0.0340
0.2233
0,0059
U-Net
23.0M
0.0223
0.1589
0,0025
0.0085
0.0583
0.0007
0.0084
0.0579
0.0007
0.6667
CNO
8.0M
0.0241
0.1724
0.0029
0.0105
0.0741
0.0009
0.0096
0.0679
0.0008
0.5600
DeepONet
3.4M
0.0637
0.4606
0.0097
0.0350
0.2502
0.0051
0.0332
0.2368
0.0048
0.3125
FNO
268.5 M
0.0426
0.3095
0.0059
0.0129
0.0892
0.0012
0.0127
0.0881
0.0012
0.5714
WDNO
91.7M
0.0369
0.2700
0,0050
0.0117
0,0817
0.0011
0.0116
0.0824
0.0011
0.5116
FSI
MWT
2.9M
0,0339
0.2487
0.0046
0.0128
0.0910
0.0010
0.0128
0.0912
0.0010
0.9992
GK To-Transfommer
67.6M
0.0307
0.2241
0,0039
0.0127
0.0916
0.0011
0.0124
0.0898
0,0010
0.6200
Transolver
4.3M
0,0305
0.2119
0.0043
0.0236
0,1567
0.0027
0.0223
0,1480
0,0025
0.1600
DPOT-S-PT
41.3M
0.0260
0.1886
0,0031
0.0105
0.0746
0.0008
0.0099
0.0701
0.0007
0.2500
DPOT-L-FT
673.5
0.0262
0.1898
0.0030
0,0099
0.0687
0.0008
0.0097
0.0668
0.0008
0.3125
ML Average
0.0337
0,2434
0.0045
0.0148
0,1036
0.0015
0.0143
0.0999
0.0015
0.4964
DMD
0.0450
0.2955
0.0060
U-Net
23.0M
0.0272
0.0745
0.0039
0.0100
0.0159
0.0011
0.0094
0.0145
0.0008
0.5116
CNO
8.0M
0.0227
0,0438
0.0028
0.0136
0,0253
0.0018
0.0114
0.0206
0.0013
0.4400
DeepONet
3.6M
0.0339
0.0565
0.0051
0.0222
0.0363
0.0028
0.0226
0.0375
0.0029
0.5185
FNO
50.4M
0.0274
0.0540
0.0037
0.0130
0,0228
0.0015
0.0120
0.0206
0,0012
0.3192
WDNO
358.4 M
0.0242
0.0490
0.0026
0.0162
0,0444
0.0018
0.0106
0.0181
0,0010
0.1035
Foil
MWT
2.9M
0.0264
0.0492
0.0036
0.0133
0,0227
0.0015
0.0125
0.0210
0,0012
0.4894
GK- NN-Transformer
50.8
0.0277
0.0512
0.0038
0.0142
0,0245
0.0018
0.0138
0.0237
0,0017
0.5918
Transolver
4.3M
0.0345
0.0465
0.0050
0.0220
0.0370
0.0027
0.0138
0.0237
0.0017
0.5918
DPOT-S-FT
41.3M
0.0221
0.0397
0.0027
0.0106
0.0166
0.0010
0.0109
0.0174
0,0012
1.0000
DPOT-L-PT
673.5M
0.0229
0.0402
0.0029
0.0105
0.0159
0.0011
0.0109
0.0161
0.0012
1.0000
Average
0.0505
0.0036
0.0146
0.0261
0.0017
0.0128
0.0213
0,0014
0.5566
DMD
0.0322
0.0520
0.0641
U-Net
2
0,0358
0.7290
0.0051
0.0216
0,5487
0.0026
0.0213
0.5403
0.0025
0.5682
CNO
8.0
0.0401
0.8274
0.0057
0.0248
0.6030
0.0032
0.0238
0.5877
0.0030
0.7083
DeepONet
3
0.0403
0.8565
0.0056
0.0229
0,5751
0.0028
0.0227
0.5723
0.0028
0.7800
FNO
67.1
0.0363
0.7606
0.0052
0.0226
0.5664
0.0027
0.0225
0.5680
0.0027
0.4286
WDNO
122.7 M
0.0439
110016
0.0061
0,0380
0.8605
0.0055
0.0314
0.7441
0.0044
0.2632
Combustion
MWT
2.9
0.0367
0.7536
0.0052
0.0221
0,5560
0.0027
0.0220
0.5549
0.0026
0.9982
GK Transformer
67.6M
0.0400
0.9143
0.0056
0.0247
0,6083
0.0031
0.0258
0.6421
0.0034
1.0000
Transolver
4.3M
0.0442
0.9056
0.0061
0.0375
0,7825
0.0050
0.0375
0.7836
0.0050
1,0000
S
41.5M
0.0393
0.7842
0.0054
0.0209
0.5149
0.0024
0.0211
0.5378
0.0024
1.0000
DPOT-L-FT
674.2 M
0.0378
0.7750
0.0053
0.0208
0.5331
0.0024
0.0206
0.5318
0.0023
0.8125
MI. Average
0.0394
0.8408
0.0055
0.0256
0,6169
0.0032
0.0249
0.6063
0.0031
0.7559
DMD
0.0914
1.3360
0.0110
4.3 IMPROVEMENT WITH SIMULATED DATA PRETRAINING
Although the previous section has shown that there exists a gap between real-world and simulated data, simulated data possess unique advantages, including a lower cost, access to a greater number of modalities, and being free from measurement-indueee noise.
This raises a key question: can simulated data enhance model performance on real-world data? Our results provide a strong affirmative answer, demonstrating clear benefits from simulated pretraining across two key aspects. First, as shown in Table the errors in the Real-world Finetuning column are lower than those of Real-world Training, revealing that simulated pretraining and then real-
8

---

Published as a conference paper at ICLR 2026
0.14
FNO
Frequerry freee
150-
IASE (log)
4x10--
0.12
DeepOWet
U-bbet
IMASE
0.30
MalT
1.0
3 1
9.08
CNO
UKiteerafomme ormme
Thanso iver
0.06
DPOTL
2x19
0.04
Simuated
Real world Task
6
10
Percentage Iterations
Round of regonssgeesivee
(b)
(c)
Figure 3: (a) Frequency errors of baselines (statisties of 10 values) on real-world vs. simulated data from Controlled Cylinder. (b) Validation RMSE curves of real-world finetuning on Combustion, with crosses marking the best RMSE of real-world training. The x-axis shows the percentage of update iterations. (c) RMSE under 1, 2, 3, 5, and 10 rounds of autoregressive evaluation on Cylinder.
world finetuning leads to better performance compared to training directly on the same amount of real-world data. On the one hand, this improvement can be attributed to the larger volume of simulated data, which exposes the model to trajectories under a wider range of system parameters. On the other hand, the simulated modalities that are unmeasured in real-world data have additional dynamic information, which is utilized through the mask wrakk-raininn strategy.
Second, the loss decreases faster during finetuning, which is evidenced by the Update Ratio metric. For most datasets and baselines, the value is less than 1, which means real-world finetuning requires fewer update iterations reach the optimal performance of real-world training. Moreover, in Figure 3b) we plot the validation RMSE curves on the Combustion dataset. It can be clearly observed that real-world finetuning achieves a much faster decrease compared real-world training, further confirming the benefit of simulated data in improving model performance.
4.4 COMPARISON BASELINES
Different scientific ML models have distinct advantages. In this subsection, we analyze and discuss the performance of baselines in terms of the trade-off between data-oriented metrics reflecting local pixel-level errors and physics-oriented metrics reflecting global fea- tures. Specifically, we select RMSE from the data-oriented metrics and Frequency Error from the physics-oriented metrics, which can reflect the model's ability to capture global periodic- ity. In order to compare the results on multiple datasets, we integrate them into Figure where the x-axis is Frequency Error and the y-axis is RMSE. Each quadrant is dataset. Proximity to the origin indicates hetter performance.
000 RMSE
410
FSI
000
Cylinder
0.2
405
001
23.8
324
385
300
300
150
10
82
2
FE
Controlled Cylinder
Foil
0071
8038
Figure 4 reveals three interesting phenomena. First, DPOT-L-FT, the large pretrained foun- dation model, achieves the best overall perfor- mance since it is closest to the origin. This reflects the benefits of large-scale PDE pre- training and a larger number of model param- eters. Second, since most models are trained with data-oriented losses (e.g., MSE), they ex- cel at local features while their performance on physics-oriented metrics involving global fea- tures is weaker. Especially, convolution- based methods (U-Net, CNO) achieve lower RMSE, the tasks resemble image processing, where con- volution has proven to be highly effective (Albawi et al., (017)). Third, due to the use of the mul- tiwavelet transform, in general, MWT exhibits advantages in learning periodicity. Therefore, it is
FNO
WDNO
U-Not
DeepONet MWT
CNO
Transolver
DPOT-FT-S
DPOT-FT-L
Figure 4: Trade-off of RMSE and Frequency Er- ror. Distant points are linked with dashed lines.
We omit the points where the DMD error is far from the center to make most of the points clearer.

---

Published as a conference paper at ICLR 2026
0.13
033
210
025
025
-423
-453
445
-459
-150
-6.5
4.35
-815
-.35
1xgee
can antten
6
415
505
675
676
(b)
Figure 5: (a) MVPEs of real-world finetuning under 10-round autoregressive evaluation Cylinder. (b) MVP of U-Net's 10-round autoregressive prediction on Cylinder.
important to select the appropriate network architecture and training strategy based on the different objectives of the specific tasks and data.
4.5 OTHER ANALYSIS
Autoregressive evaluation. To investigate the long-term performance, we conduct evaluations with 2. 3, 5, and 10 rounds of autoregressive prediction. Given training inputs and outputs of time steps, autoregressive evaluation with N rounds iteratively feeds each predicted T steps back as input, yielding predictions over NT steps. The prediction eπrоr is finally measured over the entire NT steps. Figure 33 shows results on the Cylinder dataset. While CNO performs well in single-round prediction, error grows a faster rate than other methods as the number autoregressive rounds increases. This suggests that may suffer from substantial error accumulation, leading to inaccurate long-term predictions. In addition, on the 10-round evaluation, we calculate MVPES of baselines and visualize the MVP of U-Net in Figure 5] which reveals the long-term summary statistic. From the results, we observe that DMD shows limitations under this metric, while the large-size DPOT model performs the best. More results on other datasets and metrics provided in Appendix A.2)
FNO
1.2
WOND
Nomaalize Kieed DAMSS
DeepONet
0.8
MWT
0.6
CNC
CKSTransform
Tranalver
0.2
DPOT-S-F
DPOT-L-T
Figure 6: Normalized low-, mid- and high- figh-frequency fRMSE on Foil.
Low-, mid-, and high-frequene fRMSE. We compute the normalized fRMSE within low-, mid-, and high-frequency bands, which reflects the model's ability capture system dynamics different frequency ranges. As shown in Figure on the Foil dataset, CNO's relative performance compared to other models increases as the frequency gets higher. This phenomenon observed in CNO may be related to its design principle of eliminating aliasing errors. Full results are in Appendix A.3
5 CONCLUSION
In this work, to address the critical issue of lacking real-world data, we introduce RealPDEBench, the first real-world scientific ML benchmark for complex physical system prediction. It incorporates paired real-world and simulated data, three categories of tasks, nine evaluation metries, and ten baselines. Our work underscores the differences between real-world and simulated data and takes a significant step toward bridging the gap. For limitations, and datasets will spur the development of algorithms that more effecttvvly integrate real-world and simulated data, paving the way for scientific MI. that truly applicable in real-world settings.
Appendix [F] We hope our benchmark
10

---

Published as a conference paper at ICLR 2026
ACKNOWLEDGEMENT
We thank Jiashu Pan, Long Wei, Zhe Yuan, and Tianci Bu for providing feedback on our manuscript. We also thank Chunyu Liu for discussions. We gratefully acknowledge the support of Westlake University Research Center for Industries of the Future; Westlake University Center for High- performance Computing. The content is solely the responsibility of the authors and does not neces- sarily represent the official views of the funding entities.
ETHICS STATEMENT
This work proposes benchmark for scientific Machine Learning using real-world and simulated data. All datasets are collected from physical experiments and numerical solvers, without involving sensitive information. Caution that models trained this benchmark should not be directly applied in safety-critical scenarios without further validation.
REPRODUCIBILITY STATEMENT
The code, data, checkpoints, and log files are available at https://gith:: com/ waapeeeeeWeettaakeeeRReeeeaeeeee
with full documentation to ensure transparency and reproducibiliity. We provide a unified and modular code framework, together with scripts for re- producing all experiments. We also provide numerical scripts, calibration files, processing software, and part of the raw data for validation of the accuracy of data acquisition and processing.
REFERENCES
Hassan Abdulmouti. Particle imaging velocimetry (piv) technique: principles and applications, review. Yanbu Journal of Engineering and Science, 6(1):35-65, 2021.
Saad Albawi, Tareq Abed Mohammed, and Saad Al-Zawi. Understanding of convolutional neural network. 2017 International Conference on Engineering and Technology (ICET), pp. 1-6, 2017.
Dario Alviso, Miguel Mendieta, Jorge Molina, and Juan Carlos Rolón. Flame imaging reconstruc- tion method using high resolution spectral data of oh*, ch and c2 radicals. International Journal of Thermal Sciences, 121:228-2000 2017.
Abdulgaphur Athani, Nik Nazri Nik Ghazali, Irfan Anjum Badruddin, Abdullah Y Usmani, Mo- hammad Amir, Digamber Singh, and Sanan H Khan. Image-based based hemodynamic and rheological study of patient's diseased arterial vasculatures using computational fluid dynamies (efd) and strid-structure interactions (Isi) analysis: A review: Image-based hemodynamic and rheological study of patient's diseased arterial vasculatures using computational fluid dynamics. Archives of Computational Methods in Engineering, 32(3):1427-1457, 2025.
M Susan Bloor. The transition turbulence in the wake of circular cylinder. Journal of Fluid Mechanics, 19(2):290-36) 304, 1964.
T. Bratanow. Navier-stokes equations: theory and numerical analysis. Applied Mathematical Mod- elling, 2, 1978.
Steven L Brunton and Nathan Kutz. Data-driven science and engineering: Machine learning, dynamical systems, and control. Cambridge University Press, 2022.
W Kenneth Bullivant, Tests of the naca 0025 and 0035 airfoils in the full-scale wind tunnel. Annual Report-National Advisory Committee for Aeronautics, 27:91, 1941.
Shuhao Cao. Choose a transformer: Fourier or galerkin. Advances in neural information processing systems, 34:24924-24940, 2021.
MV Casey and Wintergerste. ERCOFTAC Best Practice Guidelines for Industrial CFD. ERCOF. TAC, Brussels, Belgium, 2000.
11

---

Published as a conference paper at ICLR 2026
M Cathonnet. Advances and challenges in the chemical kinetics of combustion. In Proceedings of the European Combustion Meeting, volume 2003, 2003.
Francis F. Chen. Introduction to Plasma Physics and Controlled Fusion. New York: Plenum press, 2015.
Huaguan Chen, Yang Liu, and Hao Sun. Pinp: Physics-informed neural predictor with latent esti-
mation of fluid flows. In The Thirleenth International Conference on Learning Representations, 2025.
Wai Tong Chung, Ki Sung Jung, Jacqueline H. Chen, and Matthias Ihme. Blastnet: A call for community-inoolved big data in combustion machine learning. Applications in Energy and Com- bustion Science, 12:100087, 2022. ISSN 2666-352X. doi: https://doi.org 100087.
Paolo Conti, Mengwu Guo, Andrea Manzoni, Attilio Frangi, Steven L Brunton, and J Nathan Kutz. Multi-fidelity reduced-order surrogate modelling. Proceedings of the Royal Society A, 480(2283): 20230655, 2024.
Salvatore Cuomo, Vincenzo Schiano Di Cola, Fabio Giampaolo, Gianluigi Rozza, Maizar Raissi, and Francesco Piccialli. Scientific machine learning through physics-informed neural networks: Where we are and what's 's next. Journal of Scientific Computing, 92, 2022.
J. R. Dormand and P. J. Prince. A family of embedded runge-kutta formulae. Journal of Computa- tional and Applied Mathematics, 6: 6:19-26, 1980.
Haodong Feng, Peiyan Hu, Yue Wang, and Dixia Fan. Multimodal policies with physics-informed representations. ArXiv, abs/2410. 15250, 2024.
Haodong Feng, Peiyan Hu, Yue Wang, Dixia Fan, Tailin Wu, and Yuzhong Zhang. Physics-informed super-resolution and forecasting method based on inaccurate partial differential equations and partial observation. Physies of Fluids, 37(6), 2025a.
Haodong Feng, Haoren Zheng, Peiyan Hu, Hongyuan Liu, Chenglei Yu, Long Wei, Ruiqi Feng, Jinlong Duan, Dixia Fan, and Tailin Wu. Fluidzero: Mastering diverse tasks in fluid systems through a single generative model. 2025b.
Joel H. Ferziger and Milovan Peric. Computational methods for fluid dynamics. New York: Springer, 2002.
Mike Folk, Gerd Heber, Quincey Koziol, Elena Pourmal, and Dana Robinson. An overview of the hdf5 technology suite and its applications. In Proceedings of the EDBT/ICTT 2011 workshop on array databases, pp. 36 47. 2011.
Peter Glarborg, James A. Miller, Branko Ruscic, and Stephen J. Klippenstein. Modeling nitrogen chemistry combustion. Progress in Energy and Combustion Science, 2018
Michael Griebel, T. Dornseifer, and Tilman Neunhoeffer. Numerical Simulation in Fluid Dynamics: Practical Introduction. Society for Industrial and Applied Mathematics, 1998.
Boyce E. Griffith and Neelesh A. Patankar. Immersed methods for fluid-structure interaction. Annual review of fluid mechanics, 12:421-448. 2020.
Gaurav Gupta, Xiongye Xiao, and Paul Bogdan. Multiwavelet-based operator learning for differen- tial equations. Advances in neural information processing systems, 0124048-240666 2021.
Jayesh K Gupta and Johannes Brandstetter. Towards multisppatiotemppral-ssste generalized pde modeling. arXiv preprint arXiv:2209. 15616, 2022.
Zhongkai Hao, Zhengyi Wang, Hang Su, Chengyang Ying, Yinpeng Dong, Songming Liu, Ze Cheng, Jian Song, and Jun Zhu. Gnot: A general neural operator transformer for operator learning. In International Conference Machine Learning, pp. 125691 PMLR, 2023.
12

---

Published as a conference paper at ICLR 2026
Zhongkai Hao, Chang Su, Songming Liu, Julius Bernrr, Chengyang Ying, Hang Su, Anima Anand- kumar, Jian Song, and Jun Zhu. Dpot: Auto-regressive denoising operator transformer for large- scale pde pre-training. In International Conference Machine Learming, pp. 17616-17635. PMLR, 2024.
Guanghua He, Weijie Mo, Yun Gao, Zhigang Zhang, Jiadong Wang, Wei Wang, Pengfei Liu, and Hassan Ghassemi. Modification of effective angle of attack on hydrofoil power extraction. Ocean Engineering, 240: 109919, 2021.
J-W He, R Glowinski, R Metcalfe, A Nordlander, and Periaux. Active control and drag optimiza- tion for flow past a circular cylinder: I. oscillatory cylinder rotation. Journal of Computational Physics, 163(1):83-11)) 2000.
Maximilian Herde, Bogdan Raonic, Tobias Rohner, Roger Käppeli, Roberto Molinaro, Emmanuel de Bézenac, and Siddhartha Mishra. Poseidon: Efficient foundation models for pdes. Advances in Neural Information Processing Systems, 37:72525-72624, 2024.
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840-688 2020.
Simone Hochgreb. Mind the gap: Turbulent combustion model validation and future needs. Pro- ceedings of the Combustion Institute, 37(2):2011-2107, 2019.
Peiyan Hu, Yue Wang, and Zhi-Ming Ma. Better neural pde solvers through data-free mesh movers. In The Twelfth International Conference on Learning Representations, 2024.
Peiyan Hu, Rui Wang, Xiang Zheng. Tao Zhang, Haodong Feng, Ruiqi Feng, Long Wei, Yue Wang, Zhi-Ming Ma, and Tailin Wu. Wavelet diffusion neural operator. In The Thirteenth International Conference on Learning Representations, 2025.
Bin Huang and Jianhui Wang. Applications of physics-informed neural networks in power systems-a review. IEEE Transactions on Power Systems, 38(1):772-588, 2022.
Xinquan Huang, Wenlei Shi, Qi Meng, Yue Wang, Xiaotian Gao, Jia Zhang, and Tie- Yan Liu. Neuralstagger: accelerating physics-constrained neural pde solver with spatial-temporal decom- position. In International Conference on Machine Learning, pp. 13993-14006 PMLR, 2023.
Linlin Kang, An-Kang Gao, Fei Han, Weicheng Cui, and Xi- Yun Lu. Propulsive performance and vortex dynamics of jellyfish-like propulsion with burst-and-coast strategy. Physics of Fluids, 35 (9), 2023.
Linlin Kang, Yanru Wang, Zhongzheng Jiang, Jinhua Hao, Shiying Xiong, Dixia Fan, and Weicheng Cui. Effects of magnetic-vortee vortical interactions magnetic splitting. Journal of Fluid Mechanics, 1009:A55, 2025.
George Em Karniadakis, loannis G Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics-informed machine learning. Nature Reviews Physics, 3(6):422-440, 2021.
Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint 1rXiv:1412.69880 6980, 1412(6), 2014.
Felix Kochler, Simon Niedermayr, Nils Thuerey, et al. Apebench: A benchmark for autoregres- sive neural emulators of pdes. Advances in Neural Information Processing Systems, 37:1 20252- 120310, 2024.
Arthur G Kravchenko and Parviz Moin. Numerical studies of flow over circular cylinder at re d= 3900. Physics of fluids, 12(2):403-417, 2000.
Samvit Kumar, Sanjay Mittal, et al. Lock-in in forced vibration of a circular cylinder. Physics of Fluids, 28(11), 2016.
J Nathan Kutz, Steven Brunton, Bingni W Brunton, and Joshua L Proctor. Dynamic mode decom- position: data-driven modeling of complex systems. SIAM, 2016.
13

---

Published as a conference paper at ICLR 2026
Alexander Lavin, David Krakauer, Hector Zenil, Justin Gottschlich, Tim Mattson, Johann Brehmer, Anima Anandkumar, Sanjay Choudry, Kamil Rocki, Atilm Güneş Baydin, et al. Simulation intelligence: Towards generation of scientific methods. arXiv preprint arXhv:2112.02335, 2021.
Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, An- drew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. In The Ninth International Conference on Learning Representations, 2021.
Zongyi Li, Hongkai Zheng, Nikola Kovachki, David Jin, Haoxuan Chen, Burigede Liu, Kamyar Azizzadenesheli, and Anima Anandkumar. Physics-informed neural operator for learning partial differential equations. ACM/MMS Journal of Data Science, 1(3):1-27, 2024.
Ke Lin, Yuankun Sun, Hongyuan Liu, Yunpeng Zhu, Jiasong Wang, Michael S Triantafyllou, and Dixia Fan. Mapping the properties of wake-induced vibration on a circular cylinder. Journal of Fluid Mechanics, 1001:A33, 2024.
Chunyu Liu, Haojie Yang, Can Ruan, Liang Yu, and Xingcai Lu. Influence of swirl intensity on combustion dynamics and emissions in ammonia-enriched methane/air combustor. Physics of Fluids, 36(3), 2024a.
Jian Liu, Jianyu Wu, Hairun Xie, Jing Wang, Liu Wei, Wanli Ouyang, Junjun Jiang, Xianming Liu, SHIXIANG TANG, Miao Zhang, et al. Afbench: A large-scale benchmark for airfoil design. Advances in Neural Information Processing Systems, 37:82757-82788 2024b.
Ilya Loshchilov and Frank Hutter. Sgdr: Stochastic gradient descent with warm restarts. arXiv preprint 16Xiv: 160000088,...8,, 2016.
J.-M. Lourier, M. Stöhr, B. Noll, S. Werner, and A. Fiolitakis. Scale adaptive simulation of thermoacoustic instability in a partially premixed lean swirl combustor. Combustion and Flame, 183:343-357, 2017. ISSN 0010-2180.
Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nature machine intelligence, 3(3):218-229, 2021.
Yining Luo, Yingfa Chen, and Zhen Zhang. Cidbench: A large-scale benchmark for machine learn- ing methods fluid dynamics. arXiv preprint 1x.x::3331000963 2023.
Zekang Lyu, Tongtong Yan, Yong Qian, Liulin Cen, Dezhi Zhou, and Xingcai Lu. Experimental and numerical study of soot formation in laminar n-butyleyclohexane and n-butylbenzene diffusion flames at elevated pressures. Combustion and Flame, 13341, 2024.
Ma, G-S Karamanos, and GE Karniadakis. Dynamics and low-dimensionality of a turbulent near wake. Journal of fluid mechanics, 410:29-65, 2000.
Runze Mao, Rui Zhang, Xuan Bai, Tianhao Wu, Teng Zhang, Zhenyi Chen, Minqi Lin, Bocheng Zeng, Yangchen Xu, Yingxuan Xiang, al. Benchmarking neural surrogates on realistic spa- tiotemporal multiphysics flows. arXiv preprint arXiv:2512.18599 2025.
Carl D Meinhart, Steve T Wereley, and Juan G Santiago. A piv algorithm for estimating time- averaged velocity fields. Journal of fluids engineering. 122(():285-22)) 2000.
Juan Nathaniel, Yongquan Qu, Tung Nguyen, Sungduk Yu, Julius Busecke, Aditya Grover, and Pierre Gentine. Chaosbench: A multi-channel, physics-based benchmark for subseasonal-to- seasonal climate prediction. Advances in Neural Information Processing Systems, 37:43715- 43729, 2024.
Arnold Neumaier. Introduction to Numerical Analysis. Courier Corporation, 1987.
Ingrid Neunaber, Srikar Yadala, and Jason Hearst. Induced periodicity in wakes. Journal of Fluid Mechanics, 1022:R3, 2025.
14

---

Published as a conference paper at ICLR 2026
William L Oberkampf and Timothy G Trucano. Verification and validation in computational fluid dynamics. Progress in aerospace sciences, 38(3):209-213 2002.
Ruben Ohana, Michael McCabe, Lucas Meyer, Rudy Morel, Fruzsina Agocs, Miguel Beneitez, Mar- sha Berger, Blakesly Burkhart, Stuart Dalziel, Drummond Fielding, et al. The well: a large-scale collection of diverse physics simulations for machine learning. Advances in Neural Information Processing Systems, 37:44989-45037, 2024.
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, high- performance deep learning library. Advances neural information processing systems, 32, 2019.
Tobias Pfaff, Meire Fortunato, Alvaro based simulation with graph networks. In The Eighth International Conference on Learning Representations, 2020.
and Peter Battaglia. Learning mesh-
Thierry Poinsot and Denis Veynante. Theoretical and numerical combustion. RT Edwards, Inc., 2005.
Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Physics-informedt neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys., 378:686-7077 2019.
Bogdan Raonic, Roberto Molinaro, Tim De Ryck, Tobias Rohner, Francesca Bartolucci, Rima Alai- fari, Siddhartha Mishra, and Emmanuel de Bézenac. Convolutional neural operators for robust and accurate learning of pdes. Advances in Neural Information Processing Systems, 36:77187- 77200, 2023.
Patrick J Roache, Verification and validation in computational science and engineering, volume 895. Hermosa Albuquerque, NM, 1998.
Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedi- cal image segmentation. In International Conference on Medical image computing and computer- assisted intervention, pp. 234-241. Springer, 2015.
Khemraj Shukla, Vivek Oommen, Ahmad Peyvan, Michael Penwarden, Nicholas Plewacki, Luis Bravo, Anindya Ghoshal, Robert M Kirby, and George Em Karniadakis. Deep neural operators as accurate surrogates for shape 129:107615, 2024.
o optimization. Engineering Applications of Artificial Intelligence,
Siemens Digital Industries Software. Simcenter STAR-CCM+, version 2022.1, Siemens 2022.
Philippe R Spalart and V Venkatakrishnan. On the role and challenges of cfd in the aerospace industry. The Aeronautical Journal, 120(1223):209-232 2016.
Jihao Sun, Qiang Yang, Ningbo Zhao, Mingmin Chen, and Hongtao Zheng. Numerically study of ch4/nh3 combustion characteristics in industrial gas turbine combustor based on reduced mechanism. Fuel, 327:124897, 2022. ISSN 0016-2361.
Xu Sun, C Steve Suh, Ze-Hua Ye, and Bo Yu. Dynamics of circular cylinder with an attached splitter plate in laminar flow: A transition from vortex-induced vibration to galloping. Physics of fluids, 32(2), 2020.
Makoto
Takamoto, Timothy Praditia, Raphael Leiteritz, Daniel MacKinlay, Francesco Alesiani, Dirk Pflüger, and Mathias Niepert. Pdebench: extensive benchmark for scientific machine learning. Advances in Neural Information Processing Systems, 15:159611611, 2022.
Ronak Tali, Ali Rabeh, Cheng-Hau Yang. Mehdi Shadkhah, Samundra Karki, Abhisek Upadhyaya, Suriya Dhakshinamoorthy, Marjan Saadati, Soumik Sarkar, Adarsh Krishnamurthy, et al. Flow- bench: A large scale benchmark for flow simulation over complex geometries. arXiv preprint arXiv:2409. /8032, 2024.
Jeyan Thiyagalingam, Mallikarjun Shankar, Geoffrey Fox, and Tony Hey. Scientific machine learn- ing benchmarks. Nature Reviews Physics, 4(6):413-420, 2022.
15

---

Published as a conference paper at ICLR 2026
Artur Toshev, Gianluca Galletti, Fabian Fritz, Stefan Adami, and Nikolaus Adams. Lagrangebench: A lagrangian fluid mechanics benchmarking suite. Advances in Neural Information Processing Systems, 36:64857-648888 2023.
Amin Totounferoush, Serge Kotchourko, Michael W Mahoney, and Steffen Staab. Paving the way for scientific foundation models: enhancing generalization and robustness in pdes with constraint- aware pre-training. arXiv preprint arXiv:2503. 19081, 2025.
Stéfan van der Walt, Enthought Usa S. Chris Colbert, Gaël Varoquaux, and Inria Saclay. The numpy array: A structure for efficient numerical computation. Computing in Science & Engineering, 13: 22-30, 2011.
Denis Veynante and Luc Vervisch. Turbulent combustion modeling. Progress in energy and com- bustion science, (8(3):193-222) 266, 2002.
Haixin Wang, Yadi Cao, Zijie Huang, Yuxuan Liu, Peiyan Hu, Xiao Luo, Zezheng Song, Wanjia Zhao, Jilin Liu, Jinan Sun, et al. Recent advances on machine learning for computational fluid dynamics: A survey. arXiv preprint arXivv2408. 12171, 2024a.
Haixin Wang, Jiaxin Li, Anubhav Dwivedi, Kentaro Hara, and Tailin Wu. Beno: Boundary- embedded neural operators for elliptic pdes. In The Twelfth International Conference on Learning Representations, 2024b.
Rui Wang, Karthik Kashinath, Mustafa Mustafa, Adrian Albert, and Rose Yu, Towards physics- informed deep learning for turbulent flow prediction. In Proceedings of the 26th ACM SIGKDD international conference on knowledge discovery & data mining, pp. 1457-1466, 2020.
Yuting Weng, Han Li, Hao Zhang, Zhi X Chen, and Dezhi Zhou. Extended fourier neural operators to learn stiff chemical kinetics under unseen conditions. Combustion and Flame, 272:113847, 2025.
Gabriel D Weymouth and Bernat Font. Waterlily. jl: A differentiable and backend-agnostic julia solver for incompressible viscous flow around dynamic bodies. Computer Physics Communica- tions, pp. 109748, 2025.
Gabriel Weymouth and Dick K-P Yue. Conservative volume-of-fuu method for free-surface simulations on cartesian-grids. Journal of Computational Physics, 229(8):853-2265) 2010.
Gabriel D Weymouth and Dick KP Yue. Boundary data immersion method for cartesian-grid simula- tions of fluid-body interaction problems. Journal of Computational Physics, 230(16):6233-62277 2011.
JG Wissink and W Rodi. Numerical study the near wake of a circular cylinder. International journal of heat and floidffow, flow, 29)40:060-1177 2008.
Haixu Wu, Huakun Luo, Haowen Wang, Jianmin Wang, and Mingsheng Long. Transolver: a fast transformer solver for pdes on general geometries. In Proceedings of the 41st International Con- ference on Machine Learning, pp. 53681-537777 2024a.
Tailin Wu, Takashi Maruyama, Long Wei, Tao Zhang, Yilun Du, Gianluca laccarino, and Jure Leskovec. Compositional generative inverse design. In The Twelfth International Conference on Learning Representations, 2024b.
Kecheng Xu, Philippe Lavoie, and Pierre Sullivan. Flow reattachment on a naca 0025 airfoil using an array of microblowers. AIAA Journal, 61(6):2476-2485 2023.
Zhanhong Ye, Xiang Huang, Leheng Chen, Hongsheng Liu, Zidong Wang, and Bin Dong. Pde- former: Towards a foundation model for one-dimensional partial differential equations. In ICLR 2024 Workshop on Al/DifferentilEEquation In Science, 2024.
Weiwei Zhang, Xintao Li, Zhengyin Ye, and Yuewen Jiang. Mechanism of frequency lock-in in vortex-induced vibrations at low reynolds numbers. Journal of Fluid Mechanics, 783:72-109 2015.
16

---

Published as a conference paper at ICLR 2026
Weiwei Zhang, Xianglin Shan, Yilang Liu, Xiao Zhang, Zhenhua Wan, Xinliang Li, Xinguo Sha, Junbo Zhao, Hui Xu, Chuangxin He, et al. High-reynolds numberr turbulence database: Aeroflow- data. Scientific Data, 12(1):1500, 2025.
Hongkai Zheng, Wenda Chu, Bingliang Zhang, Zihui Wu, Austin Wang, Berthy Feng, Caifeng Zou, Yu Sun, Nikola Borislavov Kovachki, Zachary E Ross, et al. Inversebench: Benchmark- ing play diffusion priors for inverse problems in physical sciences. In The Thirteenth International Conference on Learning Representations, 2025.
Donghua Zhou, Ming Fang, Jian Feng, Jie Zhao, Ke Xu, Yingsong Li, Zhixiang Huang, and Atef Z Elsherbeni. Spatial-temporal electromagnetic field simulation based on fourier neural operator. IEEE Transactions Antennas and Propagation, 2025.
17

---

Published as a conference paper at ICLR 2026
A
OTHER EXPERIMENT RESULTS
In this section, we provide the experiment results that are not in the main text due to the page limit.
A.1 OTHER METRICS
In addition to the results reported in Ta the main text, our main experiments are also evaluated under other metrics, including MAF, R², KF, and FE as mentioned in Sec. 3.3 The results for these additional metries are summarized in Table [ We note that the KE is not applied to the Combustion dataset, because the modality in this dataset is the light intensity. Besides, the velocity field is not a key variable in this setting. From the table, overall, we find that U-Net and DPOT-L- FT perform relatively well on these metrics, showing their strong capability of modeling complex physical systems. Besides, the Combustion dataset more challenging. For instance, the R metric on this dataset is significantly lower than on the others.
Table 2: Results of other metrics. The bolded values are the best.
Simulated Training
Real-worid Training g
Real-world Finetuning
Dalaset
Baselime
MAE L
Ie"
FB
MAE 1
J
MAE
1
KB L
4
Net
0.03591
0.62955
0.00040
195.31436
0,01266
0.68363
0,60027
141.57477
0,01213
0.74257
0.00027
128.004339
CNO
0.02909
0.71633
0.00050
124.15566
0,01208
0.90334
0.00622
44.89573
0.01369)
0.91341
0.00051
201 59423
DeppONet
0,04357
0.51881
0.00050
176.41255
0,02025
0.67225
0.00440
150,94789
0.01912
0,71811
0.00449
130. 10420
HNU
0003100
0.64762
0,000
244.28989
0,01277
0.77934
0.0046
124.80349
0,01105
0.30835
0.00633
161.91525
Cylinder
WDNO
0,02762
0.68467
0.00799
0.02190
0.0013
150.92130
0.01596
0.83044
0.00554
124 44474
MWT
0.02454
0.71324
0.0047
105.17693
0.01155
0,84425
70.63197
0.01332
0.00039
95.87988
CKTTransfrmer
0.04161
0.59035
0.00056
247 50152
0.02099
0,23996
0.00322
220.35785
0,01587
0,7159
0.0038
202. 11179
Hinssolver
0.52580
0.00050
504.92477
0.02611
0.30250
860087
26486488
0,02566
020032
000055
187,44112
DROT-S-FT
001847
0.8344
0.0044
137-41856
0.01440
0.72733
000070
180 77551
000056
087681
000060
132. 15703
DROTL-FT
0.01749
0,84745
0.00070
107.55474
0,01010
0,90178
0.00006
90.55615
0,0826
039862
0.0044
78.74014
DMD
005116
0.52011
0,00500
107 47345
U.Net
0.01550
0.92443
0.00005
46.81991
0.0085
0.98747
0,00003
0,36774
0.000N3
18:91348
CNO
0.00895
0,94880
0,00005
0.0006
0.9665
0000063
16
036181
0,000L3
15.58394
Controlld
Cylinder
GK TKaandoormrr
DeepONet
WENO
MWT
0.01336
0.03755
0.04944
001304
0.86483
088544
0.83962
0.42567
0.90883
000000
0.0000
0.0007
300000
100000011000
106-0000
0.00565
0
0.00355
00000
0000555
0000505
100033
092280 900000
0,1073
0,9136
80000
200000
60000
000000
100000
100000
21.3322
21-58843
19
21-58003
10000000
6003
000000
0
0
037888
09622
00188800000
00000000000000000000000000000000
000000
0
000000
010000
000001
1000000
00000
19 200000110000
194771
20.21-886
28
35.51
20-12006
Transolver
0.01996
0.82884
600000
0100011
100005
100025
OrOIns
28. 16750
DPOT-S-FT
0.01355
0.89218
0.00005
94211255
0:00115
096588
000003
19 96995549
0H0000
0 0365788
0,00033
21 366666
DPOT-L-FT
0.01418
0.7713
0000066
31.55077
0.96394
0000003
19,87323
000430
058675
0.00013
18.11246
DMD
081194
0:77002
O000T
44 73851
Net
0.91588
0.08008
11,22311
6.9779
0,0004
7,29539
0,00440
0.96798
0,00004
6.18526
FSU
DeepONet
WDND
CNO
HND
0.1408
0.2284
0.43299
0,02455
0. 067678
0.90544
0.69099
0.3052
80000
000000
00000
0
00000000000000000000000000000000
45.6 3380085
0
000000
070027
00000000000000000000000000000000
000000
00000000000000000000000000000000
10 100006
80000
00000
1000000
12 12:30000
23
975724
2333550
20200
100000
0677000
050212
031301
199433
00000000000000000000000000000000
0
0000000
12. 12-17235 13,1238
MWT
0.02071
0. 0.79785
0,00008
1985775
0
0:00734
0991100
800006
7.53911
00X158
059135
010006
7,09237
GK-Trandormer
0.81875
883438
000013
241896448 61686649
030055
0591155
1000066
14.14439
180750
057788
010036
10. 10.72866
Trnssoiver
0001805
033091
000011
036214
800099
1307740
101204
051215
010009
9. 9.52701
DPOT-S-FT
0.1573
088510
000000
7035580
0.8135
80005
7.11500
088575
058132
0 000005
6.49055
DPOT-L-FT FT
0.01543
0.88336
0,00008
31,56328
0.00555
0.9334
0,0005
5.01443
000040
058112
0.00055
4,86275
DMD
003413
0.65550
0000100
30.34145
205. 18223000
U-Net
0.1334
033131
000016
0
6 999773
0
22 22.60
0,02277
0.99186
0000007
1463333
CNO
0.00229
0.95236
5
0:00553
0992266
1000
76:37536
0.0426
0 096784
000008
S7 55155
DeepONet
0.01305
0.89360
0.0001
0:00800
095433
8,0019
43,25836
103004
03:272
010019
47.95815
FND
0.1100
0393077
00001
3599399911
1
019828
800089
52.29553
130063
059664
010019
43.83851
Foil
GK Transoomer
WDND
MWT
001069
0.88959
0.00009
0.92888
0.94567
0.93540
0.00115
0.0016
000023
19833413
138 17717711
4124311
000059
0000100
0.9127
0 0.8368
099564
8:0010
000009
800012
50.65557
58.32386
130244
000413
038134
1596551
038560
000000
000009
010008
26,05155 16.05156
25 25888887
60,33815
Tinnsolvvr
0.01020
0.88917
0,00015
00000
800016
45
156415
000015
22:92277
DPOT-S-FT
0.95464
0,0000
02.92200
0.0025
090054
000008
15.31028
100055110000
0.0008
26.61732
DPOT-L-FT
0.00777
0.95150
0.0016
34.12406
0.00310
0.98978
8.0008
0000000100000000000000000000000
15.25477
0.00244
00000S
18.24177
DMD
0.00937
0.0088
0.00199
84 34200
Net
0.00934
0.29106
$4455788
0,90575
0.74085
26.02853
0.00542
0.34736
25.05 070--70
CNO
0011007
0.11123
103. 33561
0.00662
0.6990
38. 38.39980
0.006544
0.68844
34 42250
DeepONet
0.11171
0,10024
105 85174
0.00617
0. 077817
29.43881
0.00602
0.71397
29. 29,32702
HNO
0.00944
0.26832
0.00886
0.71771
30.65520
0,00597
0.72008
30 03.53448
Combustion
WDNO
0007007
.006951
0.01087
119941
90. 00.55599
0.00888
0.45128
$7.61599
0.00991
0.26912
13,03391
0,00562
0.73053
28.46920
0.00566
0.73146
27.91226
EK-Transfommer
0.11399
0:11814
112.22233
0.00099
0.66380
38,45433
0.K0747
0.63269
43.30125
Trimsol ver
0.1387
-007878
130 66479
0.01172
0.22220
76 6767668
0.01170
0.33352
7568771
DPOT FT
0.01096
0.14192
104.63533
0,00583
0.75845
23.07150
0.00584
0.75402
26.17408
DPOT-L-FT
0.01000
0.20883
92.19999
0.00537
0.76120
23.25353
0.00535
0.76467
22,73871
DMD
0.00201
-3.630922
120966338
A.2 AUTOREGRESSIVE EVALUATION
Below, we provide the full result of autoregressive evaluation in Table 3 (2 rounds of autoregressive
evaluation), Table 4 (3 rounds of autoregressive evaluation), and Table (1, 2, and 3 rounds of
18

---

Published as a conference paper at ICLR 2026
autoregressive evaluation under RMSE, relative L2 eπrоr, and FRMSE). From the results, several observations can be made. First, the Controlled Cylinder and Foil datasets exhibit relatively stable performance, with models such U-Net maintaining high accuracy, suggesting that their underlying dynamics are more favorable for stable long-horizon prediction. contrast, the Combustion dataset remains the most challenging: all methods achieve significantly lower R values, and erross increase more sharply. Second, U-Net variants consistently achieve superior results across most metrics and datasets, highlighting their robustness in capturing spatiotemporal dynamics. Transformer-based approaches (GK-Trae Transformer, Transolver) show competitive short-term performance but degrade faster with increasing horizons. Finally, Frequency Errors increase markedly from three autoregressive steps, indicating that the error of capturing temporal dynamics grows rapidly over time.
19

---

Published as a conference paper at ICLR 2026
Table 3: rounds of autoregressive evaluation. The bolded values are the best.
Dataset
Baseline
RMSE 4
MAE +
Rel L2 4
R²
KE L
IRMSE
FE +
U-Net 0.07535
0.01478
0.09430
0.63360
0.00070
0.00645
198.26683
CNO
0.05911
0.01962
0.15177
0.77450
0.00145
0.00597
479.91360
DeepONet
0.07612
0.02123
0.16087
0.62603
0.00115
0.00685
214,22360
FNO
0.06765
0.01362
0.09777
0.70467
0.00134
0.00632
281.23196
Cylinder
WDNO
0.06319
0.01768
0.12782
0.74232
0.00091
0.00566
337.25354
MWT
0.06178
0.01341
0,10118
0.75365
0.00079
0.00541
139,07365
GK- RerTransformer
0.10814
0.02063
1.19
0.24523
0.00071
0.00821
241.95125
Transolver
0.08928
0.02582
0.19561
0.48559
0.00096
0.00753
204.22385
DPOT-S-FT
0,05778
0.01342
0.09745
O. 0.78450
0.00096
0.00570
190.29182
DPOT-L-FT
0.05774
0.01264
0.09439
0,78480
0.00092
0.00533
137.29283
DMD
0.09943
0.03575
0,40840
0.36195
0.00131
0.00799
176.32237
U-Net 0,01083
0.00523
0.07585
0.97676
0.00008
0,00130
34.18222
CNO
0.01126
0.00565
0.08116
0.97485
0.00008
0.00135
28 28,88530
DeepONet
0.03063
0.01974
0,23983
0.81402
0.00017
0.00477
103.55875
FNO
0.01221
0.00656
0.08960
0.97047
0.00009
0.001 46
34.40803
Controlled
WDNO
0.01331
0.00720
0.09 0.09922
O. D.96490
0.00009
0.00168
S3.22416
Cylinder
MWT
0.01328
0.00726
0.09763
0.96504
0.00009
0.00156
36,93626
GK-rransformer
0.01291
0.00698
0.09513
0.96695
0.00010
0.00155
32,69817
Transolver
0.01974
0.01105
0,14127
0.92278
0.00013
0.00258
47.72284
DPOT-S-FT
0.01140
0.00590
0.08268
0.97422
0.00008
0.00138
38,86862
DPOT-L-FT
0.01
0.00573
0.08163
0,97461
0.00008
0.00137
34.77075
DMD
0.04111
0.02186
0.27235
0.66506
0.00021
0,00619
75.94245
U-Net
0.01379
0.00685
0.09313
0.96760
0.00014
0.00119
12.71709
CNO
0.01527
0.00829
10587
0. 0.96032
0.00014
0.00131
21,32656
DeepONet
0,03472
0.02122
0.24746
O. 0.79477
0.00025
0.00345
49.31825
ENO
0.01714
0.00944
0.11648
0.94998
0.00016
0.00155
23.81794
FSI
WDNO
0.01756
0.00972
0.12205
0.94749
0.00016
0.00157
28.00188
MWT
0.01817
0.00999
0.12837
0,94375
0.00016
0.00151
15.51482
GK-Transformer
0.01683
0.00930
0. 1695
0.95178
0.00016
0.00148
21.17584
Transolver
0.02358
0.01278
0.15668
0.90214
0.00009
0.00267
13.07140
DPOT-S-FT
0.01476
0.00781
0.10131
0.96289
0.00015
0.00125
13.45711
DPOT-L-FT
0.01440
0.00736
0.09714
O. 0.96467
0.00014
0.00123
10.97794
DMD
0.05420
0.02875
0.35644
O. 0.49863
0.00030
0.00524
55.02239
U-Net 0,01228
0.00333
0.01905
0.98601
0.00012
0.00080
23.86791
CNO
0.01503
0.00535
0.02 0.02667
0.97904
0.00016
0.001 22
107.73200
DeepONet
0.05055
0.01461
0.06368
O. 0.76775
0.00095
0.00401
$23. 23944
FNO
0.01582
0.00493
0.02709
0.97677
0.00016
0.00109
77.52514
WDNO
0.01409
0.00414
0.02321
0.98156
0.00013
0.00089
44.82506
MWT 0.01529
0.00454
0.02528
0.97829
0.00015
0.00104
38.69154
GK- Oo-Transformer
0.01663
0.00555
0.02851
0.97432
0.00015
0.00135
104.44638
Transolver
0.02074
0.00616
0.03521
0,96008
0.00025
0.00144
46.71509
DPOT-S-FT
0.01420
0.00420
0.02182
0.98127
0.00013
0.00108
48. 70566
DPOT-L-FT
0.01449
0.00397
0.02094
0.98049
0.00012
0.00112
40.78880
DMD
0.03335
0.01051
0.05529
0.89673
0.00029
0.00282
156.66721
U-Net 0.02279
0.00570
0.55826
O. 0.76663
0.00204
44 7058888
CNO
0.02728
0.00733
0.62687
0.58102
0.00253
67.08688
DeepONet
0.02930
0.00738
0.67170
0.51662
0.00278
88.01767
FNO
0.02422
0.00633
0.59011
0.66980
0.00221
S2.72492
Combustion
WDNO
0.03360
0.00936
0.76863
0.36430
0.00324
95.65475
MWT
0.02346
0.00594
0.57131
0.69001
0.00212
48.42057
GK- Grarafoforr
0.02755
0.00770
0.65741
0.57253
0.00257
75.15970
Transolver
0.04066
0.01277
0.82321
0.06895
0.00373
129.05060
DPOT-S-PT
0.02262
0.00614
0.55686
0.71180
0.00203
44 44.36895
DPOT-L-FT
0.02255
0.00577
0.56212
0.71376
0.00202
43.57514
DMD
0.09502
0.02172
1.38668
-.08454
0.00932
162.85385
20

---

Published as a conference paper at ICLR 2026
Table 4: 3 rounds of autoregressive evaluation. The bolded values are the best.
Dataset
Baseline
RMSE 4
MAE +
Rel L2 4
R²
KE L
[RMSE
FE 4
U-Net
0.08270
0.01679
0.11133
0.55877
0.00107
0.00473
268.25345
CNO
0,08047
0.02730
0.21099
0.58218
0.00269
0.00560
887.97833
DeepONet
0.08515
0.02318
0.17226
0,53221
0.00165
0.00509
277.13663
FNO
0.07479
0.01537
0.11204
0.63911
0.00181
0.00500
379.31039
Cylinder
WDNO
0,07129
0.02093
0.15456
0.67208
0.00127
0,00450
558.40649
MWT 0.06570
0.01511
0.11673
0.72153
0.00111
0.00404
196.52660
GK- FrrTransformer
0.10912
0.02197
0.12545
0.23177
0.00112
0.00566
284.
Transolver
0.09106
0.02808
0,21982
0.46500
0.00156
0.00528
236.661 66180
DPOT-S-FT
0,06700
0.01550
0.11305
0.71036
0.00128
0.00454
252.41632
DPOT-L-FT
0.06689
0.01480
0.10939
0.71132
0.00120
0.00425
199.98730
DMD
0.10664
0.03774
0.42629
0.26768
0.00210
0.00607
239.53703
U-Net 0,01289
0.00636
0.09091
0.96705
0.00013
0,00134
49.1 1861
CNO
0.01380
0.00711
1015
0.96221
0.00014
0.00144
45.14964
DeepONet
0.03241
0.02116
0.25293
0.79177
0.00029
0.00383
48720
FNO
0.01424
0.00769
10474
0.95981
0.00014
0.00145 48
47. 47.96572
Controlled
WDNO
0.01532
0.00833
O. 1422
0.95348
0.00014
0.00164
73.24361
Cylinder
MWT
0.01595
0.00885
0.11881
0.94953
0.00015
0.00166
58,24154
oK-rransfrrmer
0.01505
0.00824
0.11178
0.95507
0.00015
0.00157
45.33165
Transolver
0.02236
0.01267
1.1444
0.90090
0.00021
0.00243
70.27728
DPOT-S-FT
0.01344
0.00704
0.09799
0.96416
0.00013
0.00141
55.72421
DPOT-L-FT
0.01326
0.00681
0.09643
0.96514
0.00013
0.00140
$0.39636
DMD
0.04714
0.02534
0.31516
0.55938
0.00036
0.00545
109.54674
U-Net
0.01767
0.00863
0.11893
0. 0.94680
0.00023
0.00123
19.91612
CNO
0.02237
0881
0.15929
1491
0.00027
0.00151
$1,93961
DeepONet
0.03638
0.02221
0.25920
0.77434
0.00042
0.00261
76.82671
ENO
0.02078
0.01117
0.14110
0.92638
0.00026
0.00149
35.43581
FSI
WDNO
0.02150
0.01164
D.14855
0.92120
0.00026
0.00153
44.13584
MWT
0.02907
0.01424
0,21189
0.85591
0.00045
0.00178
40.14235
GK-Transformer
0.02064
0.01115
14221
0.92740
0.00027
0.00145
32. 1624
Transolver
0.02358
0.01278
0.15668
0.90214
0.00009
0.00267
13.07140
DPOT-S-FT
0.01869
0.00954
0.12678
0,94046
0.00026
0,00130
21,02419
DPOT-L-FT
0.01795
0.00891
02145
0.94505
0.00023
0.001 25
17.38784
DMD
0.06251
0.03289
0.41228
0,33119
0.00052
0.00458
82,20554
U-Net 0,01458
0.00377
0.02269
0.98026
0.00014
0.00068
31,53317
CNO
0.01816
0.00641
0.03241
0. 0.96939
0.00021
0.00108
169,928 9281 12
DeepONet
0.07452
0.02080
0.08767
O D.4853
0.00171
0.00412
896. 996.86994
FNO
0.01867
0.00560
0.03144
0.96765
0.00018
0.00089
11.04112
Foil
WDNO
0.01674
0.00463
0.02691
0.97399
0.00014
0.00075
59.70494
MWT
0.01812
0.00504
0.02961
0.96952
0.00017
0.00085
52.03396
forTransformer
0.01874
0.00617
0.03166
0.96739
0.00019
0.001
143.23317
Transolver
0.02180
0.00665
0.03694
0,95588
0.00030
0.001 14
69.33317
DPOT-S-FT
0.01690
0.00477
0.02530
0.97348
0.00016
0.00093
69, 69.95742
DPOT-L-FT
0.01740
0.00455
0.02452
0.97189
0.00015
0.00096
61.23933
DMD
0.03469
0.01169
0.05916
0.88825
0.00033
0.00226
938.98497
U-Net 0.02418
0.00599
0.58134
0.67051
0.00184
62,51865
CNO
0.03039
0.00830
0.67417
0.47944
0.00232
95.54838
DeepONet
0.03434
0.00910
0.97732
0.33515
0.00262
146.2 146.21243
FNO
0.02616
0.00675
0.62085
0.61428
0.00200
73.07064
Combustion
WDNO
0.03559
0.00982
0.79687
0.28603
0.00277
122.8840 88408
MWT
0.02521
0.00631
0.59812
0.64181
0.00192
67.06616
GK- Grarafoffrr
0,02935
0.00804
0.68410
0.51450
0.00223
94.56139
Transolver
0.04187
0.01332
0.84313
0.01181
0.00307
148.69182
DPOT-S-PT
0.02421
0.00648
0.58350
0.66966
0.00185
63,20166
DPOT-L-FT
0.02454
0.00622
0.59435
0.66066
0.00187
64.32231
DMD
0.09817
0.02263
1.42943
4,43148
0.00871
172.48363
21

---

Published as a conference paper at ICLR 2026
Table 5: 1, 2, and 3 rounds of autoregressive evaluation under RMSE, relative L2 error, and fRMSE. The bolded values are the best.
Round Rel L2
2 Round
3 Round
Dataset
Bascline
RMSE
IRMSE
RMSE +
Rel L/2
FRMSE
RMSE 4
Rel L2
FRMSE
U-Net
0.0632
0.0728
0.0097
0.0754
0.0943
0.0065
0.0827
0.1113
0.0047
CNO
0.0403
0.1013
0.0066
0,0591
0,1518
0.0060
0.0805
0.2110
0,0056
DeepONet
0.0661
0.1503
0.0107
0.0761
0.1609
0.0069
0.0852
0.1723
0.0051
FNO
0.0545
0.0780
0.0087
0.0677
0.0978
0.0063
0.0748
0.1120
0,0050
Cylinder
WDNO
0.0513
0.0969
0.0073
0.0632
0.1278
0.0057
0.0713
0.1546
0.0045
MWT
0.0584
0.0850
0.0088
0
0.1012
0.0054
0.0657
0.1167
0.0040
GK TrkFfrnffrmmrr
0.0995
0.0908
0.0146
0,10
0.1135
0.0082
0.1091
0.1285
0.0057
Transolver
0.0965
0.1723
0.0144
0.0893
0.1956
0.0075
0.0911
0.2198
0,0053
DPOT-S-FT
0.0440
0.0766
0.0067
0.0578
0.0975
0.0057
0.0670
0.2198
0.0053
DPOT-L-FT
0.0394
0.0733
0.0059
0.0577
0.0944
0,0053
0 06669
0.1094
0,0043
DMD
0,0862
0.3590
0.0114
0.0994
0.4084
0.0080
0.1066
0.4263
0.0061
U-Net
0,0079
0,0543
0.0IM9
0.0108
0.0759
0
0.0129
0,0909
0.0013
CNO
0.0080
0.0812
0.0014
0.1008
0.0014
DeepONet
0.0291
0.2284
0,0052
0.0306
0.2398
0.0048
0.0324
0.2529
0.0038
FNO
0.0094
0.0702
0.0896
0.0015
0.1047
0.0015
Controlled
WDNO
0.0101
0.0752
0.0013
0.0133
0.0993
0.0017
0.0153
0.1143
0.0016
Cylinder
MWT
0.0101
0.0752
0.0013
0.0133
0.0976
0.0016
0.0100
0.1188
0.0017
GK-Transfomer
0.0101
0.0748
0.0012
0.0129
0.0951
0.0016
0.0151
0.1118
0.0016
Transolver
0.0168
0.1176
0.0022
0.0197
0.1413
0.0026
0.0224
0.1614
0.0024
DPOT-S-FT
0.0085
0.0615
0.0010
0.01 14
0,0827
0.0014
0.0134
0.0980
0.0014
DPOT-L-FT
0.0085
0.0611
0.0010
0.0113
0.0816
0.0014
0.0133
0.0964
0.0014
DMD
0.0340
0.2233
0.0059
0,0411
0,2724
0.0062
0.0471
0.3152
0,0055
U-Net
0.0084
0.0579
0.0007
0.0138
0.0931
0
0.0177
0,1189
0.0012
CNO
0.0096
0.0679
0.0008
0.0153
0,1059
0.0013
0.0224
0.1553
0.0015
DccpONet
0.0332
0.2368
0.0048
0.0347
0.2475
0.0035
0 03034
0,2592
0,0026
FNO
0.0127
0.0881
0.0012
0.0171
0,1165
0.0016
0 0208
0.1411
0.0015
FSI
WDNO
0.0116
0.0824
0.0011
0.0176
0.1221
0.0016
0.0215
0.1486
0,0015
MWT
0.0128
0.0912
0.0010
0.0182
0,1284
0.0015
0.0291
0.2119
0.0018
GK-Transformer
0.0124
0.0898
0.0010
0.0168
0.1170
0.0015
0.0206
0,1422
0.0015
Transolver
0.0223
0.1480
0.0025
0.0236
0.1567
0.0027
0.0236
0.1567
0.0027
DPOT-S-FT
0.0099
0.0701
0.0007
0.0148
0,1013
0.0013
0.1268
0.0013
DPOT-L-FT
0.0097
0.0668
0,0008
0.0144
0.0971
0.0012
0.0180
0.1205
0.0013
DMD
0.0450
0.2955
0.0060
0.0542
0.3564
0.0052
0 0.0625
0.4123
0.0046
U-Net
0.0094
0.0145
0.00088
0.0191
0.0008
0.0227
0.0007
CNO
0.0114
0.0206
0.0013
0.0150
0.0267
0.0012
0.0182
0.0324
0,0011
DeepONet
0.0226
0.0375
0.0029
0.0506
0.0637
0.0040
0.0877
0.0041
FNO
0.0120
0.0206
0.0012
0.0158
0.0271
0.0011
0.0187
0.0314
0.0009
Foil
WDNO
0.0106
0.0181
0.0010
0.0141
0.0232
0.0009
0.0167
0.0269
0.0008
MWT
0.0125
0.0210
0.0012
0.0153
0.0253
0.0010
0.0181
0.0296
0,0009
GK-Transformer
0.0138
0.0237
0.0017
0.0166
0.0285
0.0014
0.0187
0.0317
0.0011
Transolver
0.0138
0.0237
0.0017
0.0207
0.0352
0.0014
0.0218
0.0369
0.0011
DPOT-S-FT
0.0109
0.0174
0.0012
0.0142
0.0218
1.0011
0.0169
0.0253
0.0009
L-
0.0109
0.0161
0.0012
0.0145
0,0209
0.0011
0.0174
0.0245
0,0010
DMD
0.0322
0.0520
0.0041
0.0334
0.0553
0.0028
0.0347
0.0592
0.0023
U-Net
0.0213
0.5403
0.0025
0.0228
0,5583
0.0020
0.0242
0.5813
0.0018
CNO
0.0238
0.5877
0.0030
0.6269
0.0025
0.0304
0.6742
0.0023
DeepONet
0.0227
0.5723
0.0028
0.0293
0.6717
0.0028
0.0343
0.9773
0.0026
FNO
0.0225
0.5680
0.5901
0.0022
0.6209
0.0020
Combustion
WDNO
0.0314
0.7441
0.0044
0.0336
0.7686
0.0032
0.0356
0.7969
0.0028
MWT
0.0220
0.0235
0.5713
0.0021
0.0252
0.5981
0.0019
GK Transformer
0.0258
0.6421
0.0034
0.0276
0.6574
0.0026
0.0294
0.6841
0.0022
Transolver
0.0375
0.7836
0.0050
0.8232
0.0037
0.8431
0.0031
DPOT S FT
0.0211
0.5378
0.0024
0.0226
0.5569
0.0020
0.0242
0.5835
0.0019
DPOT-L-FT
0.0206
0,5318
0.0023
0.0226
0.5621
0.0020
0.0245
0.5944
0.0019
DMD
0.0914
1.3360
0.01
0.0950
1.3847
0.0093
0.0982
1.4294
0.0087
22

---

Published as a conference paper at ICLR 2026
In addition, apply MVPE the 10-round evaluation. The visualizations of all baselines provided in Figure 7 and g From the visualizations, we observe that the velocity field u casier to capture, while the predictions of velocity harder.
300RG47
Ol-Taasbme
WNT
Figure 7: Mean velocity profile predicted by baselines' 10-round autoregressive evaluation on Cylinder.
23

---

Published as a conference paper at ICLR 2026
Sound
20146
2
30
1984
Figure 8: Mean velocity profile of predicted by baselines' 10-round autoregressive evaluation on Cylinder.
24

---

Published as a conference paper at ICLR 2026
A.3 Low-, MID- AND FICH-FREEUENCY FRMSE
We provide the full results of low-, mid-, and fighffreueecy fRMSE in Table including results of all baselines on all datasets. From the results, we find that on the Controlled Cylinder dataset and the FSI dataset, although the U-Net does perform the best under low-frequency fRSME, out- performs under Mid- and High- tiehufrreueecy IRMSE. This displays that the U-Net excels at capturing the information with higher frequency.
Table 6: Low, mid, high frequency error. The bolded values are the best.
Dataset
Baseline
Low-fRan Now-RRMSE 1
Mid-IRMSE
High-IRMSE
U-Net
0.01835
0.00774
0.00375
CNO
0,00916
0.00626
0.00449
DeepONet
0.01887
0,00887
0.00496
Cylinder
FNO
0.01241
0.00905
0.00439
WDNO
0.01203
0.00655
0.00368
MWT
0.01 145
0.00947
0.00519
GK-Transfonner
0.02855
0.01 130
0.00504
Transolver
0.02746
146
DPOT-S-FT
0,01081
0.00562
0.00411 1
DPOT-L-FT
0.00962
0.00509
0.00316
DMD
0.01182
0.01455
0.00688
U-Net
0.00079
0.00102
0.00101
0.00103
Controlled
DeepONet
0.00350
0.00642
0.00626
Cylinder
0.00094
0.00120
0.00115
WDNO
0:00129
0.00131
0.00128
MWT
0.00085
0.00117
0.00116
Transformer
0.00097
0.00129
Transolver
0.00203
0.00238
0.00227
DPOT-S-FT
0.00087
0,00108
0.00105
DPOT-L-FT
0.00089
0.00107
0.00104
DMD
0.00370
0.00837
0.00685
0.00076
CNO
0.00057
0.00084
0.00091
DeepONet
0.00476
0.00540
0.00391
FSI
FNO
0.00119
0.00117
0.00112
WDNO
0.00091
0.00108
0.00115
0.00066
0.00104
GK-TTransfomeer
0.00096
0.00102
0.00104
Transolver
0.00277
0.00239
DPOT-S-FT
0,00056
0.00079
0.00085
DPOT-L-FT
0.00058
0.00080
0.00086
DMD
0.00698
0.00617
0.00468
U-Net
0
0.00085
0.00077
CNO
0.00184
0.00124
0.00099
DeepONet
0.00373
0.00299
0.00193
Foil
FNO
0.00142
0.00122
0.00109
WDNO
0.00099
0.00098
0.00086
MWT
0.00117
0.00131
0.001 19
OK-Transfomner
0.00156
0.00123
Transolver
0.00243
0.00247
0.00176
DPOT-S-FT
0.00159
0.00106
0.00093
DPOT-L-FT
0.00182
0.00098
0.00089
DMD
0.00411
0.00478
0.00316
U-Net
0.00257
0.00279
0.00201
CNO
0.00344
0.00319
0.00217
DeepONet
0.00289
0.00308
0.00216
Combustion
FNO
0.00307
0.00295
0.00209
WDNO
0.00615
0.00443
0.00257
0.00290
GK-Transformer
0.00423
0.00349
0.00229
Transolver
0.007
0.00495
0.00275
DPOT-S-FT
0.00230
0.00273
0.00208
DPOT-L-FT
0.002229
0.00260
0.00197
DMD
0.01526
0.00986
0.00832
25

---

Published as a conference paper at ICLR 2026
B DATASET DETAILS
B.I DATA FORMAT
Details of individual datasets are given in Table In the table, we report the number of trajectories, the number of frames, the resolution, the memory of the data, and modalities, where U, is the velocity field, p is pressure, and the luminescence intensity. The modalities of simulated combustion data include absolute pressure, chemistry heat release rate, mole fraction of CH2, CO, CO2, H2O, NHg, NHg, and OH, temperature, u, v, w, p. and velocity magnitude.
Table 7: Overview of datasets. n_traj is the number of trajectories. n_f n_frame is the number of frames.
Resolution
Modalities Simulated Real-world
Dataset
n.traj n.frame
Af (s)
Simulated
Real-world
Memory (GB)
Cylinder
92 2
3990
2.5 x 10-3
64 x
128 256
190.50
W, V.P
M,
Controlled Cylinder
96
3990
2.5 10-3
64 x 128
128 256
187.08
u. 12
u,
FSI
51
2173
10
128
128
128 128
94.73
u, 1:
1r, u.
Foil
3990
2.5 10-3
128
256
128 256
335.64
a, v.P
Combustion
30 x
2001
2.5 10
128
128
128 128
110.12
see lext
I
B.2 DATA COLLECTION
data through hardware experiments, we consider multiple different phys- ical parameters, and each dataset is set with parameters based on hardware capabilities, adjustable parameters, and physically meaningful ranges. We correspond all the physical parameters of the simulated data with the real-world data one by one, and ensure the convergence and approximation of the numerical simulation to the real-world experiments by adjusting the spatial calculation do- main and time step. The boundary conditions of fluid systems are the no-slip boundary condition on the solid/fluid interface, a uniform flow inlet condition, and a zero gradient exit condition. The configuration of combustion includes two separate inlets for air and the fuel mixture. Both inlets are defined with mass flow inlet boundary conditions, while the exit is a pressure outlet. The no-slip condition is applied to all wall surfaces. These conditions correspond to our experimental measure- ment environment. Please note that all data, including real-world and simulation data, are collected after stabilization to prevent inaccuracies caused by physical parameter conversion. The hardware settings and physical parameters for each of our datasets are in the following paragraphs.
When collecting
As for the fluid systems, we calculate the simulated data using CFD (Lilypad and Waterlily D based on the parameters of real-world data. For the combustion system, a three-dimensional. implicit, unsteady Large Eddy Simulation (LES) is employed to simulate thermoacoustic instabilities in swirl-stabilized combustor. The simulations are performed using the CFD software STAR-CCME 2022.1 (Siemens Digital Industries Software Siemens 2022). The computational framework utilizes a pressure-basee segregated solver for the coupled solution of the governing equations. Near- wall flow physics are captured using an all y+ wall treatment. For modeling turbulent combustion, the Eddy Dissipation Concept (EDC) is applied to account for finite-rate chemistry effects within the turbulent flame structure (Lourier et al., 2017). The combustion chemistry is described by a reduced methane co- co-firing, comprising 38 species and 184 elementary reactions (Sun et al. 2022). The stiff chemical kinctics are efficiently integrated using the CVODE solver.
chemical mechanism for ammonia- aummonia-methane
The computational domain features two separate inlets, defined as mass flow inlets for the air and the fuel mixture, respectively, and outlet specified as a pressure outlet. No-slip conditions are applied to all walls. A series of operating conditions is simulated to systematically examine the effects of fuel composition and equivalence ratio. The investigated matrix includes pure CH2 and NH, /CH, blends with ammonia fractions ranging from 20% to 80% by volume. For each fuel composition, the equivalence ratio varied, typically from fuel-lean fuel-rich conditions (c.g.. d= 0.75-1.3),
hutp::/guhuu..com wwt.:///aattaaaaaaaea Wit.mm/llaaaalllllyla//aaailroee
26

---

Published as a conference paper at ICLR 2026
under constant backpressure of bar. These conditions are designed to correspond directly to the experimental measurement environment, facilitating a thorough validation of the numerical model.
Cylinder. The sampling frequency is 400Hz, which means it can take 400 photos per second. The duration is 20 seconds. In order to increase the gap between two neighboring time steps and demonstrate the predictive ability of models, we down-sample these data from 8000 time steps to 4000 time steps. In addition, to reduce noise in the data, we apply time-averaged filtering (Meinhart al. 2000), with sliding window size of 10 time steps, and take the average within this sliding window. Regarding physical parameters, the diameter of the cylinder is 30mm. We change different Reynolds numbers because the fixed cylinder only has this physical quantity that can be changed. Specifically, we change the Reynolds number by altering the incoming flow velocity, with range of 1800 - 12000. They are the minimum and maximum flow velocities that the equipment can achieve. Through PIV technique, we measure the velocity (in x- and y- directions) of the flow field (Abdulmouti] (02T). We conduct experimental data collection approximately every 100 intervals within this range. However, due to data loss and poor data quality during the experimental process, we selected 92 trajectories from them.
Controlled Cylinder. Both the sampling frequency and duration are the same as those of the above dataset. Since we introduce an external control variable, we set the Reynolds number to 1781, 2625, 3562, 4406, 5343, 6281, 7125, 8062, 9000, 9843. Please note, we change the Reynold number by changing the incoming flow velocity, and the Reynold number is calculated by LU/V, where T. is the characteristic Length, U is the incoming flow velocity, and is the kinematic viscosity. For the control variable, we use the sinusoid function for forced vibration control, which uses different control frequencies, including 0.5Hz, 0.6Hz, 0.7Hz, 0.8Hz, 0.9Hz, 1.0Hz, 1.1Hz, 1.2Hz, 1.3Hz, and 1.4Hz. In the end, we removed low-quality data from 100 measurements and obtained 96 trajectories.
FSI. Due to FSI's need to capture higher frequency fluid structure coupling phenomena, we conduct the experiments with a sampling time of 500Hz and duration of 8s. We connect two cylinders in series, with the front cylinder fixed to generate the Karman vortex street, and the rear cylinder installed on a smooth rail that can move in the y-direction randomly under the fluid force receives. We change the mass ratio of the rear cylinder (18.2 and 20.8), which is achieved by 3D printing the new cylinder. In addition, we also change the Reynolds number within 3272, 3545, 3955, 4091, 4636, 5045, 5318, 6682, and 9068. For each experiment, we measure three times, starting from different times to change the initial conditions. One more thing, our damping ratio is fixed at 0.8. Based on the above parameters, we finally collect 51 trajectories.
Foil. In this experiment, our sampling frequency and duration are the same as those of the Cylinder. Differently, our foil is 3D structure. Before, we approximate the 2D fluid field by making the cylinder have a universal diameter above and below, as this can be seen as an infinitely long cylinder. But for the foil, the cross-section we 3D-print is different from the top and bottom, and we use the NACA0025 foil (Bullivant 1941, Xu et al. 2023). where the top chord length is 100mm and the minimum bottom chord length is 20mm. We choose shoot the cross- scoss-n at 50mm. The cross- section captured in this way has a 3D effect. Then, we consider the angle of attack (aoa) of the foil, which directly affects the changes in the fluid field (He et al., 202T). The values are 0, 5, 10, 15, and 20. Similarly, the Reynolds numbers are 2968, 3750, 4531, 5312, 6093, 6875, 7656, 8437, 9218, 10000, 10781, 1 1562, 12343, 13125, 13906, 14687, 15468, 16250, and 17031.
Combustion. We use another set of equipment in the combustion experiment. We measure the luminescence intensity of combustion through flame CL imaging technique (Alviso et al. 2017). The sampling frequency is 4000Hz and the duration is 1s. The setting is the combustion of NHg and CH, mixed with air in different proportions. The ratios of CH, are 100%, 80%, 60%, 40%, and 20%, while the ratios of NH₃ corresponding 0, 20%, 40%, 60%, and 80%. The equivalence ratios are 0.75, 0.85, 0.9, 1, 1.05, 1.1, 1.2, 1.25, and 1.3.
B.3 DETAILS OF DATA MEASUREMENT
Circulating Water Tunnel. All fluid experiments were performed in a closed-loop Circulating Wa- ter Tunnel (CWT), shown in Figure 9 The facility, with overall dimensions of 4.85 m (L) 1.65 m (W) x 1.72 m (H), comprises the primary flow sections, a circulating pump coupled with a variable speed drive, and interconnecting piping. The main test section is constructed from high-
27

---

Published as a conference paper at ICLR 2026
(a)
(b)
Laser Shooting Surtic
Camera
Figure 9: Schematic illustration of the Circulating Water Tunnel platform and the PIV measurement system installation. (a) A schematic of the water tunnel. (b) Experiment equipment.
(a)
(b)
Clrestar Magset Steel Plate
Lead Call
Micro
Jasi Rosiet
Cylinder
Cylluder
Figure 10: (a) A schematic the FSI experimental set-up. (b) The experimental setup with free- vibration response of a cylinder in an oscillating cylinder wake.
transparency, highly Dbrasion-resistane acrylic, providing superior optical access for flow visualiza- tion and laser-based diagnostics. This section features internal working dimensions of 1.00 m (L) 0.30 m (W) 0.30 m (H). key characteristic of the facility is its exceptional flow quality; within the operational velocity range of 0.01 m/s to 0.5 m/s, the free-stream turbulence intensity in the test section is maintained below 1%. This highly quiescent and stable flow environment requisite for conducting nigh-precision measurements utilizing high-speed camera systems, laser diagnostics (e.g., PIV), and sensitive force or pressure transducers.
Cylinder. For the static cylinder experiments. cylinder with outer diameter of D 30 mm was rigidly fixed along the central axis the CWT test section. The structure was fabricated from high- rigidity, lightweight photopolymer resin via 3D printing. To suppress three-dimensional end effects and approximate 2D flow conditions, the gap between the cylinder's ss lower end and the glass floor of the section was precisely controlled mm. The primary experimental parameter was the Reynolds number (Re). By continuously adjust- ing the inflow velocity, Re range of 1800-12000 was investigated, corresponding to the minimum and maximum stable flow velocities of the facility. For PIV measurements, the horizontal laser sheet was positioned at a height of 150 mm from the test section floor. This location corresponds to the mid-span plane of both the test section and the high-aspect-ratio cylinder, capturing the most representative two-dimensional flow structures.
mm, resulting an immersion depth of / 298
Controlled Cylinder. For the controlled cylinder experiments, the structure was integrated into three-axis (XYZ) servo platform for active control(see Figure 2c This platform was configured to execute cite-ddgrre-ooffreeddoo (1-DOF) transverse (Y-axis) motion, while the X-axis (stream- wise) and Z-axis (vertical) remained rigidly fixed. The cylinder (D = 30 mm, = 298 mm, h 2 mm) precisely driven to perform forced harmonic oscillations following the trajectory yt) = A sin((2 with the amplitude A fixed at 15 mm (A/D=0.5). This active control system, featuring positioning accuracy of 0.1 mm and control frequency of 100 Hz, ensured high-fidelity
28

---

Published as a conference paper at ICLR 2026
reproduction of the motion profile. Under this configuration, the wake of the oscillating cylinder was systematically investigated at ten discrete Reynolds numbers (Re=1781, 2625, 3562. 4406, 5343, 6281, 7125, 8062, 9000, and 9843). These Re values were achieved by varying the inflow velocity while keeping the non-dimensional amplitude (A/D) constant. PIV measurements were likewise conducted at the mid-span plane (z 150 mm).
FSL. The FSI experiments were conducted utilizing a tandem-cylinder arrangement. Both the up- stream and downstream cylinders were fabricated from high-rigidity, lightweight photopolymer resin via 3D printing, each with an outer diameter of D = 30 mm. To suppress three-dimensional end effects, the gap between the lower end of the cylinders and the glass floor was held constant at h 2 mm, ensuring an immersion depth of H = 298 mm. The central axes of both cylinders were aligned with the longitudinal centerline of the test section, and the streamwise (center-to-cented spacing was fixed at x = 4D. In this configuration, the upstream cylinder was rigidly fixed to serve as a static disturbance body. The downstream cylinder was designed as a high-precision single- orgree-of-freedoon (I-DOF) vibrating system. Its upper end was connected via a support plate to an air bearing, powered by a 0.5 MPa air supply. This structure strictly constrained the cylinder's 's mo- tion to the transverse (cross- flow) direction while providing mear-frictionles, support. The dynamic characteristics (mass, stiffness, and damping) of the vibrating system were precisely controllable: (1) Stiffness: The restoring force was provided by two linear springs mounted symmetrically. The stiffness coefficients were pre-calibrated to ensure the system's 's natural frequency in air remained constant 0.6 Hz, irrespective of the mass configuration. (2) Mass: The total oscillating mass (m) was controlled using loading device (by adding or removing lead beads). The displaced fluid mass was m 210.6 g, defining the non-dimensional mass ratio as m, m/m (3) Damping: Structural damping was managed using non-contact eddy current damping mechanism. This appa- ratus (see Figure T0p) utilizes a micro-actuaton stage (resolution 0.01 mm) to adjust the gap between permanent magnets and a copper plate, thereby enabling precise tuning of the damping level. The actual damping ratio for each configuration was calibrated via free-decay tests conducted in air.
Foil. For the foil experiments, a chree-dimensional tapered hydrofoil model based on the NACA0025
profile was utilized. The model was fabricated from high-rigidity. lightweight photopolymer resin via 3D printing. It featured span of 298 mm, root chord of 100 mm (at the top), and a tip chord of 20 mm (at the bottom). The model was rigidly fixed in the center of the CWT test sec- tion (see Figure 2ap. The primary experimental parameters were the Angle of Attack (a) and the Reynolds number (Re). The angle of attack was systematically varied across five conditions: o==0 09, 5-, 10 15,, and 20 For each angle of attack, 19 discrete Reynolds numbers were investi- gated by adjusting the inflow velocity, covering a range of Re=2968-1100 1703168 PIV measurements were likewise conducted at the plane of z = 50 mm.
Particle Image Velocimetry (PIV) System. Flow visualization and quantitative mcasurements were performed using Time-Resolved Particle Image Velocimetry (TR-PIV) system. The flow was seeded with hollow glass spheres (MV-H0520, China) with a nominal diameter of 10 ym and a density of 1.05 g/cm", closely matching the density of water to ensure high flow-following fidelity. The illumination unit was a 10W continuous-wav (CW) laser (SM-SEMM--532mm- 10W, China), which emitted a horizontal laser sheet (approx. 2 mm thick) into the measurement area from the side of the CWT after being shaped by an optical system. The image acquisition unit was a high- speed CMOS camera (Photron, FASTCAM Mini UX50, Japan) positioned beneath the tunnel floor, capturing images at a resolution of 1280 X 1024 pixels. The sampling frequency was set to 500 fps for the FSI experiment and 400 fps for the Cylinder, Controlled Cylinder, and Foil experiments. For the FSI experiment, acquired image sequences were processed using the commercial software MicroVee 3.6.5 [ Velocity fields were calculated using a multi-pass iterative cross- correlaton al- gorithm, with an initial interrogation area (IA) of 32 x 32 pixels and a final IA of 16 x 16 pixels, both with 50% overlap. Image sequences from the Cylinder, Controlled Cylinder, and Foil data were processed using the open-source software PIVIab 3.10 Multipass FFT window deformation algorithm. This iterative procedure involved
3l Velocity fields were computed using
initial pass (Pass I) with 32 x 32 pixel IA (50% overlap), followed by a final pass (Pass 2) with a 24 x 24 pixel IA (50% overlap). A 3-point Gaussian fitting algorithm was employed for sub-pixel displace- ment estimation. Following computation, raw vector fields underwent rigorous post post-processing
https://piv.com.sgg/
29

---

Published as a conference paper at ICLR 2026
routine to reject spurious vectors. This process included global rectangular velocity limit deter- mined via histogram analysis, followed sequentially by a standard deviation filter (threshold=8) and local median filter (threshold=3). Finally, any voids created by vector removal were filled using linear interpolation. Given the critical importance of velocity calibration in PIV post-processing, performed a verification step beyond standard spatial calibration. Specifically; we compared the PIV-derived velocity in the free-stream region (areas undisturbed by the cylinder or foil) with the pre-calibrated bulk velocity of the cireulating water tunnel to further confirm the accuracy of the flow measurements.
To facilitate validation and reproducibility, we have released subset of the raw particle image sequences and calibration files in the database) along with the specific parameters for the Multipass FFT window deformation and multi-pass iterative cross-correlation algorithms. Furthermore, the raw velocity vector field data derived from the software processing has also been made available.
Combustion. The combustion dynamics are characterized primarily through high-speed OH* chemiluminescence imaging, which serves the principal diagnostic technique for capturing flame structure and heat release distribution. high-speed camera (Photron Mini AX 200) coupled with an image intensifier (EyeiTS-D-HHB-- records the OH chemiluninescence signals through UV lens (Nikon PF10545MF-UVV and a 310 = 10 nm bandpass filter, with the system capturing images at 2000 frames per second across 70 x 90 mm? field of view. The instantancous OH* intensity provides a direct indicator of local heat release rate, enabling detailed analysis of flame stabilization mechanisms and transient flame motions under varying operating conditions.
B.4 COMPARISON BETWEEN SIMULATED AND REAL-WORLD DATA
In this section, we quantify the discrepancy between simulated and real-world data. For the fluid datasets, we use MVPE to measure the gap and visualize MVPs. Specifically, we evaluate MVPE over the full temporal window of each trajectory, with the results summarized in Table g The corresponding MVP visualizations are shown in Figure IT The results reveal that the time-average velocity is close to the real-world data.
As for the Cylinder dataset, also calculate the 200-step MVPE of simulated data and predictions from the baselines. Specifically, we compute MVPE over a 200-step temporal window. Model pre- dictions obtained by running rounds of autoregression to predict 200 steps. For the simulated and real data, we separately compute the norm of the velocity field, select for each the time point at which the norm attains its minimum the starting point, and then extract the subsequent 200 time steps as the time window. The resulting MVPE for the simulated data is 0.05650, and Table [B.4] reports the MVPE of each model under two training settings: Simulated training and Real-world finetuning. We observe that the MVPE of models pretrained on simulated data can be either larger or smaller than the MVPE of the simulated data itself. Smaller MVPE of models may be because the models input the same initial state as real-world data, while the alignment of simulated data and real-world data is not as strict as the models'. However, after finetuning on real-world data, all mod- els achieve lower MVPE than the simulated data, highlighting both the importance of real-world measurements and the potential of deep learning models.
Table 8: MVPE of simulated data (full trajectory).
Cylinder Controlled Cylinder
FSI
Foil
0.08718
0.06985
0.1 1440
0.08653
Table 9: MVPE of baselines (200 steps).
FND
WDND
UNe
DeepONer
MWT
CNO
GK- GK-Trunsfemmer
Transolver
DPOT-SPT
DPOT-L-FT
DMD
Simulated training
0.07418
0.06330
0.03137
0.05295
0.03547
0.06409 0,004888
0.05431
0.08466
0.03378
0.03944
Real world finetuning
0,01317
002181
0.01408
0.01887
0.01821
0.01485
0.02574
0.01363
0.02500
0.10668
As for the combustion data, we implement frequency-domain analysis, focusing on OH radical mole fraction dynamics. The experimental instantaneous OH chemiluninescence signal is provided by
30

---

Published as a conference paper at ICLR 2026
Diagnostle
1.00
hrod
0.75
3.0
6.50
8 3.00
8.00
0.00
+ Tenge
2 6.00
-159
-450
-0.30
-6.0
+8.5
-1.0
-300
+1.00
-1.00
0.2
03
10
20
1/9
Probe Based Diagnostic
23
1.00
1.20
Tegg.
3.75
6.75
6.75
333
0:3
0:25
6.2
-125
-425
-0.25
-1.2
-3.75
-675
-6.75
+6.75
Bryer
Tagel
Diagnostic
1.00
1.00
1.30
hrd
0.75
3.50
8.50
0.50
6.50
3.00
6.00
0.00
-+ Terge
22 6.70
-250
-650
-6.00
-6.00
-100
110
3.00
1.00
6.5
0.2
0.5
15
0.
16
15
680
Figure 11: Mean velocity profile simulated u on Cylinder.
measurement, while the equivalent transient signal extracted from the LES results by performing spatial integration of the instantaneous OH concentration field. Both time series are detrended to remove the DC component, and transformed into the frequency domain using the Fast Fourier Transform (FFT), showing the dominant frequencies of the combustion instability. The visualization result is provided in Figure [2]
C
CODE FRAMEWORK
C.1
SCRIPTS
The training script's form is follows:
31

---

Published as a conference paper at ICLR 2026
100
(a)
(b)
Figure 12: Frequency analysis of the Combustion dataset. (a) Real-world data. (b) Simulated data.
python train.py \
--config IG_PATH
-trrrrinnne daldddatttt_yeee type TRAIN DATA TYPE
--is_ finetune IS FINETUNE
where CONFTG_PATH is the path of the baseline's config file, TRATN_DATA_TYPE. is real or numerical, and INEEINEE is True or False. TRAINDAATALTPPP=ummeel and IS.FINETUNE-Faise corresponds to simulated training. TRAIN.DATA.TYPE and IS.FINETUNE-FaLs corresponds to real-world training, and IS S..FINETUNE=Tzue corresponds to real-world finetuning.
and
The evaluation script is
python eval.py \
--config CONEIG PATH \
--chhhcpinntppeteecc CHECKPOINT PATH
where CONFIG_PATH the path of the baseline's config file, and CHECKPOINT_PATH is the path of the checkpoint.
C.2 DATA PRE-, POST-PROCESSING, OPTIMIZER AND SCHEDULER
Our code framework supports three types of data pre- and post-processing: none (no normaliza- tion), Gaussian (standardization maximum absolute value).
and unit variance), and range (normalization by the
Besides, code framework adopts the Adam optimizer (Kingma & Ba 2014), and the learning rate scheduler supports two strategies: the step decay scheduler and cosine annealing scheduler Loshchilov & Hutter 2016).
C.3 DATASET MODULE
All datasets are loaded using unified module RealDataset, whose parameters are detailed be- low. To support evaluations under both in-domain and oft-oo-domain (OOD) settings, we provide flexible 'test_mode' option in the 'RealDataset' module. Users specify 'test_mode' as 'all', in dist', 'out dist', 'seen', or "unseen" to control which parameter regimes are included in the test set. Here, 'in dist' refers to parameters in the distribution of the training set, while 'out dist' refers
32

---

Published as a conference paper at ICLR 2026
to the cut-of-distributtoon parameters. The 'seen' mode evaluates model performance parameter regimes observed during training, while the 'unseen' mode enables OOD testing by selecting data from parameter regimes that do not appear in the training set. This design allows users conve- niently assess the model's generalization capability.
class RealDataset (Dataset)
def i
self, dataset__nmm,
dataset_root,
dataset_type,
mode, mask_prob,
in_step,
out_step,
K_autoregressive,
interval,
train_ratio,
split_numerical,
trunk_length,
noise_scale,
L_sin_out_ddisrrioutt
n_sim_frame,
sub_s_real-1,
sub_s_numerical-l,
noise_type=
size=,, optical_sigma-1.0
()
dataset_name: name of the dataset
dataset_root: root path of the dataset dataset_type: real numerical mode: train | val test test_mode: all in dist out_dist | seen | unseen mask_prob: probaoility of masking the unmeasured modalities, only for numerical.
datasets
in_step: number of steps for input out_step! number of steps for output N_ seauoregrrsssvee: number interval: interval of the sliding window train_ratio: ratio of training data aplit_numerical: split numerical data into training, validation and test data not trunk_length: length of the trunk for splitting simulation into training and
autoregressive times
test data
noise_scale: scale: scale of the noise added to namerical data sismmiinddiii distrioutionn number of simulations for in-distribution test
__sin__our_dissribuiiio number of simulations for out- distribution test
n_sin_frame: number of frames in each simulation sub_s_real; spatial sub-sampling factor for real data jub_s_numericall spatial sub-sampling factor for numerical data
noise_type type of noise, gaussian I poisson | optical pptical.kernel_iee size of the kernel for optical noise optical_sigma: standard deviation of the kernel for optical noise *..
33

---

Published as a conference paper at ICLR 2026
C.4 MODEL MODULE
We define a unified Model class, from which all baselines are derived. This class provides the following functions.
class Moael Mon.Me) def iniiiiittt (self, ...):
Initialize the model with given parameters
def (self, x): Forward pass of the model
def train_loss (self, input, target): Compute the training less for a batch
def Load_ checkpent (self, bheckpoint_path. device); Load checkpoints from the path
D EXPERIMENT DETAILS
D.1 MASK-TRAINE STRATEGY
Because the simulated data contain modalities than the real-world data, design specific strategies to utilize the additional modalities. Specifically, we employ the mask- trainin strategy. During simulated training, Consequently, the trained model can be directly applied to real-world test data by simply concate- nating zero vectors place of the unobserved modalities. With training on the additional modalities, the model can learn more information about the dynamics.
randomly mask the additional modalities with fixed probability.
D.2 COMBUSTION SURROGATE MODEL TRAINING
For the Combustion dataset, a particular challenge is that the observed modalities in real-world data directly available in the simulated data. To address this, we train additional surrogate model whose input is the simulated modalities and output is the corresponding real- world modalities each time step. The surrogate model is trained on paired simulated and real-world data from the same periods. We choose the U-Net the surrogate model due to its outstanding performance. During simulated training on this dataset, the simulated data are concatenated with the output of the surrogate model.
E BASELINE MODELS
E. DMD
The Dynamic Mode Decomposition (DMD) traditional method that extracts spatiotemporal coherent structures from time-series flow field data through a data-driven matrix decomposition approach. DMD processes volumetric flow data by reshaping the spatiotemporal field into snapshot matrices X, (first n 1 snapshots) and X, (last - snapshots), where each column represents spatial state given time step. It performs singular value decomposition (SVD) on X, to obtain X, UEVT, with optional rank truncation for dimensionality reduction. The low-dimensional UT U,,VV VE-1, followed by eigenvalue decomposition to
evolution operator is computed as extract eigenvalues X; and eigenvectors W. DMD modes are constructed via V = V where each mode t/: represents spatial coherent structure. Modal amplitudes b are determined by leaat-squares fitting of the initial condition X onto the DMD modes. Modes are sorted in descending order by amplitude magnitude |b). and the top Rowods 10 modes with the largest amplitudes are selected and retained for prediction. Future states are predicted using the linear superposition formula x() Σ; bath: exp(A;-1)
34

---

Published as a conference paper at ICLR 2026
E.2 U-NET
The 3D U-Net processes volumetric data through an encoder-decoder structure with skip connec- tions. Its encoder module successive blocks of two 3D convolutional layers (kernel K, padding P. ReLU activation) followed by max pooling (stride/kernel S), progressively halving spatiotempo- ral resolution while doubling channel counts. A bottleneck of two convolutional layers bridges the decoder. which employs transposed convolutions (stride/kernel S) for upsampling, feature concate- nation via skip connections, and two convolutional layers per block. The output layer uses x x convolution to map features to target fluid state variables, enabling effective spatiotemporal feature extraction for medical imaging and fluid dynamics applications. In our configuration, the tunable parameters include the number of encoder-decoder layers, the batch size, the learning rate, and the total number of update iterations.
E.3 CNO
The Convolutional Neural Operator (CNO) is deep learning framework designed to learn map- pings between function spaces through hierarchical convolutional architectures. It extends operator learning by incorporating spectrally motivated filters, which ensure stable frequency representa- tions across scales. The network follows an encoder-decoder structure with skip connections and bottleneck composed of residual blocks. The lifting and projection layers map input fields into a higher-dimensional latent space and back to the output space. Each CNO block consists of a 3D) convolution, optional batch normalization, and filtered activation, where upsampling and down- sampling are handled inside the activation functions. In our configuration, the tunable parameters of the CNO model include the number of encoder-decoder layers, the number of residual blocks per level and in the bottleneck, the channel multiplier that controls feature width, as well as filter-related hyperparameters.
E.4 DEEPONET
Deep Operator Network (DeepONet) is a deep learning network designed to approximate a nonlinear continuous operator, which models complex relationships between functions via two sub-networks. The branch net processes the input function to extract the input features, and the trunk net encodes the location coordinates for the output space. The outputs the two sub-networks are combined by dot product generate the final results. In our baseline, the code framework of DeepONet uses 3D Convolutional Neural Network (CNN) in the branch net and Fully Connected Layers in the trunk net. In configuration, the tunable parameters of the DeepONet model include the dropout rate and the feature dimension (p) of the outputs from both the trunk net and branch net.
E.5 FNO
The Fourier Neural Operator (FNO) is deep learning framework designed to learn mappings be- tween infinite-dimensional function spaces. It parameterizes the integral kernel in the Fourier do- main, where inputs are processed by successive Fourier layers that apply linear operations to able efficient convolution. This design naturally supports zero-shot super-resolution. After several Fourier layers, there are two linear layers. In our configuration, the tunable parameters of the FNO model include the truncation level of frequencies (modesl, modes2, modes3), the number of Fourier layers, and the width of the linear layer.
E.6 WDNO
Wavelet Diffusion Neural Operators (WDNOs) aim to model physical systems with generative mod- els. The diffusion model a generative framework that learns to transform random noise into data samples by reversing a gradual noising process (Ho et al. 2020). WDNOs combine the wavelet transform with diffusion models to capture the abrupt changes complex physical systems, while leveraging the diffusion models' strong ability to model high-dimensional distributions. In our code framework, we take the U-Net as the denoising network. In the code framework of WDNO, the tunable parameters include the U-Net's base feature dimension, the U-Net's channel multipliers for each stage, the schedule of diffusion models, the type of wavelet basis, and the padding mode of wavelet transform.
35

---

Published as a conference paper at ICLR 2026
E.7 MWT
The Multiresolution Wavelet Transform Neural Operator Guupt et al. 2021) learns mappings be- tween function spaces by combining wavelet-based multiseale decomposition with sparse Fourier transforms. At each resolution level, MWT applies wavelet filters to extract coarse and fine-scale features, processes them via frequency- and space-domain kernels, and reconstructs the output using inverse transforms. This design enables efficient modeling of both global structures and localized variations. In our setup, we use a wavelet filter size of k 3 and a spectral width a = 5 to bal- ance resolution and frequency modeling. The channel multiplier is set to c = 4 to increase latent capacity, and we stack π = 4 multiwavelet operator blocks. The coarsest level is = O, allow- ing full-resolution output. We use Legendre polynomials to construct the wavelet basis due to their favorable numerical properties.
E.8 OK-TRANSFORMER
The Galerkin Transformer is attention-base operator learning framework that removes the soft- max in self-attention and admits a projection- based interpretation. In our baseline, we follow the original Galerkin Transformer architecture, while modifying its output regressor: instead of simple Fourier transform, we employ a Fourier Neural Operator (FNO) layer. This replacement enhances the model's ability to capture volumetric features and recover fine-scale structures, while the rest of the GT design remains consistent with the original implementation. The tunable parameters of this model include the number of layers, the number of attention heads, the hidden size, and the parameters of the FNO regressor.
E.9 TRANSOLVER
The Transformer-based Solver (Transolver) is a neural operator framework designed to solve partial differential equations (PDEs) by leveraging the global receptive field of transformer architectures. Instead of relying on localized convolutional kernels or spectral truncations, Transolver employs multi-head self-attention to capture long-range dependencies in the solution space, making it par- ticularly effective for modeling nonlocal interactions. The architecture typically consists of stacked transformer encoder blocks, each containing self-attention, feed-forward layers, and residual con- nections, combined with positional encodings to handle spatial coordinates. To enhance numerical stability, normalization layers and gating mechanisms are integrated across layers. In configu- ration, the tunable parameters of the Transolver model include the number of transformer encoder layers, the number of attention heads, the hidden dimension of each layer, the choice of positional encoding scheme, and the dropout rate used for regularization.
E.10 DPOT
The Denoising Pre-training Operator Transformer (DPOT) is a scalable neural operator framework for learning foundation model for spatiotemporal PDE mappings. It combines an auto-regressive denoising objective with a Fourier-attention Transformer backbone for robust, transferable repre- sentations across diverse PDE systems. During pre-training, DPOT predicts the next timestep from the previous T frames with injected Gaussian noise to mitigate train-test mismatch and enhance long-horizon stability. The architecture includes a patch embedding layer with positional encoding, a temporal aggregation layer using Fourier features to compress T input frames, and stacked multi- head Fourier attention blocks that apply learnable nonlinear transformations in the frequency domain for efficient global mixing. In our benchmark, we use the official small (30M) and large (509M) pre- trained DPOT models. To adapt to varying dataset channels while reusing pretrained input/output projections (designed for 4 channels). we pad inputs/outputs with all-ones for datasets with fewer than 4 channels, following the paper; for more than 4 channels, we reinitialize the respective pro- jection layer. Original training is at 128x 128 resolution, so we preprocess all data to 128 128 via Fourier-domain serr-padding/trnnnee dictions back to original resolution identically during training and inference. In our configuration we load the pretrained DPOT models and leave model structural hyperparameters such as the num- ber of Fourier attention layers, attention dimension, MLP dimension, or number of heads they are and only tune the optimization hyperparameters for finetuning, such as learning rate and total learning steps, to preserve the integrity of the pretrained model.
resize (as in the paper) before input, and postprocess pre-
36

---

Published as a conference paper at ICLR 2026
F LIMITATIONS AND FUTURE WORKS
While RealPDEBench provides the benchmark with paired real-world and simulated datasets, its scope is currently limited. Extending the benchmark other domains such electromagnetics. structural mechanics, or aerodynamics would broaden its applicability. In addition, although we provide multiple physics-orienter oriented metrics, no dedicated metrics have been introduced specifically for the Combustion system, which could further strengthen physics fidelity assessments. Another limitation is that the current benchmark does not systematically explore strong out-o--distribution regimes. Future work will expand RealPDEBench to cover additional physical scenarios, introduce other metrics, and design out-of-distribution tasks. We also maintain a long-term development view towards more data and models in the future, thereby enabling more comprchensive evaluation of scientific ML in real-world contexts.
G VISUALIZATION
In this section, we visualize one sample's results of all baselines on all datasets (real-world data finetuning setting), including ground truth and predictions.
H THE USE LARGE LANGUAGE MODELS (LLMS)
In this paper, employ LLMs general-purpose assist tools for text refinement and language polishing. All core research ideas, datasets, and scientific conclusions presented in this paper are original contributions.
37

---

Published as a conference paper at ICLR 2026
Ground Truth
Indelar 1:13
Inecian 1-18
DMD
wolria 1:11
meiiian 1:18
CNO
hodd 1:
Incc
Pedaas 1:11
DeepONet
FNO
Profccon 160
hobicte 1013
Frotcder 14
WDNO
MWT
Graarrssfrrm
Indiiian
Pac/din 6-33
Prdicor 1:18
Transolver
Trebobe
Frafettn
Trafiden 1:15
DPOT-S-FT
DPOT-L-FT
Figure 13: Visualization of results (u) on Cylinder dataset.
38

---

Published as a conference paper at ICLR 2026
Swend Fudn
Phick witt
Ground
DMD
PodeSon
U-Net
CNO
DeepONet
FNO
1e11
WDNO
MWT
GrTTransformrr
Transolver
Pach 1nd
1611
DPOT-S-FT
Indttou
mnc.cin 145
Oon 1413
DPOT-L-FT
Figure 14: Visualization of results (v) on Cylinder dataset.
39

---

Published as a conference paper at ICLR 2026
Ground Truth
1413
Peddici
U-Net
CNO
Peclda Ltd
DeepONet
Poddoos
FNO
WONO
Pir 1xt
hoiddio
Frw
Pedidios Er
holtouod
GK-Transformer
Transolver
DPOT-S-FT
DPOT-L-FT
Figure 15: Visualization of results (u) on Controlled Cylinder dataset.
40

---

Published as a conference paper at ICLR 2026
ureed RuB
uncind Dr
Ureed RUS 00.23
umard
coe
Ground Truth
DMD
cooe
U-Net
ccs
CNO
DeepONet
FNO
140
Ecoo
WDNO
Lee
GK- Tr-rrrssformer
Proditin
Fradctte
Transolver
vnc.tenn
DPOT-S-FT
Prefictr
Iraccden
DPOT-L-FT
Figure 16: Visualization of results (v) on Controlled Cylinder dataset.
41

---

Published as a conference paper at ICLR 2026
Ground Truth
Peoder brt
DMD
U-Net
CNO
DeepONet
FNO
WDNO
MWT
GK-Transformer
Nate
Transolver
Aacua 1.a
DPOT-S-FT
DPOT-L-FT
Figure 17: Visualization of results (u) on FSI dataset.
42

---

Published as a conference paper at ICLR 2026
Ground Truth
DMD
U-Net
CNO
Rubiion
DeepONet
FNO
WDNO
Paeste Fea
MWT
Recdus Ci
Iroooo Sioa
GK- TK-rrnsffrrerr
Praccsn Bice
Transolver
DPOT-S-FT
DPOT-L-FT
Figure 18: Visualization of results (v) on FSI dataset.
43

---

Published as a conference paper at ICLR 2026
Ground Truth
DMD
U-Net
CNO
hobdoos
Frodcte
DeepONet
Nodiito
FNO
M1
1413
WDNO
MWT
Treccde
Frodcte 5:13
Nedideen 1-11
GK GrrTrassformer
Transolver
1011
DPOT-S-FT
M
DPOT-L-FT
Figure 19; Visualization of results (1) on Foil dataset.
44

---

Published as a conference paper at ICLR 2026
Orcen
Emcel Trat
Omoed Park 9x11
Oreed Srlb 1013
Ground
Tasto
U-Net
CNO
madie 1-17
DeepONet
FNO
Prodatin
Nobiii 3x11
WDNO
Ieddoos
Peddoor
maddden
MWT
Prndution
Transolver
meadii 1.12
DPOT-S-FT
Proddos
PAaCIA 2012
Peccio 111
DPOT-L-FT
Figure 20: Visualization of results Foil dataset.
45

---

Published as a conference paper at ICLR 2026
CRLACTUU SHLE
Ground Truth
DMD
U-Net
4t
CNO
Pecoton Fra
DeepONet
FNO
MWT
GK- Transformer
Transolver
un
DPOT-S-FT
DPOT-L-FT
Figure 21: Visualization of results (7) on Combustion dataset.
46


Reviews:
[{"review_id": "F57t6Rab9l", "reviewer": "Reviewer_6AF9", "summary": "This paper proposes a benchmark that pairs real measurements with matched numerical simulations across five scenarios (cylinder wake, controlled cylinder, FSI, foil, and combustion; governing equations span Navier-Stokes, coupled FSI, and reactive Navier-Stokes with species transport). The benchmark defines three training regimes (train on simulation, train on real, pretrain on simulation then finetune on real), includes a mix of pixel and physics flavored metrics, and evaluates a set of neural PDE baselines including a pretrained foundation model. The headline empirical messages are: there is a nontrivial gap between simulation and laboratory data; pretraining on simulation generally helps downstream on real; and the codebase makes it straightforward to add models or datasets.\n\nI think this is timely and potentially useful. If we are serious about sim2real for scientific ML, we need carefully curated real data and a shared protocol. However, the current paper mixes benchmarking with a sim2real narrative in ways that are a bit loose, the experimental details are too thin for others to trust or extend the datasets, and some of the metrics and literature framing are not well aligned with fluid mechanics and combustion practice.\n\nAm open to potentially increasing my score if the authors narrow the claims, remove the robotics sim2real framing argument and instead point to real sim2real problems in fluid dynamics, add significantly more context to place this in the existing fluid dynamics literature, and substantially improve the experimental documentation and physics grounded evaluation.", "questions": "[Several questions discussed above]", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The benchmark collects paired real and simulated trajectories for several nontrivial systems instead of yet another synthetic-only PDE suite. This is likely to be useful for the community.\n- The split of training regimes (simulation only, real only, pretrain on simulation then finetune on real) is useful, and the pretraining result is consistent with what many of us have seen in practice.\n- The code appears modular enough to add a new dataset or baseline without painful surgery, and using a single file format lowers friction for adoption.\n- Including both data oriented metrics and physics oriented diagnostics is better than reporting only RMSE. The autoregressive evaluation option is also a good idea.\n- The combustion scenario is ambitious and, if documented properly, could become a valuable stress test beyond the usual laminar toy problems.\n- The baseline measurements for their benchmark are very extensive.", "weaknesses": "Major concerns:\n\n- The documentation about the experiment is unacceptably thin in its current form. If the experimental data was created or modified from another source, you need to cite it. If someone else created the dataset for you, you need them to write documentation for it. The current documentation on experimental data generation (which, of course, can be included in the appendix) is simply unacceptable for publication, especially for a paper which is supposed to be about this very dataset.\n- The paper motivates the sim2real gap by citing mostly work from robotics, which I found very strange, almost as if the authors are guessing there is a sim2real gap in fluids, without actually surveying the literature. Robotics has a much different sim2real gap than turbulence research does. In fluids, the sources of discrepancy, data acquisition, and noise models can be quite different. Please ground the narrative in fluids and combustion references. If you are addressing the sim2real gap in fluids, you must speak from the context of the fluids community, and discuss the ways in which the fluids community has quantified this gap.\n- Please state clearly whether you will release raw data (e.g., the PIV frames), calibration files, and the full processing scripts, not just the final HDF5 arrays, so that it can be checked by others. If raw data cannot be released, say so and justify it. Benchmarks live or die by their data hygiene, and biases in a benchmark can leak into biases in the community's preferred models.\n- Several figures quantify differences using frequency or Fourier errors over image-like arrays. This is admittedly a start, but it is not a physics grounded measure of mismatch between experiment and simulation, and certainly not something people in the fluids community would actually use as a robust measure of discrepancies between simulation and real data. In fact, for several real-world experimental problems, there is a difference here that is simply due to the nature of the real-world experiment, yet the simulation and experiment can actually have no discrepancy. This is because you would only care about some summary statistic, and not care about some wave mode that you know does not affect your statistic. Yet, your metric would completely miss this. I recommend reviewing and citing the fluids literature and how people measure discrepancies between simulations and real data. Note that this is a problem people have studied for literally decades in the fluids community. These additions would make your claims about a \"gap\" more convincing.\n- Simulation contains modalities that are not observed in the lab, and the current strategy randomly masks channels and adds noise. That is a start, but it does not reflect the actual sensor physics. Please consider sensor specific degradations (camera noise models, optical blur, saturation, PIV algorithmic artifacts) and state explicitly which channels are used for training and which are hidden. It would also help to define tasks that force parity, for example training all models only on the modalities that the lab provides.\n- The paper claims to be the first benchmark that integrates real-world measurements with paired numerical simulations across complex physical systems. Within fluids, this is far from true; as just one demonstration, \"ERCOFTAC\" has hosted combined experimental and numerical reference cases since 1995 across a wide range of flows. Please narrow the novelty claim to something that accurately represents existing datasets and benchmarks, and perhaps cite existing databases.\n\n\nAdditional comments and suggestions\n\n- The update ratio metric is interesting, but it conflates pretraining data scale with optimization effects.\n- Please make the train, validation, and test splits explicit at the parameter level so that generalization across Reynolds number, control frequency, mass ratio, or equivalence ratio is clear. I consider those to be most interesting axes. And if you already do this, show it more prominently.\n- The autoregressive evaluation stops very early. If you want to make claims about stability, show longer horizons and add probe based diagnostics, not only field RMSE.\n- The baselines are modern ML models, which is fine for a benchmark, but the story would be stronger if you add one or two domain baselines for each scenario (for example a simple reduced order model, or even a physics based filter) so readers have a calibration point.\n- Throughout the paper there are small terminology issues. I would prefer \"numerical error\" over \"computational error\" (computational error sounds like a code bug). Be precise about whether errors arise from discretization, closure modeling, boundary conditions, or measurement.\n- Not a criticism, but I think it would be better to rename the benchmark. \"RealBench\" seems far too broad and will clash with many domains. Something like \"RealPDEBench\" or \"RealFlowBench\" might be more appropriate.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "8245mXvN5V", "reviewer": "Reviewer_kvd1", "summary": "The paper describes a new physics-oriented benchmark and dataset that focuses on the gap between simulated and real experimental data. \n\nThe paper presents $5$ applications and provides metrics for evaluation, alongside classical and recent benchmark methods. \n\nThe description of the \"TASK DEFINITION\" should be improved and linked to the experiments, for example, in Table.1. \n\nThe dataset is not well described; a summary table is necessary (how many samples, what is the size of each sample, what is the total real time duration, step size, dimension of the problem 1d, 2d, 3d, ...). \n\nThe authors provide code to run the experiment, some sample data (real and simulated), but do not provide the script to generate the numerical data.", "questions": "One possible source of difference between experimental and numerical experiments is the measurement noise, but i can assume there could be a larger difference, for example, if the state is not directly measurable (pressure is not available, but only velocity).\n\nCould the author expand and position this paper in this context? What are the possible main and most critical differences between experiments and numerical simulations?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "This dataset addresses the important aspect of bridging the gap between the simulated (numerical) and the physical system. \n\nWhile is not possible to cover a large experimental setup, the authors provide $5$ taks with both numerical and physical data.\n\nThis work will therefore help in evaluating new models, even if may not cover all possible scenarios.", "weaknesses": "It is hard to say, but it is not possible to cover all possible physical experimental conditions. Nevertheless, the paper is a good contribution in the right direction. \n\nThe main point is that the numerical generation scripts are missing, therefore not possible to extend the data (at least numerical) to other scenarios. \n\nOn the experimental side, I am not able to judge if the information is sufficient. \n\nI found the task description disconnected to the actual experiments. I would encourage the authors to improve that section.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "MJSr1mbZan", "reviewer": "Reviewer_1VNT", "summary": "The paper introduces RealBench, which evaluates the generalizability of simulated-data trained ML models on real-world measurements. It is well written, and covers an obvious gap in the literature. The authors clearly put significant thought into solving this issue, and it appears that the paper is as close to reproducible as is possible. II see they even posted their code on anonymous github. I looked through their code and it has the same quality/usability as PDEBench, which has set the standard for this kind of release.", "questions": "I don't have any questions.", "limitations": "", "rating": 10, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "The paper is original, and tackles a difficult challenge of pairing real and simulated data. This a clear gap in the literature, where simulated data was the only solution before. This provides a unique insight into many of the claims made on various architectures aimed at training surrogates for PDEs. It is comprehensive, well written, and thoughtfully formulated (especially the figures). It appears to achieve the maximum possible level of reproducibility through the use of the anonymous github repo.", "weaknesses": "The only weakness is the obvious one - out of domain regimes are not covered. However, this is probably the biggest area of weakness for this area of study as a whole. The complexity of fluid dynamics makes that a separate challenge entirely (one I dont see being solved any time soon). Surrogates typically cover some precise range of reynolds numbers around a specific geometry. It is the nature of this domain.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "PZwCwlLtxZ", "reviewer": "Reviewer_YXXo", "summary": "This paper introduces a new dataset featuring paired simulation and experimental data for a variety of systems. The authors then benchmark a variety of state-of-the-art models on this data exploring the impact of simulated data on learning the real-world dynamics.", "questions": "1. Often one of the cited limitations of numerical simulation is the difficulty of simulating regimes (Reynolds/schmidt/ect) that occur commonly in real world settings. For CFD, the datasets currently contain mostly DNS data. Have you considered further comparisons between experimental data in regimes where DNS struggles and the more approximate simulations used to model them?\n2. Do any of the models evaluated compare favorably to the numerical simulation for predicting the state of fields? Is the pairing sufficiently close that this is a question it is possible to ask with this data?", "limitations": "", "rating": 10, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "This type of dataset is so valuable to the community that I would strongly recommend acceptance even if the paper itself had major issues. Almost all work in this space today uses simulation data exclusively which is a space where learned models can at best offer speed improvements. This work and future datasets like it can offer a chance to explore in which regimes we may be able to achieve improved accuracy over numerical simulation and could great inform research directions over time.\n\nThat said, the paper itself is quite strong. Some particular reasons:\n1. The presentation is excellent. It explains the problem very cleanly and describes prior work and the intended goal of this dataset.\n2. The appendix is quite extensive providing data on the models and data generation process. In general, the division between main text and appendix content feels very well planned. \n3. Experimental fluids data is extremely rare and valuable to the community for developing models that can operate independently of the standard numerical solver framework.", "weaknesses": "The benchmarking has a few issues, but I'm largely reading this as a dataset paper, so I'm not marking down for them. However, in the interest of improving the paper, I will still list them out here.\n\nMajor:\n1. The major missing component of this submission is an evaluation of how accurately the numerical simulation models the real scenario. If this is possible from the data, providing this information would drastically increase the potential impact of the paper. \n2. The model comparisons aren't very convincing due to the vastly different scales between them. It would improve the submission as a benchmarking effort to apply some form of normalization - FLOPs, parameter count, run time on fixed hardware - but I'm largely treating this as a dataset paper so not marking off for this. \n3. Similarly, one-step prediction is a useful reference for how well the models do what they're trained for, but longer rollout evaluations are more reflective of real tasks. I'd add more details on longer autoregressive rollouts if possible. \n4. Frequency domain error (in space) is generally going to be more informative when normalized. In 4.5 the text notes that a decrease of error in the high frequency is remarkable, but it's actually very much expected when comparing absolute metrics. Most fluid systems follow a polynomial decay law in spatial frequency, so the values being compared should be much smaller. It is actually concerning (for the baselines, not this submission) that this seems to be rare.\n\nMinor notes\n1. 3.1 prediction task - this is all discretized data, so it feels inaccurate to describe the task as a mapping between the continuous spaces. \n2. (paragraph beginning at 155) It would make sense to also include a canonical validation/test set of simulated data to evaluate the level of overfitting to simulation data and the overall sim2real gap. \n3. Table 4 is pretty hard to read. I don't know that this is the best way to demonstrate this data.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Oral)