Lost in Latent Space: An Empirical Study of Latent Diffusion Models for Physics Emulation
Authors: François Rozet
Ruben Ohana
Michael McCabe
Gilles Louppe
Francois Lanusse
Shirley Ho
Venue: NeurIPS 2025 poster ()
Keywords: latent
diffusion models
autoencoders
physics
emulation
compression
fluid dynamics
PDE
Primary area: machine_learning_for_sciences

Abstract: The steep computational cost of diffusion models at inference hinders their use as fast physics emulators. In the context of image and video generation, this computational drawback has been addressed by generating in the latent space of an autoencoder instead of the pixel space. In this work, we investigate whether a similar strategy can be effectively applied to the emulation of dynamical systems and at what cost. We find that the accuracy of latent-space emulation is surprisingly robust to a wide range of compression rates (up to 1000x). We also show that diffusion-based emulators are consistently more accurate than non-generative counterparts and compensate for uncertainty in their predictions with greater diversity. Finally, we cover practical design choices, spanning from architectures to optimizers, that we found critical to train latent-space emulators.

Lost in Latent Space: An Empirical Study of Latent Diffusion Models for Physics Emulation
François Rozet 1,3
Ruben Ohana 1.2
Michael McCabel
Gilles Louppe
François Lanusse 1.2.6
Shirley Ho2.2.4.5
'Polymathic AI
University of Liège *New York University Princeton University
Flatiron Institute
Universitéé Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM
Abstract
The steep computational cost of diffusion models at inference hinders their use fast physics emulators. In the context of image and video generation, this com- putational drawback has been addressed by generating in the latent space of an autoencoder instead the pixel space. In this work, we investigate whether a simi- lar strategy can be effectively applied to the emulation of dynamical systems and at what cost. We find that the accuracy of latent-space emulation surprisingly robust wide range of compression rates (up 1000x). We also show that diffusion- based emulators are consistently more accurate than non-generative counterparts and compensate for uncertainty in their predictions with greater diversity. Finally, we cover practical design choices, spanning from architectures to optimizers, that we found critical to train latent-space emulators.
Introduction
Numerical simulations of dynamical systems at the core of many scientific and engineering disciplines. Solving partial differential equations (PDEs) that describe the dynamics of physical phenomena enables, among others, weather forecasts [1, 21. predictions of solar wind and flares [3 [3-5], or control of plasma in fusion reactors [6, 7]. These simulations typically operate on fine-grained spatial and temporal grids and require significant computational resources for high-fidelity results.
To address this limitation, promising strategy to develop neural network- emulators make predictions orders of magnitude faster than traditional numerical solvers. The typical approach [8-17] is to consider the dynamics as a function f(x") = +++1 that evolves the state x' of the system and to train a neural network fo fx) to approximate that function. In the context of PDEs, this network is sometimes called a neural solver [11, 18, 19]. After training, the autoregressive application of the solver, rollout, emulates the dynamics. However, recent studies [11, 18 21] reveal that, while neural solvers demonstrate impressive accuracy for short-term prediction, errors accumulate over the course of the rollout, leading distribution shifts between training and inference. This phenomenon even more severe for stochastic or undetermined systems, where it not possible to predict the next state given the previous one(s) with certainty. Instead of modeling the uncertainty, neural solvers produce a single point estimate, usually the mean, instead of a distribution.
The natural choice to alleviate these issues are generative models, in particular diffusion models. which have shown remarkable results in recent years. Following their success, diffusion models have been applied emulation tasks [18, 19, 22 22-25] for which they were found to mitigate the rollout instability of non- non-generative emulators. However, diffusion models are much more expensive than deterministic alternatives at inference, due to their iterative sampling process, which defeats the purpose of using an emulator. To address this computational drawback, many works in the image and
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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video generation literature [26-32] consider generating in the latent space of an autoencoder, This approach has been adapted with success the problem of emulating dynamical systems [33-37], sometimes even outperforming pixel-space emulation. In this work, we seek to answer a simple question: What is the impact of latent-space compression on emulation accuracy? To this end, we train and systematically evaluate latent-space emulators across wide range of compression rates for challenging dynamical systems from TheWell [38]. Our results indicate that
i. Latent diffusion based emulation is surprisingly robust to the compression rate, even when autoencoder reconstruction quality greatly degrades.
ii Latent-space emulators match exceed the accuracy of pixel-space emulators, while using fewer parameters and less training compute.
i.. Diffusion- based emulators consistently outperform their con-generative counterparts both accuracy and plausibility of the emulated dynamics.
Finally, we dedicate part of this manuscript to design choices. We discuss architectural and modeling decisions for autoencoders and diffusion models that enable stable training of latent-space emulators under high compression. To encourage further research this direction, we provide the code for all experiments at https: //gi h/g.thub....t.g///g/ggg thub. com/polllll//c pttppp/atcacaalll./ along with pre-trained model weights.
2 Diffusion models
The primary purpose of diffusion models (DMs) [39, 40], also known as score-based generative models [41, 42], is to generate plausible data from a distribution p(x) of interest. Formally, continuous- time diffusion models define a series of increasingly noisy distributions
p(x,) = f p(zz x) p(z) dr - / N(as a x. oz dz
such that the ratio a1/a, E RR is monotonically decreasing with the time / E [0, 1). For such series, there exists family reverse-time stochastic differential equations (SDEs) [42-44]
1 1 n²² o Ve, log p(z)) dt + 1 dw:
dæ.
fi 21
(2)
where 7) 2 0 is a parameter controlling stochasticity, the coefficients fe and St are derived from on and σ, [42-44], and for which the variable x; follows p(z,). In other words, can draw noise samples æ ~ p(x1) xs (0, 1) and obtain data samples zg ~> p(zo) p(x) by solving Eq. (2) from t = 1 to 0. For high-dimensional samples, the terminal signal-to-noise ratio es/0s should be at or very close to zero [45]. this work, we adopt the rectified flow [28, 46, 47] noise schedule, for which 0ę 1 and 0e t.
Denoising score matching practice, the score function Vg, log in Eq. (2) is unknown, but can be approximated by neural network trained via denoising equivalent parameterizations and objectives have been proposed for this task [40-42, 47, 50, 51]. In this work, we adopt the denoiser parameterization do (Z,, t) and its objective [51]
matching [48, 49]. Several
D (Te, 6) -x112 =1[2]]
arg min Ep(x)p((p(es11s)
(3)
for which the optimal denoiser is the mean E[x x2] of p( wx). Importantly, B[x xre] is linked to the score function through Tweedie's formula (52-55)
Et of V., log log
Eir xr]
at
which allows to use sα(21) o,
- (ds(2,))) Oce vz,) estimate in Eq. (2).
3 Methodology
In this section, we detail and motivate our experimental methodology for investigating the impact of compression on the accuracy of latent-space emulators. To summarize, consider three challenging

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Diffusion model
Diffusion model
Encoder
Figure 1. Illustraiion of the latent-space emulation process. At each step of the autoregressive rollout, the diffussin model generates the next n 4 latent states zi+li+ given the current state 2' and the simulation parameters 0. After rollout, the generated latent states are decoded to pixel space.
datasets from Well [38]. For each dataset, we first train a series of autoencoders with varying com- pression rates. These autoencoders learn to map high-dimensional physical states x² R H'x HwwWa. C
-ww-dimessional latent representations 2 E R we train two emulators operating in the latent space: a diffusion model (generative) and a neural solver (non-gend noon-generative). Both are trained to predict the next latent states 2*+ls+tn given the current latent state 2 and simulation parameters 0. This technique, known as temporal bundling [1 1J. mitigates the accumulation of errors during rollout by decreasing the number of required autoregressive steps. After training, latent-space emulators are used to produce autoregressive rollouts 2 z1:h starting from known initial state zd (x() and simulation parameters 0, which are then decoded to the pixel space as ø' (2*))).
X Subsequently, for each autoencoder,
3.1 Datasets
To study the effects of extreme compression rates, the datasets we consider should be high-dimensional and contain large amounts of data. Intuitively, the effective size of the dataset decreases in latent space, making overfitting more likely at fixed model capacity. According to these criteria, we select three datasets from Well [38]. Additional details are provided in Appendix B.
Euler Multi Multi-Quadrants The Fuler equations model the behavior of compressible non-viscous fluids. In this dataset, the initial state presents multiple discontinuities which result in interacting shock waves as the system evolves for 100 steps. The 2d state of the system is represented with three scalar fields (energy, density, pressure) and one vector field (momentum) discretized on 512 512 grid, for total Cpixel 5 channels. Each simulation has either periodic or open boundary conditions and a different heat capacity 7, which constitutes their parameters 0. We set time stride A between consecutive states x² and #+1 such that the simulation time T = A.
Boyleigh-Bénarc (RB) The Rayleigh-Bénard convection phenomenon occurs when an horizontal layer of fluid is heated from below and cooled from above. Over the 200 simulation steps, the temperature difference leads to the formation of convection currents where cooler fluid sinks and warmer fluid rises. The 2d state of the system is represented with two scalar fields (buoyancy, pressure) and one vector field (velocity) discretized on a 512 x 128 grid, for a total of Cpixe 4 channels. Each simulation has different Rayleigh and Prandtl numbers as parameters 0. We set time stride A = 1.
Turbulence Gravity Cooling (TGC) The interstellar medium can be modeled as turbulent fluid subject to gravity and radiative cooling. Starting from an homogeneous state, dense filaments form in the fluid, leading to the birth of stars. The 3d state of the system is represented with three scalar fields (density, pressure, temperature) and one vector field (velocity) discretized on a 64 x 64 x 64 grid, for total of Cpisel 6 channels. Each simulation has different initial conditions function of their density, temperature, and metallicity. We set a time stride A 1.
3.2 Autoencoders
To isolate the effect of compression, use a consistent autoencoder architecture and training setup across datasets and compression rates. We focus on compressing individual states sf into latent states z' Ey(x') which are reconstructed as øt (z(2f).

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Architecture We adopt convolution- autoencoder architecture similar to the one used by Rombach et al. [26], which we adapt perform well under high compression rates. Specifically, inspired by Chen et al. [31], we initialize the downsampling and upsampling layers near identity, which enables training deeper architectures with complex latent representations, while preserving reconstruction quality. For 2d datasets (Euler and RB), we set the spatial downsampling factor r = 32 for all autoencoders, meaning that a 32 x 32 patch in pixel space corresponds to one token in latent space. For 3d datasets (TGC), we set varying the number of channels per token in the latent representation. For instance, with the Euler dataset, autoencoder with Clatent = 64 latent channels f32c64 in the notations of Chen et al. [31] - transforms the input state with shape 512 x 512 x to a latent state with shape 16 x 16 x 64, yielding compression rate of 80. This setup ensures that the architectural capacity remains similar for all autoencoders and allows for fair comparison across compression rates. Further details as well short ablation study provided in Appendix B.
8. The compression rate is then controlled solely by
Training Latent diffusion models [26] often rely on a Kullback-Leibler (KL) divergence penalty to encourage latents follow standard Gaussian distribution. However, this term is typically down-weighted by several orders of magnitude to prevent severe reconstruction degradation. As such, the KL penalty acts more weak regularization than a proper variational objective [56] and post-hoc standardization of latents often necessary. We replace this KI. penalty with deterministic saturating function
(5)
VI+=//B2
applied to the encoder's output. In our experiments, we choose the bound B = 5 to mimic the range of standard Gaussian distribution. We find this approach simpler and more effective at structuring the latent space, without introducing a tradeoff' between regularization and reconstruction quality, We additionally omit perceptual [57] and adversarial [58, 59] loss terms, as they are designed for natural images where human perception is the primary target, unlike physics. The training objective thus simplifies to a reconstruction loss
arg min Ep(a)) [6(x, (x(B)(
(6)
The loss l is typically variation of Ia L-2 regression, which discuss in Appendix B. Finally, we find that preconditioned optimizers [60-62] greatly accelerate the training convergence of autoen- coders compared to the widespread Adam [63] optimizer (see Table 4). We adopt the PSGD [60] implementation in the heavyball [64] library for its fewer number of tunable hyper-parameters and lower memory footprint than SOAP [62].
3.3 Diffusion models
We train diffusion models to predict the next n latent states 2++1 given the current state of and simulation parameters 0, that is to generate from p(2++1a++ 2²,0). We parameterize our diffusion models with denoiser do(2)=** b, ,,tt) whose task is to denoise sequences of noisy states =t p(2 =²) (2 04 2 T) given the parameters f of the simulation. Conditioning with respect to known elements in the sequence zii+n is tackled with a binary mask b {0, 1] concate- nated to the input, as in MCVD [65]. For instance, b = 01, .,... indicates that the first element 2' is known, while b (I....,Io 1) indicates that the first n 1 elements 2hit are known. Known elements are provided to the denoiser without noise.
Denoiser Transformer b
Figure 2. Illustration of the denoiser's 's in- puts and outputs, while generating from pl iilitn 22 2).
Drawing inspiration from recent successes in latent image generation [27-31], we use transformer-based architecture for the denoiser. We incorporate several architectural refinements shown to improve performance and stability, including query-key normalization [66], rotary positional embedding (RoPE) [67, 68), and value residual learning [69]. The transformer operates on the spatial jistn and temporal axes of the input 2t while the parameters 0 and diffusion time modulate the
Architecture

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transformer blocks. Thanks to the considerable (r 32) spatial downsampling performed by the autoencoder, we are able apply full spatio-temporss attention, avoiding the need for sparse attention patterns [70-72]. Finally, we fix the token embedding size (1024) and the number of transformer blocks (16) for all diffusion models. The only architectural variation stems from the number of input and output channels dictated by the corresponding autoencoder.
Training
As in Section 2, diffusion models are trained via denoising score matching [48, 49]
arg min 2 (p()) || gg @ b iann
C (1 - b). b. 0). 2 :::+ 1
noisy
with the exception that the data does not come from the pixel-space distribution p(O, x _::2) but from the latent-space distribution p(0, 21:2) determined by the encoder Eg. Following Voleti al. [65], we randomly sample the binary mask b ~ p(b) during training to cover several conditioning tasks, including prediction with variable-length context p(2tteitt 2(1-6-1).
Sampling After training, we sample from the learned distribution by solving Eq. (2) with ry 0, which corresponds the probability flow ODE [42]. To this end, we implement 3rd order Adams- Bashforth multi-step integration method, as proposed by Zhang et al. [73]. Intuitively, this method leverages information from previous integration steps improve accuracy. We find this approach highly effective, producing high-quality samples with significantly fewer neural function evaluations (NFEs) than other widespread samplers [50, 51].
3.4
Neural solvers
We train neural solvers to perform the same task as diffusion models. Unlike the latter, how- ever, solvers do not generate from p(2 p(=++1+4 2t, 0), but produce point estimate fø(=i, 0) s 24, instead. We also train pixel-space neural solver, for which 2 x², baseline.
E 21
Architecture For latent-space neural solvers, we use the same transformer- architecture as for diffusion models. The only notable difference is that transformer blocks are only modulated with respect to the simulation parameters 0. For the pixel-space neural solver, we keep the same architecture, but group the pixels into 16 16 patches, in vision transformers [74]. We also double the token embedding size (2048) such that the pixel-space neural solver has roughly two times more trainable parameters than autoencoder and latent-space emulator combined.
Training
Neural solvers are trained via mean regression
arg min (1()
[2 ob,,,)) .itt
(8)
Apart from the training objective, the training configuration (optimizer, learning rate schedule, batch size. epochs, masking.... ...) for neural solvers is strictly the same as for diffusion models.
3.5 Evaluation metrics
We consider several metrics for evaluation, each serving different purpose. We report these metrics either at a lead time x A or averaged over a lead time horizon a If the states x' present several fields, the metric is first computed each field separately, then averaged.
RMSE The root mean squared error (RMSE) and its normalized variants are widespread metrics quantify the point-wise accuracy of an emulation [21, 38, 75]. Following Ohana et al. [38], we pick the cariance-normalized RMSE (VRMSE) over the more common normalized RMSE (NRMSE), as the latter down-weing weights errors in non-negative fields such pressure and density. Formally, for two spatial fields u and v, the VRMSE is defined as
((u v)2)
VRMSF(u, v)
(u - (u)) +
where (-) denotes the spatial mean operator and 10 G is numerical stability term.
5

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Power spectrum RMSE For chaotic systems such turbulent fluids, it is typically intractable to achieve accurate long-term emulation very small errors can lead entirely different trajectories later on. In this case, instead of reproducing the exact trajectory, emulators should generate diverse trajectories that remain statistically plausible. Intuitively, even though structures are wrongly located, the types of patterns and their distribution should stay similar [76]. Following Ohana et al. [38], we assess statistical plausibility by comparing the power spectra the ground-truth and emulated trajectories. For two spatial fields and we compute the isotropic power spectra Pa and pa and split them into three frequency bands (low, mid and high) evenly distributed log-space. We report the RMSE of the relative power spectra Pc/ps over each band.
Spread-skill ratio In earth sciences [25, 75], the skill an ensemble of particles defined as the RMSE of the ensemble mean. The spread is defined as the ensemble standard deviation. Under these definitions and the assumption of perfect forecast where ensemble particles are exchangeable, Fortin et al. [75] show that
(10) This motivates the use of the (corrected) spread-skill ratio as metric. Intuitively, if the ratio is smaller than one, the ensemble is biased or under-dispersed. If the ratio is larger than one, the ensemble over-dispersed. It should be noted, however, that spread-skil -kill ratio of is necessary but insufficient condition for perfect forecast.
Skill R /K+1/K Spread
4 Results
We start with the evaluation of the autoencoders. For datasets, we train three autoencoders with respectively 64, 16, and 4 latent channels. These correspond to compression rates of 80, 320 and 1280 for the Euler dataset, 64, 256, and 1024 for the RB dataset, and 48, 192, 768 for the TGC dataset, respectively. In the following, we refer to models by their compression rate. Additional experimental details are provided in Section and Appendix B.
For each autoencoder, we evaluate the reconstruction it D of all states of 64 test trajectories 2.11, As expected, when the compression rate increases, the reconstruction quality degrades, as reported in Figure 3. For the Euler dataset, the reconstruction error grows with the lead time due to wavefront interactions and rising high-frequency content. For the RB dataset, the reconstruction error peaks mid-simulation during the transition from low to high-turbulence regime. Similar trends can be observed for the power spectrum RMSE in Tables 8, 9 and 10, where the high- frequency band intuitively expect from lossy compression.
affected by compression. These results far align with what practitioners
the evaluation of the emulators. For each autoencoder, train latent-space emulators: a diffusion model and a neural solver. Starting from the initial state 20 E (x.)) and simulation parameters of 64 test trajectories 001,, each emulator produces 16 distinct autoregressive rollouts 2 !:t.. which then decoded to the pixel space i² (2')") Note that for neural solvers, all 16 rollouts are identical. We compute the metrics of each prediction a² against the ground-truth state xf.
We
0.36
0.14
0.12
320
236
0.30
192
1280
0.10
1024
0.10
VRMSE
0.08
0.08
VRMSE
0.20
0.15
0.04
0.06
0.00
0.00
0.00
01:22
21:60
61:100
01:20
21:00
61:180
01:10
11:30
31-50
Load time
Figure 3. Average VRMSE of the autoencoder reconstruction different compression rates and lead time horizons for the Euler (left), RB (center) and TGC (right) datasets. The compression rate has clear impact reconstruction quality.

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Sis in & S
Ground-rruth
Cound-truth
Autoencoder S-tt
Autoenender
Diffision S
Neural solver Diffusion
Neuna salver $280
20
60
16 32
61 128
Figure 4. Examples of latent-space emulation for the Euler (left) and Cayleigh-Bénart (right) datasets. Even for large compression rates (+), latent-space emulators are able to reproduce the dynamics surprisingly faithfully, despite significant reconstruction artifacts. For Euler, wavefronts are accurately propagated until the end of the simulation, while vortices are well located, but distorted. For Rayleigh- Bénard, diffusion-based emulators produce plumes that grow at the correct pace but diverge from the ground-truth. Similar observations can be made in Figures 10 to 21.
As expected from imperfect emulators, the emulation error grows with the lead time, as shown in Figures 5 and 8. However, the point-wise error of diffusion models, as measured by the VRMSE, does not grow (Euler, TGC) and sometimes decreases (RB) with higher compression rates. Even extreme (> 1000) compression rates, latent-space emulators outperform the baseline pixel-space neural solver, despite the latter benefiting from more parameters and training compute. Similar observations be made with the power spectrum RMSE over low and mid-frequency bands. High- frequency content, however, appears limited by the autoencoder's reconstruction capabilities. We confirm this hypothesis by recomputing the metrics relative the auto-encoded state D (z"))) which we report in Figure 9. This time, the power spectrum RMSE of the diffusion models is low for mid and frehuffryy bands. These findings support a puzzling narrative: emulation accuracy exhibits strong resilience to latent-space compression, starkly contrasting with the clear degradation in reconstruction quality.
Our experiments also provide a direct comparison between gen- erative (diffusion) and deterministic (neural solver) approaches to emulation within a latent space. Figures 8 and 9 indicate that diffusion-based based emulators are consistently more accurate than their deterministic counterparts and generate trajectories that are statistically more plausible terms of power spectrum. This can be observed qualitatively in Figure 4 or Figures 10 21 in Appendix In addition, the spread-skill ratio of dif- fusion models is close to 1, suggesting that the ensemble of trajectories they produce are reasonably well calibrated in terms of uncertainty. However, the ratio slightly decreases with the compression rate. This phenomenon is partially explained by the smoothing effect of L2-ddrven compression, and is therefore less severe in Figure 9, Nonetheless, remains present and could be sign of overfitting due to the reduced amount of training data in latent space.
Table Inference time per state for the Euler dataset, including genera- tion and decoding.
Method
Space
Time
simulator
pixel
O(10s)
neural solver
pixel
56 ms
neural solver
latent
13 ms
diffusion diffusion
pixel
((1s)
latent
84 ms
7

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320
diffusion
80
1240
neural solver
Power spectrum RMSE
1.00
0.8
VRMSE
0.75
1000
0.2
0.50
0.25
0.00
0.4
1.00
Spread/Sh /Skil
0.75
0.0
0.50
0.4
0.25
0.00
Lead time
Lead time
Figure 5. Average evaluation metrics of latent-space emulation for the Euler dataset. As expected from imperfect emulators, the emulation error grows with the lead time. However, the compression rate has little to no impact on diffusion- maasion-bbaedd emulation accuracy, beside high-frequency content. The spread-skill ratio [25, 75] drops slightly with the compression rate, which could be sign of overfitting. Diffusion-barn emulators are consistently more accurate than neural solvers.
In terms of computational cost, although they remain slower than latent-space neural solvers, latent- space diffusion models are much faster than their pixel-space counterparts and competitive with pixel-space neural solvers (see Table 1). With our latent diffusion models, generating and decoding full (100 simulation steps. 7 autoregressive steps) Euler trajectory takes 3 seconds on single A100 GPU, compared roughly I CPU-hour with the original numerical simulation [38, 77].
final advantage of diffusion models lies in their capacity to incorporate additional information during sampling via guidance methods [42, 78- 78-81]. For example, if partial noisy state observations are available, we can guide the emulation such that it remains consistent with these observations. We provide illustrative example in Figure where guidance is performed with the MMPS [78] method. Thanks to the additional information in the observations, the emulation diverges less from the ground-truth.
Related work
Data-driven emulation of dynamical systems has become prominent research area [8-17] with diverse applications, including accelerating fluid simulations on uniform meshes using convolutional networks [8, 12], emulating various physics on non-uniform meshes with graph neural networks [9- 11, 14], and solving partial differential equations with neural operators [13, 21, 82 22-84.. However, McCabe al. [15] and Herde et al. [16] highlight the large data requirements these methods and propose pre-training on multiple data-abundant physics before fine-tuning data-scarce ones to improve data efficiency and generalization. Our experiments similarly suggest that large datasets are
needed to train latent-space emulators.
parallel line of work, related reduced-order modeling [85], focuses on learning low-dimensional representations of high-ddiensiioal system states. Within this latent space, dynamics can be emulated more efficiently [86-94]. Various embedding approaches have been explored: convolutional autoen- coders for uniform meshes [88, 89], graph-based autoencoders for non-uniform meshes [90], and implicit neural representations for discretization- free states [34, 92]. Koopman operator theory [95] has also been integrated into autoencoder training to promote linear latent dynamics [91, 96]. Other approaches to enhance latent predictability include regularizing temporal derivatives [97], jointly optimizing the decoder and latent emulator [98], and self-supervised prediction [99]. While our work adopts this latent emulation paradigm, we not impose structural biases on the latent space beside reconstruction quality.
8

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Observation
Observatinn
Ground troth
Ground-truft
MANAX
Gaided diffusion Set
Guudeed diffusion Sea
Diffusion *ei
DDffiusioon S
16 32
128
5
25
45
Figure 6. Example of guided latent-space emulation for the RB (left) and TGC (right) datasets. The observations are the states downsampled by factor 16 for RB and stripe along the domain boundaries for TGC. Guidance is performed using the MMPS [78] method. Thanks to the additional information in the observations, the emulation diverges less from the ground-truth.
persistent challenge in neural emulation is ensuring temporal stability. Many models, while accurate for short-term prediction, exhibit long-term instabilities as errors accumulate, pushing the predictions of the training data distribution [21]. Several strategies have been proposed to mitigate this issue: autoregressive unrolling during training [11, 86, 100], architectural modifications [21, 83], noise injection [12], and post-processing [18, 101]. Generative models, particularly diffusion models, have recently emerged as a promising approach to address this problem [18, 19, 22-25] as they produce statistically plausible states, even when they diverge from the ground-truth solution.
While more accurate and stable, diffusion models are computationally expensive at inference. Draw- ing inspiration from latent space generation in computer vision [26-32], recent studies have applied latent diffusion models to emulate dynamical systems: Gao et al. [33] address short-term precipitation descriptions, Du al. [34]
forecasting, Zhou et [35] generate trajectories conditioned on generate trajectories within an implicit neural representation, and Li et al. [36] combine a state-wise autoencoder with a spatiotemporal diffusion transformer [27] for autoregressive emulation, simi- lar to our approach. These studies report favorable or competitive results against pixel-space and deterministic baselines, consistent with our observations.
Discussion
Our results reveal key insights about latent physics emulation. First, diffusion-based emulation accuracy surprisingly robust latent-space compression, with performance remaining constant or even improving when autoencoder reconstruction quality significantly deteriorates. This observation consistent with the latent generative modeling literature [26, 56], where compression serves dual purpose: reducing dimensionality and filtering out perceptually irrelevant patterns that might distract from semantically meaningful information. Our experiments support this hypothesis as latent-space emulators outperform their pixel-space counterparts despite using fewer parameters and requiring less training compute. Yao al. [102] similarly demonstrate that higher compression can sometimes improve generation quality despite degrading reconstruction. While our findings seem to violate the famous data processing inequality, they are well aligned with the theory usable information [103], where learned representation hold more V-information from the point

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of view of computationally constrained observer. Second, diffusion iuffusion-based generative emulators consistently achieve higher ensemble accuracy than deterministic neural solvers while producing diverse, statistically plausible trajectories. This supports the idea that generative models mitigate distribution shift [18, 19, 22-25]. However, at the first prediction step, before distribution shift can take effect, diffusion models are already more accurate than deterministic neural solvers. This suggests an inherent modeling advantage, possibly lying in the iterative nature of diffusion sampling.
Despite the finite number of datasets, we believe that our findings are likely to generalize well across the broader spectrum of fluid dynamics. The Euler, RB and TGC datasets represent distinct fluid regimes that cover many key challenges in dynamical systems emulation: nonlinearities, multi-scale interactions, and complex similar conclusions for other fluid dynamics problems. However, we exercise caution about extending these conclusions beyond fluids. Systems governed by fundamentally different physics, such as chemical or quantum phenomena, may respond unpredictably to latent compression. Probing these boundaries represents an important direction for future research. Our empirical findings also prompt the need for thcoretical explanations, which we leave to future work.
peatio-temporal patterns. In addition, previous studies [33-36] come to
Apart from datasets, if compute resources were not limiting factor, our study could be extended along several dimensions, although we anticipate that additional experiments would not fundamen- tally alter our conclusions. First, we could investigate techniques for improving the structure of the latent representation, such as incorporating Koopman innsiired losses [91, 96], regularizing temporal derivatives [97]. or training shallow auxiliary decoders [102, 104]. Second, we could probe the behav- ior of different embedding strategies under high compression, including spatio-temporal embeddings [34, 35, 105), implicit neural representations [34, 92], and masked auto-encoders (104, 106). Third, we could add the capability to trade speed for accuracy, analogous to running numerical solvers at finer resolutions, by training an auto-encoder with an adaptive latent dimensionality [107-109]. Forth, we could study the effects of autoencoder and emulator capacity by scaling either up or down their number of trainable parameters. Each of these directions represents a substantial computational investment, particularly given the scale of our datasets and models, but would help establish best practices for satent-space emulation.
Nevertheless, our findings lead to clear recommendations for practitioners wishing to implement physics emulators. First, try latent-space approaches before pixel-space emulation. The former offer reduced computational requirements, lower memory footprint, and comparable or better accuracy across a wide range of compression rates. Second, prefer iiffusion-based emulators over determin- istic neural solvers. Latent diffusion models provide more accurate, diverse and stable long-term trajectories, while narrowing the inference speed gap significantly.
Our experiments, however, reveal important considerations about dataset scale when training latent- space emulators. The decreasing spread-skill ratio observed higher compression rates suggests potential overfitting. This makes intuitive sense: as compression increases, the effective size of the dataset in latent space decreases, making overfitting more likely fixed model capacity. Bench- marking latent emulators on smaller (10-100 GB) datasets like those used by Kohl et al. [19] could therefore yield misleading results. In addition, because the latent space is designed to preserve pixel space content, observing overfitting in this compressed representation suggests that pixel-space models encounter similar issues that remain undetected. This points towards the need for large training datasets or mixtures of datasets used to pre-train emulators before fine-tuning targeted physics, as advocated by McCabe et al. [15] and Herde et al. [16].
Acknowledgments and Disclosure of Funding
We thank Géraud Krawezik and the Scientific Computing Core at the Flatiron Institute, division of the Simons Foundation, for the compute facilities and support. We gratefully acknowledge use of the research computing resources of the Empire Al Consortium, Inc., with support from the State of New York, the Simons Foundation, and the Secunda Family Foundation. Polymathic AI acknowledges funding from the Simons Foundation and Schmidt Sciences, LLC. François Rozet is research fellow of the -FRRS (Belgium) and acknowledges its financial support.
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A Spread / Skill
The skill [25, 75] of ensemble of K particles Uk is defined as the RMSE of the ensemble mean
(
1
Skill
(11)
k=1
where (+) denotes the spatial mean operator. The spread defined the ensemble standard deviation
K
K
Spread =
(12)
K
K
Under these definitions and the assumption of perfect forecast where ensemble particles exchangeable, Fortin et al. [75] show that
K +1 K
Skill
Spread.
(13)
This motivates the use of the (corrected) spread-skill ratio a metric. Intuitively, if the ratio smaller than one, the ensemble is biased or under-dispersed. If the ratio is larger than one, the ensemble over-dispersed. It should be noted however, that a spread-skill ratio of is a necessary but not sufficient condition for perfect forecast.
23

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B Experiment details
Datasets For all datasets, each field is standardized with respect its mean and variance over the training set. For Euler, the non-negative scalar fields (energy, density, pressure) are transformed with нy log(2 1) before standardization. For TGC, the non-negative scalar fields (density, pressure, temperature) are transformed with + log(z + 10 before standardization. When the states are illustrated graphically, as Figure 1, we represent the density field for Euler, the buoyancy field for RB, and slice of the temperature field for TGC.
Table 2. Details of the selected datasets. We refer the reader Ohana al. [38] for more information.
Euler Multi- Quadrants
Rayleigh- Bénard
Turbulence Gravity Cooling
Software
Clawpack [77] 5243 GB
Dedalus [110]
ASURA-FDPS [111]
Size
367 GB
849 GB
energy, density, pressure, velocity
buoyancy, pressure, momentum
density, pressure, temperature, velocity
Fields
Channels Cpicel
Resolution
512 x 512 Uniform
512 128 Chebyshev 1750
x 64
Discretization
Uniform 2700 50
Trajectories
10000
Time steps L
100
200
Stride A
heat capacity % boundary conditions
Rayleigh number, Prandu number
hydrogen density Po- temperature To, metallicity
0
Autoencoders The encoder Es and decoder De are convolutional networks with residual blocks [112], SiLU [113] activation functions and layer normalization [114]. The output of the encoder is transformed with saturating function (see Section 3). We provide a schematic illustration of the autoencoder architecture Figure 7. Following McCabe al. [15], we use feegh-weighted loss MSE (VMSE)
and choose the
((u - u)
VMSE(u, v) ((u (u) + +E
(14)
averaged over fields, where c = 10-2 mitigates training instabilities. We train the encoder and decoder jointly for 1024 256 steps of the PSGD [60] optimizer. To mitigate overfitting we use random spatial axes permutations, flips and rolls as data augmentation. Each autoencoder takes I (RB), 2 (Euler) or (TGC) days to train on H100 GPUs Other hyperparameters are provided in Table 3.
The entire dataset encoded with each trained autoencoder and the resulting latent trajectories cached permanently disk. The latter then be used to train latent-space emulators, without needing to load and encode high-dimensional samples on the fly. Depending on hardware and data dimensionality, this approach can make a huge difference in I/O efficiency.
Caching
The denoiser de and neural solver fa are transformers with query-key normalization [66], rotary positional embedding (RoPE) [67, 68], and value residual learning [69]. The 16 blocks are modulated by the simulation parameters 0 and the diffusion time t, as in diffusion transformers [27]. We train the emulator for 4096 x 64 steps of the Adam [63] optimizer. Each latent- space emulator takes 2 (RB) or 5 (Euler, TGC) days to train on 8 H100 GPUs. Each pixel-space emulator takes 5 (RB) or 10 (Euler) days train 16 H100 GPUs. We do not train pixel-space emulator for TGC. Other hyperparameters are provided in Table 5.
Emulators
During training we randomly sample the binary mask b. The number of context elements follows Poisson distribution Pois(A = 2) truncated between and n. Hence, the masks & take the form
b
1. 0, O,....))
(15)
implicitely defining distribution p(b).
24

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Table 3. Hyperparameters for the autoencoders.
Euler & RB
TGC
Architecture
Conv
Conv
Parameters
3.1 10a
7.2 10
Pixel shape
Opiset x
Cpisel X W x Z
Latent shape
Chueent X 32
W
Cintent
W
Residual blocks level
(3, 3, 3, 3, 3)
(3, 3, 3, 3) (64, 256, 512, 1024)
Channels per level
(64, 128, 256, 512, 768, 1024)
Kernel size
3 3
x SiLU
Activation
SiLU LayerNorm 0.05
Normalization
LayerNorm 0.05
Dropout
Loss
VMSE
VMSE
Optimizer
PSGD
PSGD
Learning rale
10-6
10-5
Weight decay
0.0
0.0
Scheduler
cosine
cosine
Gradient norm clipping
1.0
1.0
Batch size
64
64
Steps epoch
256
256
Epochs
1024
1024
GPUs
Table 4. Short ablation study on the autoencoder architecture and training configurations. We pick the Bayleigh-Bénar dataset and an architecture with 64 latent channels to perform this study. The two major modifications that we propose are (1) the initialization of the downsampling and upsampling layers near identity, inspired by Chen et al. [31], and (2) the use of a preconditioned optimizer, PSGD [60], instead of Adam [63]. We report the mean absolute error (MAE) on the validation set during training. The combination of both proposed modifications leads to order(s) of magnitude faster convergence.
Epoch
Optimizer Id. init
Time
10
100
1000
Adam
w/o
0.065
0.029
0.017
19h
Adam
w/
0.039
0.023
0.014
19h
PSGD
0.023
0.015
0.011
25
Evaluation For each dataset, we randomly select 64 trajectories 2:22 with various parameters in the test set. For each latent-space emulator, encode the initial state zg Eg (2o) and produce 16 distinct autoregressive rollouts 21:4, For the diffusion models, sampling is performed with 16 steps of the 3rd order Adams-Bashfore Bashforth multi-step integration method [73]. The metrics (VRMSE, power spectrum RMSE, spread-skill ratio) are then measured between the predicted states 2² (2')") and the ground-truth states rf or the auto-encoded states D. (x²)).
25

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Conv2d (Ca +64)
Residual block
Down block (C,+
Down black (64-1228)
LayerNorm
Residual block
Down block (128-226))
Conv2d (C+C)
Space ToDephh (2x2)
Down block (256-+112)
Conv2d (4C,
Down block (512-7768)
Conv2d (C-C)
Down block (768-1024)
Residual
Conv2d C1024
Saturation
Identity weight initialization
E(x)
Conv2d (024)
Residual block x3
Up block (C,)
block (1024-768)
Conv2d (CC)
Up block (768-512)
ToptheoSpace (2x2)
Up block (512-2566)
Residual block x3
Up block (256-1128)
Up block (128-644
Conv2d (64-0
D(2)
Figure 7. Schematic representation of the autoencoder architecture. Downsampling (resp. upsam- pling) is performed with a space-to-depth (resp. depth-to-space) operation followed (resp. preceded) with convolution initialized near identity.
Table 5. Hyperparameters for the emulators.
Latent Latent-space
Pixel-space
Architecture
Transformer
Transformer
Parameters
2.2 10a
8.6 x 10
Input shape
Chsten X (y
#1
x W
Cpisal
(n + 1) x H x W
Patch size
1 x
x 16
Tokens
(n + 1) 32
!!
x 32
w
(n + 1) x 16 T6
J1
W
Embedding size
1024
2048 16
Blocks
16 Absolute RoPE
Positional embedding
Absolute + RoPE
Activation
SiLU
SiLU
Normalization
LayerNorm 0.05
LayerNorm 0.05
Dropout
Optimizer
Adam
Adam 10
Learning rate
10 4
Weight decay
0.0
0.0
Scheduler
cosine
cosine
Gradient norm clipping
1.0
1.0
Batch size
256
256
Steps per epoch
64
Epochs
4096
4096
GPUs
8
16
26

---

C Additional emulation results
320
diffueion
1250
wollour
Power spectrum RMSE
1.00
VRMSE
3003
0.50
0.00
Spread (Ski)
0.70
0.25
0
20
40
60
80
100
20
60
80
100
256
difluiion
Power spectrum RMSE
1.00
0.6
VIMSS
0.2
0.50
0.6
0.00
0.4
1.00
0.2
Spread /Skill
0.75
0.6
0.20
0.27
150
175
50
125
1.50
Litne
Leaed
48
708
diftusion
Power spretrro RMSE
1.00
0.5
0.4
VIMME
0.2
0.50
0.0
0.25
0.5
0.00
0.4
1.00
0.2
Spread/Skill
0.76
0.0
0.5
0.4
0.25
0.2
Lead time
Load time
Figure 8. Average evaluation metrics of latent-space emulation for the Euler (top), RB (center) and TGC (bottom) datasets. As expected from imperfect emulators, the emulation error grows with the lead time. However, increasing the compression rate does not degrade (Euler, TGC) and sometimes improves (RB) the accuracy of diffusion models. The spread-skill ratio [25, 75] drops slightly with the compression rate, which could be sign of overfitting. Diffusion-ban based emulators consistently more accurate than neural solvers.
27

---

520
diffxxion
1250
solvur
Power spertrom RMSE
1.00
0.6
VIMSE
0.50
0.00
Spreed/Skei Skkil
0.75
0.26
0
20
40
60
80
100
20
40
60
80
100
256
dillusion
Power spectrum RMSE
1.00
0.6
VIMSS
0.2
0.50
0.6
0.00
0.4
0.2
Spread /Skill
0.75
0.6
0.20
25
76
100
125
150
175
25
50
75
100
125
150
176
Load
48
708
diftusion
Power spectrom RMSE
1.00
0.5
0.4
VRMSE
0.2
0.50
0.0
0.5
0.00
0.4
1.00
0.2
Spread/ (5kill
0.76
0.0
0.5
0.50
0.4
0.25
0.2
0.00
0.0
Lead time
Lead time
Figure 9. Average evaluation metrics of latent-space emulation relative to the auto-encoded states D (x")) for the Euler (top), RB (center) and TGC (bottom) datasets. As expected from imperfect emulators, the emulation does not degrade (Euler, TGC) and sometimes improves (RB) the accuracy diffusion models. The spread-skill ratio [25, 75] drops slightly with the compression rate, which could be sign of overfitting. Diffusion based emulators are consistently more accurate than neural solvers.
grows with the lead time. However, increasing the compression rate
28

---

Table 6. Average VRMSE of autoencoder reconstruction and latent-space emulation at different compression rates (+) and lead time horizons for the Euler, RB and TGC datasets. Increasing the compression rate has a clear impact on reconstruction quality, but does not degrade significantly (Euler, TGC) and sometimes improves (RB) the accuracy of diffusion models.
Euler
RB
Method
Method
1:20
21:60
61:100
21:60
61:180
80
0.011
0.014
0.020
64
0.023
0.033
0.019
autoencoder
0,023
0.041
0.061
autoencoder
0.039
0.064
0.042
1280
0.060
0.107
0.144
1024
0.070
0.124
0.092
0.075
0.199
0.395
64
0.171
0.582
0.704
diffusion
0.070
0.192
0.371
diffusion
0.141
0.509
0.683
1280
0.093
0.217
0.400
1024
0.146
0.457
0.670
0.138
0.397
1.102
0.185
0.681
0.918
0.077
0.232
0.500
64
0.244
0.761
0.968
neural solver
neural solver
320
0.080
0.232
0.476
256
0.197
0.716
0.945
1280
0.137
0.314
0.592
1024
0.195
0.665
0.903
TGC
Method
1:10
11:20
21:50
48
0.151
0.116
0.129
autoencoder
192
0.229
0.175
0.189
768
0.338
0.272
0.276
48
0.296
0.522
0.673
diffusion
192
0.342
0.527
0.665
768
0.425
0.575
0.694
48
0,302
0.599
0.826
192
0.361
0.632
0.835
neural solver
768
0,462
0.710
0.920
Table 7. Average VRMSE of latent-space emulation at different context lengths (c) and lead time horizons for the Euler, RB and TGC datasets. We can test different context lengths without retraining as models were trained for different conditioning tasks (see Section 3). Perhaps surprisingly, context lengths does not have a significant impact on emulation accuracy.
Fuler
Method
Method
1:20
21:60
61:100
1:20
21:60
61:180
1
0.085
0.204
0.393
0.152
0.510
0.683
diffusion
2
0.074
0.200
0.383
diffusion
0.150
0.511
0.685
0.078 3
0.203
0.389
0.157
0.527
0.689
0.108
0.266
0.526
0.208
0.705
0.932
neural solver
2
0.092
0.253
0.513
neural solver
0.209
0.708
0.943
3
0.094
0.260
0.529
0.220
0.728
0.940
TGC
Method
1:10
11:20
21:50
0.362
0.550
0.681
diffusion
2
0.351
0.535
0.669
3
0.350
0.539
0.683
0.376
0.632
0.837
neural solver
2
0.371
0.641
0.855
3
0.378
0.669
0.888
29

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Table 8. Average power spectrum RMSE of autoencoder reconstruction and latent-space emulation at different compression rates (+) and lead time horizons for the Fuler dataset. The high-freque hegh-freuuency content of diffusion-based emulators limited by the autoencoder's reconstruction capabilities.
Low
Mid
High
Method
1:20
21:60
61:100
1:20
21:60
61:100
1:20
21:60
61:100
80
0.001
0.001
0.001
0.006
0.008
0.014
0.072
0.069
0.096
320
0.002
0.003
0.004
0.022
0.047
0.085
0.112
0.141
0.240
autoencoder
1280
0.009
0.017
0.025
0,074
0.167
0.264
0.240
0.355
0,577
80
0.017
0.063
0.168
0.054
0.100
0.178
0.112
0.116
0.184
320
0.014
0.058
0.157
0.052
0.102
0.171
0.128
0.155
0.275
diffusion
1280
0.019
0.065
0.163
0.096
0.187
0.300
0.246
0.349
0.569
1
0.046
0.128
0.339
0.227
0.297
0.754
0.821
0.984
2.666
80
0,021
0.074
0.212
0,085
0.151
0.245
0.164
0.173
0.249
neural solver
320
0.020
0.075
0.204
0.074
0.144
0.234
0.151
0.169
0.271
1280
0,045
0.116
0.274
0.131
0.227
0.349
0.283
0.345
0.545
Table Average power spectrum RMSE of autoencoder reconstruction and latent-space emulation different compression rates (+) and lead time horizons for the Rayleigh-Benan dataset. The freeefryy content of diffusion-based emulators is limited by the autoencoder's reconstruction
capabilities.
Low
Mid
High
Method
1:20
21:60
61:180
1:20
21:60
61:180
1:20
21:60
61:180
0.043
0.004
0.001
0.011
0.013
0.012
0.026
0.159
0.148
256
0,061
0.011
0.004
0.028
0,080
0,075
0.050
0.220
0.212
autoencoder
1024
0.121
0.033
0.018
0,063
0.186
0.197
0.076
0.294
0.294
64
1.751
0,850
0,386
0.197
0,266
0.159
0.054
0.220
0.199
256
0.328
0.399
0.396
0.084
0.195
0.177
0.065
0.239
0.232
diffusion
1024
0.193
0.292
0.344
0.095
0.243
0.240
0.083
0.314
0.297
1
0.467
0.520
0.650
0.151
0.255
0.264
0.076
0.232
0.242
64
3.625
0.915
0.566
0.268
0.351
0.275
0.099
0.279
0.257
neural solver
256
0.575
0.675
0.526
0.165
0.317
0.264
0.091
0.275
0.257
1024
0.285
0.496
0.560
0.152
0.338
0.321
0.090
0.320
0.326
Table 10. Average power spectrum RMSE of autoencoder reconstruction and latent-space emulation different compression rates (+) and lead time horizons for the TGC dataset. The high-frequenc) content of diffusion- based emulators is limited by the autoencoder' 's reconstruction capabilities.
Low
Mid
High
Method
1:10
11:30
31:50
1:10
11:30
31:50
1:10
11:30
31:50
48
0.011
0.016
0.025
0.023
0.026
0.044
0.275
0.188
0.195
192
0.028
0.033
0.045
0.108
0.091
0.114
0.359
0.273
0.282
autoencoder
768
0.072
0.068
0.080
0.285
0.235
0.254
0.454
0.476
0.367
48
0.064
0.185
0.319
0.058
0.128
0.220
0.296
0.247
0.331
192
0.069
0.191
0.311
0.128
0.164
0.252
0.369
0.316
0.384
diffusion
768
0.107
0.294
0.425
0.289
0.305
0.360
0.456
0.419
0.444
48
0.070
0.221
0.424
0.110
0.197
0.324
0.357
0.320
0.427
192
0.086
0.228
0.402
0.172
0.201
0.295
0.391
0.317
0.395
neural solver
768
0.138
0.277
0.465
0.322
0.305
0.407
0.471
0.418
0.493
30

---

Table 11. Average sliced earth mover's distance (SEMD) [115, 116] of the density field of autoencoder reconstruction and latent-space emulation at different compression rates (+) and lead time horizons for the Euler dataset. The SEMD is small and is not significantly impacted by the compression rate, especially for diffusion models. For reference, the density fields of two consccutive states x' and xf+1 have typical SEMD of 0.0025. Why this metric? The Euler equations in aerodynamics to model flow around objects and one is typically interested in the global fluid displacement. The rationale for using this metric is that small drift in the density field would not significantly affect the (S)EMD, while could affect point-wiss wise metrics heavily.
sometimes used
EMD (density field)
Method
1:20
21:60
61:100
80
0.0000
0.0000
0.0000
autoencoder
320
0.0001
0.0001
0.0001
1280
0.0002
0.0003
0.0005
80
0.0004
0,0010
0.0023
diffusion
320
0.0003
0.0009
0.0022
1280
0.0004
0.0010
0.0023
0.0011
0,0031
0.0066
80
0.0005
0.0012
0.0028
neural solver
320
0.0004
0.0012
0.0027
1280
0.0008
0.0020
0.0041
Table 12. Average Wasserstein distance of the distribution of vertical velocity values of autoencoder reconstruction and latent-space emulation at different compression rates (+) and lead time horizons for the RB dataset. The Wasserstein distance is smaller for diffusion models and decreases with the compression rate. For reference, the distributions of vertical velocity values of two consecutive states x² and *++1 have a typical Wasserstein distance of 0.004. Why this metric? One interesting quantity in buoyancy-driven convection is the growth speed of plumes the fluid. The distribution of the (vertical) velocity values is good summary statistic for tracking the growth of plumes.
Wasserstein (vertical velocity field)
Method
1:20
21:60
61:180
64
0.0000
0.0002
0.0002
autoencoder
256
0.0001
0.0007
0.0005
1024
0.0002
0.0020
0.0018
64
0.0003
0.0104
0.0141
diffusion
0.0003
0.0092
0.0141
1024
0.0004
0.0063
0.0139
0.0003
0.0153
0.0247
64
0.0009
0.0272
0.0223
neural solver
256
0.0007
0.0197
0.0187
1024
0.0007
0.0157
0.0206
31

---

Table 13. Average Wasserstein distance of the distribution of density values of autoencoder recon- struction and latent-space emulation at different compression rates (+) and lead time horizons for the TGC dataset. The Wasserstein distance is smaller for diffusion models, but grows significantly with the lead time, even for the autoencoder reconstruction. For reference, the distributions of density values of consecutive states z' and x have a typical Wasserstein distance of 0.01. Why this metric? In the interstellar medium, gravity forms clusters of matter that eventually lead to the birst of stars. The kind of clusters (compact, diffuse, or anything in between) and their proportions is of interest for domain-scientiss, The distribution of the density values is good summary statistic for clustering dynamics.
Wasserstein (density field)
Method
1:10
11:30
31:50
48
0.0034
0.0048
0.0089
autoencoder
192
0.0082
0.0110
0.0183
768
0.0181
0.0236
0.0338
48
0.0044
0.0138
0.0266
diffusion
192
0.0089
0.0172
0,0310
768
0.0186
0.0274
0.0425
48
0.0074
0.0253
0.0524
192
0.0091
0.0171
0.0329
neural solver
768
0.0220
0.0296
0.0492
Table 14. Average VRMSE results from various studies using TheWell [38] datasets. Even though our latent diffusion models (LDMs) outperform most published baselines, we emphasize that do not position our models as state-of-the-art, notably due to the discrepancies in parameters count, training and evaluation.
Source
Method
Dataset
Parameters
Lead time
VRMSE
Ohana al. [38]
FNO
Euler
107
6:12
1.13
Ohana et al. [38]
U-Net
Fuler
x 107
6:12
1.02
Ours
VIT
Fuler
8.6 x 10
1:20
0.138
Ours
LDM +80
Euler
3.1
10 + 2.2 x 10
1:20
0.075
Ohana et al. [38]
FNO
RB
2 x 10"
6:12
10+
Ohana al. [38]
U-Net
RB
2 107
6:12
10+
Nguyen et al. [117]
PhysiX
4.5 x 10
2:8
1.067
Wu et al. [118]
TANTE
RB
108
1:16
0.609
Mukhopadhyay al. [119]
VIT CSM
RB
108
10
0.140
Ours
x 10a
0.185
Ours
LDM +256
RB
3.1
10 + 2.2 x 108
1:20
0.141
Ohana et al. [38]
FNO
TGC
x 2 x 10"
6:12
3.55
Ohana et al. [38]
U-Net
TGC
6:12
7.14
Mukhopadhyay al. [119]
VIT CKM
TGC
108
0.527
Ours
LDM +48
TGC
7.2 x 10 + 2.2 x 1
1:10
0.296
32

---

Ground-rroth
Aufoenooder
Dilfosion
Diffasson +rn
Ground--tull
Neural solver +
Neural sulver
20
60
Creund-trulh
Ground
Autvencoder Thba
Autoenewder Tne
Dilfosion Fa
Dilluxion
Dilfosionn 550
Dilfusion 1399
Neural solver 1aa
Neural solver
20
60
100
20
60
T= 100
Figure 10. Examples of emulation at different compression rates (+) for the Fuler dataset. In this simulation, the system has open boundary conditions.
33

---

Ground-troth
Aufounsoder
Dilfosioon
Ground--tol -tob
Diffasson +-n
Neural solver +
Neural salver
20
Ground troumd-trul -rulh
Ground-tral stulh
Autvencoder Fha)
Autoencoder
Dilfosion Fia)
Dillusion
Dilfosionn +30
Dilfusion 138
Neural solver 1rp
Neural solver
T= 20
60
100
20
60
100
Figure 11. Examples of emulation at different compression rates (+)) for the Fuler dataset. In this simulation, the system has periodic boundary conditions.
34

---

Ground-rroth
Autoencoder
Dilfosioo
Grourdd-tral
Diffusson
Neural solver +
Neural sulver
20
60
Ground Ceeuun-trull trulh
Ground-trull Soudl
Autvencoder Hip)
Autoencoder
Dilfosion Fro
Dilluxion 1980
Dilfosiion F30
Dilfusion 180
Neural solver saa
Neural solver 1980
= 20
60
100
20
60
=
100
Figure 12. Examples of emulation different compression rates (+)) for the Fuler dataset. In this simulation, the system has periodic boundary conditions.
35

---

Ground-troth
Aufoencoder
Dilfcsioon
Ground-ttul
Diffasson
Neural solver +
Neural salver
20
20
Ground -tulh
Ground-trull -tuuh
Autvencoder faba
Autoenculer 1940
Dilfosion Faa
Dilluxion 1980
Dilfosionn 120
Dilfusion 1986
Neural solver sa
Neural solver 1980
20
60
100
20
60
100
Figure 13. Examples of emulation different compression rates (+) for the Fuler dataset. In this simulation, the system has periodic boundary conditions.
36

---

Ground trutis
Autoemooder "c
Defunsio '4
Ground
SORSA
Diffusion
Netral solver 21
MM mm
Neural solver 'ee
16
32
48
64
128
128
RMM MM
Ground truth
Ground
Autoenzoder 900
Autoencidee
M S
Deffusinn 230
Difiusioo 'noe
Difiusion 100
Duffuciinn 1981
w
20
Neural solver 5064
Neuaa solver
0
16
32
48
128
0
16
32
48
64
T=
128
Figure 14. Examples of emulation at different compression rates (+) for the Rayleigh Bénard dataset. In this simulation, the fluid is in a low-turbulence regime (Ra 10%).
Bayleigh-Bénard
37

---

Ground trutis
Autoenooder "cl
Defuntio 61
Diffusion "ct
Ground
Nearal solver 21
Neural solver 'eL
16
32
48
64
128
Ground truth
Ground
Autoenzoder 900
Autoenceder 100
Deffusinn Ex0
Difiusion 100
Diffusion 200
Duffucion 1984
Neural solver 2064
Neual solver 1011
0
16
32
48
64
128
16
32
48
64
128
Figure 15. Examples of emulation at different compression rates (+) for the Rayleigh Bénard dataset. In this simulation, the fluid is in a high-turbulence regime (Ra 108).
Bhyleigh-Bénnrd
38

---

Ground truths
Autoenooder "c
Deffusion
MMM
Diffucion
Ground
Netral solver 21
Neural solver '6L
64
128
Ground trith
Ground
Autoenzoder 900
Autoencedder
Diffuson "30
Difiusion 1006.
Difission 200
Duffucion 1981
Neural 5064
Neurll solver
0
16
128
16
32
48
64
128
Figure 16. Examples of emulation at different compression rates (+) for the Rayleigh Bénard dataset. In this simulation, the fluid is in a low-turbulence regime (Ra 10%).
Bayleigh-Bénard
39

---

Ground trutis
Autoenorder "0
Defuusion 64
Diffucion
Ground
Netral solver 21
Neural solver 'E.
16
64
128
Ground trith
Ground
900
Autoenceder (00
Diffuson -20
DifFusion 1006.
Difission 200
Diffucion 1004
Neural 2004
Neual solver
0
16
128
16
32
64
128
Figure 17. Examples of emulation at different compression rates (+) for the Rayleigh In this simulation, the fluid is in a high-turbulence regime (Ra 108).
Rayleigh-Bénard Bénard dataset.
40

---

Ground-troth
Autouncodrr
Dilffasoon
Diffusson
Neural salver
5
Ground Srulh
Ground-rrul
Autoencoder "1u
Antoenceder
Dilfosion F1
Diffusion
Dilfosion 300
Dilfosion
Neural solver Sa
Neural solver
T=5
7 25
45
25
45
Figure 18. Examples of emulation at different compression rates (+) for the TGC dataset. In this simulation, the initial density is low and the initial temperature is low (po = 0.445, To 10.0).
41

---

Ground-troth
Autouncoder
Dilfcsiion
Diffusion T
Neural salver
Ground trulh
Creund-rrulh
Auivencoder Tr
Antoenceder
Dilfosion Fii
Diffusion
Dilfosion se
Dilfosion
Neural solver su
Neural solver
25
45
25
45
Figure 19. Examples of emulation at different compression rates (+) for the TGC dataset. In this simulation, the initial density is medium and the initial temperature is high (po = 4.45, To 1000.0).
42

---

Ground-troth
Autonnoder
Dilfcsioo
Diffosson
Neural salver
Ground-trul Atudh
Ground
Autoencoder "1u2
Auloenceler
Dilfosion F1i
Diffusion
Dilfosion 50
Dilfasion
Neural solver Se
Neural solver
5
25
45
25
45
Figure 20. Examples of emulation at different compression rates (+) for the TGC dataset. In this simulation, the initial density is high and the initial temperature is low (po = 44.5, To = 10.0).
43

---

Ground-troth
Autouncoder
Dilfcsioon
Diffusion
Neural salver
Ground-trull -tulh
Ground
Aulvencoder -rg
Antoenceder
Dilfosion F1i2
Diffusion
Dilfosion 300
Dilfasion
Neural solver se
Neural solver
==5 5
7=25 25
45
25
45
Figure 21. Examples of emulation at different compression rates (+) for the TGC dataset. In this simulation, the initial density is high and the initial temperature is medium (po = 44.5, To 100.0).
44

---

D Latent space analysis
In this section, we conduct a short analysis of the learned latent representations. We are notably interested in the separability of the latent representation with respect to different parameteres A.
For our first experiment, select random initial state x! from the test split of the Euler dataset.
We compute the initial state z' Ex(x') for the +so auto-encoder. For each heat capacity emulator. E {1.2, 1.3, 1.4, 1.5, 1.6). we generate one latent trajectory 2 11: with the diffusion-baas Afterwards compute the Euclidean distance || 2/112 for each pair (May 7h) of heat capacities. We report the results Table 15 and represent the trajectories in Figure 22. As expected, trajectories with similar heat capacity ? are close each others.
For our second experiment, we compute the latent representations z' Ex(x") E 116x4x64 of the 64 auto-encoder for randomly selected states x' of the Rayleigh-Bénn Bénaad dataset. We then train small multi-layer perceptron (MIP) to predict the simulation parameters 0 (Rayleigh and Prandtl numbers) from the latent state's central token 2" [8. 2] TR,61, We extract the activations of the MLP's last layer and visualize them with t-SNE [120] in Figure 23. We observe that t-SNE [1 20] continuously separates latent states with respect to their parameters indicating that our auto-encoders learn to distinguish between different physics. We further validate this result by computing the pairwise Bures- Wasserstein distances [121] between the distributions central tokens 2² [8, 2] for different Rayleigh and Prandtl numbers. The distances, reported in Tables 16 and 17, are anti-correlated with the similarity of simulation parameters B.
Figure 22. Example emulated trajectories with different heat capacities 7 E {1.2, 1.3, L.,, 1.6) but starting at the same initial state x for the Euler dataset. The energy field is visualized instead of the density field to emphasize the differences.
45

---

Table 15. Euclidean distance matrix between emulated trajectories with different heat capacities E [1.2, 1.3, 1.4, 1.5, 1.6] but starting at the same initial state x² for the Euler dataset.
T=1 1
1.2
1.3
1.4
1,5
1.6
20
1.2
1.3
1.4
1.5
1.6
1.2
0.00
26.61
32.45
38.46
46.28
1.2
0.00
55.95
64.92
71.06
78.85
1.3
26.61
0.00
14.72
22.09
32.33
1.3
55.95
0.00
38.93
55.03
66.19
1.4
32.45
14.72
0.00
15.26
25.62
1.4
64.92
38.93
0.00
44.00
59.93
1.5
38.46
22.09
15.26
0.00
18.52
1.5
71.06
55.03
44.00
0.00
52.14
46.28
32.33
25.62
18.52
0.00
1.6
78.85
66.19
59.93
52.14
0.00
60
1.2
1.3
1.4
1.5
1.6
100
1.2
1.3
1.4
1.5
1.6
1.2
0.00
74.68
84.37
90.41
96.04
1.2
0.00
74.71
82.09
90.16
94.79
1.3
74.68
0.00
67.06
75.22
82.20
1.3
74.71
0.00
66.72
74.16
81.68
84.37
67.06
0.00
67.42
76.49
1.4
82.09
66.72
0.00
67.51
74.72
1.5
90.41
75.22
67.42
0.00
71.58
1.5
90.16
74.16
67.51
0.00
69.75
1.6
96.04
82.20
76.49
71.58
0.00
1.6
94.79
81.68
74.72
69.75
0.00
40
20
20
-0
-0
0.1
0.2
40
-40
1e+07
1.0
1c+08
2.0
-00
le- 10
10.0
200
Figure 23. t-SNE [120] visualization of the latent states z' -x.('") The projections are colored with respect to their Rayleigh (left) and Prandtl (right) numbers.
Table 16. Bures- Wasserstein distance matrix between the distributions of latent states 2¹ (xxt) with different Rayleigh numbers.
106
107
10a
10"
10
106
0.000
1.045
1.708
2.279
2.489
107
1.045
0.000
0.965
1.537
1.794
108
1.708
0.965
0.000
0.915
1.180
109
2.279
1.537
0.915
0.000
0.714
10
2.489
1.794
1.180
0.714
0.000
Table 17. Bures- Wasserstein distance matrix between the distributions of latent states zt (x)) with different Prandtl numbers.
0.1
0.2
0.5
1.0
2.0
5.0
10.0
0.1
0.000
1.367
2.042
2.631
3.244
3.884
4.210
0.2
1.367
0.000
1.269
1.839
2.381
3.007
3.331
0.5
2.042
1.269
0.000
0.986
1.479
2.093
2.398
1.0
2,631
1.839
0.986
0.000
0.930
1.472
1.766
2.0
3.244
2.381
1.479
0.930
0.000
0.988
1.251
5.0
3.884
3.007
2.093
1.472
0.988
0.000
0.711
10.0
4.210
3.331
2.398
1.766
1.251
0.711
0.000
46


Reviews:
[{"review_id": "A4liq30Axp", "reviewer": "Reviewer_pmvt", "summary": "This paper compares deterministic neural solvers with latent diffusion models (LDM) for solving partial differential equations (PDEs). Specifically, it compresses 2D image-grid physical simulation data into a latent space using an autoencoder, then reconstructs it through a diffusion model operating within this latent space. The study evaluates performance across varying compression rates and claims that LDM demonstrates robust performance and computational efficiency even at high compression rates.", "questions": "Please see the above weaknesses.", "limitations": "Yes", "rating": 2, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 1, "strengths_and_weaknesses": "**Strengths**\n- Recently, there has been active research applying generative models, especially latent diffusion models, in the field of physics simulation. This paper aligns with that trend by conducting an empirical comparison.\n\n- The systematic experimentation with varying compression rates is meaningful in terms of practical relevance.\n\n- Addressing the trade-off between efficiency and compression through latent space is an important consideration for large-scale simulations.\n\n- The attempt to empirically summarize the potential of LDM-based PDE simulations can be viewed positively.\n\n**Weaknesses**\n- Diffusion-based generative models have already been applied to PDE problems in the literature, so the novelty of this paper is very limited.\n\n- Baseline comparisons are highly restricted; additional comparisons with various state-of-the-art methods beyond deterministic neural solvers are needed.\n\n- The fundamental reasons why LDM exhibits superior performance over deterministic solvers across various compression rates are inadequately analyzed.\n\n- It is unclear what practical impact robustness across different compression rates would have in actual simulation applications.\n\n- The proposed method should be tested beyond simple 2D grid-based experiments, such as on irregular meshes or 3D simulations.\n\n- Further validation might be necessary to confirm whether the efficiency advantages can extend to real-world, large-scale problems, and discussion on this aspect is currently lacking.", "was_revised": true, "final_justification": "After the author rebuttal and discussions, I do agree that the paper provides a thorough empirical analysis of LDMs for physics emulation. My concern was that its scope is narrowly confined to LDMs, despite there are many generative models with competitive potential in this domain (e.g., GAN-based surrogates [1], VAE-based architectures [2], normalizing flows [3], score-based generative models [4][5], etc). This is why I thought the paper limits the generalizability of its findings, and asked for additional experiments with more generative models beyond LDMs. Even though the authors said \"*We do not train pixel-space diffusion baselines as they are too expensive...*\" but I believe this is not true because, as I wrote in the response, LDMs can be applied in a latent space obtained via an AutoEncoder, rather than pixel-space, which naturally means that other generative models like VAE, GAN and Normalizing Flow could also have been applied in that space.\n\nHowever, I truly agree that this paper delivers nice work within the boundary of LDMs for physics emulation. While I will not change my original rating, I can agree with acceptance if AC and the other reviewers recommend this paper.\n\n[1] Kastner, Patrick, and Timur Dogan. \"A GAN-based surrogate model for instantaneous urban wind flow prediction.\" Building and Environment 242 (2023).\n\n[2] Clavier, B., et al. \"Generative-machine-learning surrogate model of plasma turbulence.\" Physical Review E (2025).\n\n[3] Yang, Minglei, et al. \"Conditional pseudo-reversible normalizing flow for surrogate modeling in quantifying uncertainty propagation.\" Journal of Machine Learning for Modeling and Computing 6.4 (2025).\n\n[4] Lippe, Phillip, et al. \"Pde-refiner: Achieving accurate long rollouts with neural pde solvers.\" NeurIPS (2023).\n\n[5] Rühling Cachay, Salva, et al. \"Dyffusion: A dynamics-informed diffusion model for spatiotemporal forecasting.\" NeurIPS (2023).", "rebuttal": "Thank you for your review and the legitimate concerns you have raised. We are thankful for the opportunity to improve and clarify our manuscript.\n\n**W1 (Novelty)** We would like to emphasize that the main takeaway of our work is that **the accuracy of latent diffusion-based emulation is robust to a wide range of compression rates**. This result is unexpected and has practical relevance, as you acknowledge in your review. We are not making any claims regarding the novelty of our method and do not position our models as state-of-the-art.\n\n**W2 (Baselines)** At the time of submission, no emulation results for the datasets of TheWell were available in the literature, other than the preliminary baselines from TheWell's paper [38]. Training our own competing baselines would be unreasonably expensive and lead to fairness concerns. Nevertheless, we agree that a more comprehensive set of baselines would help the reader assess the significance of our results within the context of data-driven emulators. Fortunately, in the past months, several studies have published results for TheWell datasets, which we compile here and will add to the manuscript.\n\n| Source | Method | Dataset | Parameters | lead time -> VRMSE |\n|:---|:---:|:---:|:---:|:---:|\n| Ohana et al. [38] | FNO | Euler | ~20M | 6:12 -> 1.13 |\n| Ohana et al. [38] | U-Net | Euler | ~20M | 6:12 -> 1.02 |\n| Ours | ViT | Euler | 860M | 1:20 -> 0.138 |\n| Ours | LDM $\\div_{80}$ | Euler | 240M + 220M | 1:20 -> 0.075 |\n| Ohana et al. [38] | FNO | RB | ~20M | 6:12 -> 10+ |\n| Ohana et al. [38] | U-Net | RB | ~20M | 6:12 -> 10+ |\n| Nguyen et al. (2025) | PhysiX | RB | 4.5B | 2:8 -> 1.067 |\n| Wu et al. (2025) | COAST | RB | ? | 5 -> 0.282 |\n| Mukhopadhyay et al. (2025) | ViT + CSM | RB | 100M | 10 -> 0.140 |\n| Ours | ViT | RB | 860M | 1:20 -> 0.210 |\n| Ours | LDM $\\div_{64}$ | RB | 240M + 220M | 1:20 -> 0.162 |\n| Ohana et al. [38] | FNO | TGC | ~20M | 6:12 -> 3.55 |\n| Ohana et al. [38] | U-Net | TGC | ~20M | 6:12 -> 7.14 |\n| Mukhopadhyay et al. (2025) | ViT + CKM | TGC | 100M | 10 -> 0.527 |\n| Ours | LDM $\\div_{48}$ | TGC | 720M + 220M | 1:10 -> 0.297 |\n\n> Table A. Average VRMSE results from different studies using TheWell datasets. Even though our latent diffusion models (LDMs) outperform most published baselines, we emphasize that we are not making any claims regarding the novelty of our method and do not position our models as state-of-the-art, notably due to the discrepancies in parameters count, training and evaluation.\n\n**W3 (Explanations)** Indeed, our empirical findings raise many questions, which prompt the need for theoretical explanations. We believe that this is precisely the goal of empirical research, and hence a strength of our work. Nevertheless, we agree that the lack of explanation could leave the reader puzzled. We provide two non-exclusive hypotheses that could explain our findings:\n\n1. As explained in Section 5, many models, while accurate for short-term prediction, exhibit long-term instabilities as errors accumulate, pushing the predictions out of the training data distribution. Generative models, particularly diffusion models, partially address this problem [18, 19, 22–25] as they produce statistically plausible states, even when they diverge from the ground-truth solution.\n\n    However, distribution shift doesn't explain everything. At the first prediction step, before distribution shift can take effect, diffusion models are already slightly better than deterministic neural solvers. This could be explained by the iterative nature of DM sampling which allows the model to refine its prediction. Validating this hypothesis is an interesting avenue for future research.\n\n2. For many physics, and especially fluids, dynamics are mainly explained by large-scale/low-frequency content, while small-scale/high-frequency content influences the (very) long-term evolution of the system. Learning to process small-scale content and predict its influence on long-term dynamics might require very large networks, as well as huge amounts of data and compute. When the latter are limited, it becomes more tractable to focus on larger scales. Therefore, by discarding some of the small-scale content, latent diffusion models are able to learn and reproduce the dynamics more easily than pixel-space models.\n\nWe would appreciate your feedback on these hypotheses and whether we should include them in the manuscript.\n\n**W4 (Applications)** We see several use cases for using (large) compression in simulation applications. First, some fields (weather, astro) are collecting a lot more data that we can currently process or store. Our findings suggest that we could store or transfer this data in a compressed form, without losing (too much) information/predictability. Second, robustness to compression also means that practitioners do not have to worry (too much) about the compression rate they use: doubling or halving the compression rate will lead to similar emulation accuracy.\n\n**W5 (Dataset Diversity)** We follow your suggestion and add an additional 3D dataset to our study. This dataset (`turbulence_gravity_cooling` from TheWell [38]) models the interstellar medium as a turbulent fluid subject to gravity and radiative cooling. The 3d state of the system is discretized on a 64x64x64 grid. As for the other datasets, we train several auto-encoders with increasing compression rates (from 48 to 768) and latent emulators. The results (see table below) are consistent with the previous experiments and strengthen our conclusions: latent diffusion emulators are robust to compression. Latent neural solvers, conversely, are not as robust.\n\n| Model | VRMSE 01:10 | VRMSE 11:30 | VRMSE 31:50 |\n|:----|:---:|:---:|:---:|\n| Diffusion $\\div_{48}$ | 0.253 | 0.514 | 0.667 |\n| Diffusion $\\div_{192}$ | 0.256 | 0.506 | 0.647 |\n| Diffusion $\\div_{768}$ | 0.267 | 0.535 | 0.663 |\n| Neural solver $\\div_{48}$ | 0.259 | 0.598 | 0.834 |\n| Neural solver $\\div_{192}$ | 0.283 | 0.634 | 0.856 |\n| Neural solver $\\div_{768}$ | 0.339 | 0.708 | 0.94 |\n\n> Table B. Average VRMSE of latent-space emulation at different compression rates (÷) relative to the auto-encoded states $D_\\psi(E_\\psi(x^i))$ for the TGC dataset.\n\nUnfortunately, we cannot provide plots during the rebuttal, as per NeurIPS guidelines. Further results will be provided in the camera-ready version.\n\n**W6 (Real-World Problems)** Euler and Rayleigh-Bénard are some of the most challenging physics used in the field of data-driven emulators at the moment. They are commonly used for verifying/benchmarking numerical solvers and demonstrate behaviors often found in nature. The new 3D dataset (TGC) also has real-world applications and was originally generated for a domain-science publication by Hirashima et al. (2023).\n\n- Hirashima et al., 2023. \"3D-Spatiotemporal forecasting the expansion of supernova shells using deep learning towards high-resolution galaxy simulations\"\n\nWe believe that this rebuttal addresses most of your concerns and, therefore, kindly ask you to reconsider your score."}, {"review_id": "wEG0FCkUnM", "reviewer": "Reviewer_52YY", "summary": "The paper studies how physical emulators can exploit a latent space representationn for achieving an effective speedup. For this, the authors focus on two fluiddynamics datasets and compare different autoencoders (with different compression factors) for latent space representation, as well as diffusion models and a traditional neural solver for the prediction. As a baseline, they consider a neural solver in pixel space. The main result of the paper is to show empirically that latent space representations are robust to compression (up to reasonable compression factors) and that diffusion models are effective in reducing the compounding error problem. This study suggest practitioners to adopt latent space representations and diffusion models as backbone for future research in learning-based physical emulators.", "questions": "1. Can the authors elaborate on why the two datasets presented should be sufficient to present a case for the general family of physics emulators? Or otherwise back up their claims with experiments on additional datasets?\n2. Is the stride in the datasets introduced because of limitations of the methods during training? Is this something avoidable? Or can these models only work with sufficiently coarse time-steps?\n3. Can the authors assess the prediction results with a downstream application metric? E.g., if one is analyzing an airfoil, they might be interested in the resulting lift and drag coefficients. For the datasets you considered, are the predictions accurate enough to be used in downstream applications? This is a much more important metric than VRMSE, power spectrum similarity, etc.\n4. Why did the authors use 32x32 patches per token in the latent space model but 16x16 in the pixel space model?\n5. Why did the authors introduce the binary mask? Can they argue about the benefits and necessity of training in a different scenario of what the downstream application will be? They write a sentence about this yielding benefits, did they run some ablation studies?\n6. They mention the benefits for their sampling strategy: Can they provide evidence? Making these claims without evidence to back up may steer future research in directions not necessarily thoroughly tested.\n7. Can the binary mask train the model for “inpainting”? i.e., filling in between a coarser time-scale?\n8. Why did the authors not consider a pixel-space diffusion model? What would the performances be?", "limitations": "In general, the paper discuss its limitations and does not overpromise, which I appreciate. There are however some points that need some extra careful discussion, in my opinion. I believe that by discussing more openly these limitations would position the paper more strongly.\n\n1. The authors only consider two, 2D, fluiddynamics dataset (This point is mentioned by the author, but I think they should expand on it). They argue these are different, but fluiddynamics phenomena are also very rich. Since the study is empirical, providing sufficient empirical evidence is necessary for backing up the claims presented. Furthermore, the title of the work hints at physics emulation in general. The PDE governing fluiddynamics are general, but e.g. a primarily viscous fluid flow is excluded from the study.\n2. The quality of the prediction is assessed in terms of a few, different numerical metrics which make sense but are hard to evaluate: What does 0.01 VRMSE mean for the simulation considered? The qualitative images shown seem promising, but it is also clear that the models cut off higher frequency components. Are these important in downstream applications? These questions remain unanswered in the paper, and it remains unclear for which types of applications (if not all) it is a good idea to directly focus on latent space prediction without worrying (too much) about compression.\n3. Typically, physics simulator have tradeoff speed-accuracy. It is unclear in the paper where the physics emulator proposed are placing themselves in this axis. How do the performances compare with a lower accuracy, higher speed simulator? Is there a “knob” an user can tune in these physics emulators to reduce speed and improve the accuracy (and vice versa)? These questions remain open, leaving the study somewhat incomplete.\n4. There is a vast literature on latent representations (which the authors rightfully cite). But the paper main contribution is an empirical study of latent space for physics emulation, so not discussing and studying thesed representations seem incomplete.", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "The paper is very well written, focused on a single issue, and has extensive numerical comparisons. It tackles an important and timely problem, and tries to do it empirically but with rigor and only simplifications due to the computational requirements needed for such a task. Given the intensive computational effort, sharing these results could possibly benefit the community and sparing people to repeat the same ablation studies.\n\nAlthough I believe the problem is important and I appreciate the focus of the paper, I believe there are also several weaknesses to be addressed:\n\n1. Scope\n\n- Dataset Diversity: Only two 2D fluid‐dynamics datasets are used, insufficient to support claims about physics emulators in general; key regimes (e.g., viscous flows, 3D phenomena) are omitted.\n- Time-Resolution Constraints: The reason for temporal down‐sampling (striding) is unclear (whether it’s a model limitation or training convenience) and it’s unknown if finer time‐steps would break the approach. A thorough empirical validation should nail all these details.\n\n2. Method\n\n- Patch-Size discrepancy: Latent-space tokens use 32×32 patches, while pixel-space tokens use 16×16, with no rationale or ablation study provided.\n- Binary-mask: Introduced without experiments backing up the claimed benefit.\n- Sampling strategy claims: Benefits are asserted but unquantified, with no experiments to back up the sampling scheme’s impact.\n- Omission of pixel-space Diffusion: Only latent-space diffusion is explored; a pixel-space diffusion model baseline is never evaluated.\n\n3. Metrics\n\nLack of downstream-task metrics: The study relies on VRMSE and power-spectrum similarity without assessing application-relevant quantities (e.g., in airfoil, lift/drag), so practical adequacy is unknown.\n\n4. Comparisons and baselines\n\nSpeed–accuracy trade‐off: No positioning of the emulator on the fidelity vs. runtime axis; no comparison to lower‐fidelity, faster simulators or a mechanism (“knob”) to adjust speed vs. accuracy. These are instead readily available in physics simulators.\n\n5. Literature and context\n\nLatent-representation survey: Despite citing latent-space literature, there is no critical engagement or experimental comparison of different representation strategies, leaving the empirical study feeling incomplete.", "was_revised": true, "final_justification": "The authors thoroughly addressed all my concerns. The paper is an empirical study (very clear in the title). Considering it as such, it would be a great contribution to NeurIPS and it is very well developed.\nI strongly recommend acceptance", "rebuttal": "Thank you for the quality of your review and interest in our work. We greatly appreciate that you read our manuscript with such attention to details. Your concerns are legitimate and we will address them here and in the manuscript.\n\n**Q1 & L1 (Dataset Diversity)** We follow your suggestion and add an additional 3D dataset to our study. This dataset (`turbulence_gravity_cooling` from TheWell [38]) models the interstellar medium as a turbulent fluid subject to gravity and radiative cooling. The 3d state of the system is discretized on a 64x64x64 grid. As for the other datasets, we train several autoencoders with increasing compression rates (from 48 to 768) and latent emulators. The results (see table below) are consistent with the previous experiments and strengthen our conclusions: latent diffusion emulators are robust to compression. Latent neural solvers, conversely, are not as robust.\n\n| Model | VRMSE 01:10 | VRMSE 11:30 | VRMSE 31:50 |\n|:----|:---:|:---:|:---:|\n| Diffusion $\\div_{48}$ | 0.253 | 0.514 | 0.667 |\n| Diffusion $\\div_{192}$ | 0.256 | 0.506 | 0.647 |\n| Diffusion $\\div_{768}$ | 0.267 | 0.535 | 0.663 |\n| Neural solver $\\div_{48}$ | 0.259 | 0.598 | 0.834 |\n| Neural solver $\\div_{192}$ | 0.283 | 0.634 | 0.856 |\n| Neural solver $\\div_{768}$ | 0.339 | 0.708 | 0.94 |\n\n> Table B. Average VRMSE of latent-space emulation at different compression rates (÷) relative to the auto-encoded states $D_\\psi(E_\\psi(x^i))$ for the TGC dataset.\n\nUnfortunately, we cannot provide plots during the rebuttal, as per NeurIPS guidelines. Further results will be provided in the camera-ready version.\n\nWe would also like to mention that the Rayleigh-Bénard dataset models a **viscous** fluid.\n\n**Q2 (Time-Resolution Constraints)** We apologize for the confusion and thank you for pointing this out. The time stride was introduced strictly for convenience: with larger strides, emulation is faster. We conduct the new experiment (see above) without time stride ($\\Delta = 1$). We also started to retrain our models with $\\Delta = 1$. At the time of this rebuttal, only some Rayleigh-Bénard results are available.\n\n| Model | VRMSE 01:20 | VRMSE 21:60 | VRMSE 61:160 |\n|:----|:---:|:---:|:---:|\n| Diffusion $\\div_{64}$ | 0.18 | 0.527 | 0.654 |\n| Diffusion $\\div_{256}$ | 0.19 | 0.528 | 0.647 |\n| Diffusion $\\div_{1024}$ | 0.22 | 0.536 | 0.652 |\n\n> Table C. Average VRMSE of latent-space emulation at different compression rates (÷) for the Rayleigh-Bénard dataset and with a time stride $\\Delta = 1$. The results are very similar to $\\Delta = 4$ with $\\Delta = 1$, despite the larger ($\\times 4$) number of rollout steps.\n\n**Q3 & L2 (Metrics)** (V)RMSE and power spectrum analysis are standard metrics in the data-driven emulation literature. The spread-skill ratio is also standard in Earth sciences. We are not aware of other widespread metrics compatible with the datasets we consider. We would appreciate if you could provide downstream metrics that we could compute in our datasets.\n\nRegarding the high-frequency content, this is definitely true. The use of a deterministic decoder for the auto-encoder leads to a lack of small-scale features. We further evaluate our latent models relative to the auto-encoded states $D_\\psi(E_\\psi(x^i))$ to demonstrate that this lack is mainly due to the decoder, and not the latent emulation.\n\nWe see many applications for latent (compressed) emulation. In weather forecasting, as the lead time increases, large-scale trends and features become more relevant for meteorologist. In engineering/manufacturing, large deformations are typically more important than small ones. In astrophysics, you could care more about total light flux than flux variations over a star surface.\n\n**Q4 (Patch-size Discrepancy)** This discrepancy comes from our desire to give strictly more capacity to the pixel-space models than latent-space models. With 16x16 patches, the pixel-space model has more tokens ($\\frac{HW}{16^2}$ vs $\\frac{HW}{32^2}$) and more dimensions per token (2048 vs 1024). We also train the pixel-space models with double the compute and for up to 2 times longer. As such, one cannot argue in good faith that our assessment was unfair against the pixel-space baseline.\n\nWe would also like to point out that ViTs typically perform better with smaller patches, as demonstrated by Wang et al. (2025) and Mukhopadhyay et al. (2025).\n\n- Wang et al. (2025). \"Scaling Laws in Patchification: An Image Is Worth 50,176 Tokens And More\"\n- Mukhopadhyay et al. (2025). \"Controllable Patching for Compute-Adaptive Surrogate Modeling of Partial Differential Equations\"\n\n**Q5 (Binary Mask)** The binary mask is necessary to perform multi-task conditioning. In our case, the tasks are forward and backward temporal prediction, with $c \\geq 1$ context states. We vary $c$ in our experiments to determine whether context length has an impact on emulation accuracy. The results, in Table 7, indicate that it does not.\n\nRegarding the benefits, our claim comes from our preliminary experiments and several references that report similar benefits while using randomized/variable masking strategies:\n\n- Zheng et al. (2023). \"Fast Training of Diffusion Models with Masked Transformers\"\n- Voleti et al. (2022). \"MCVD: Masked Conditional Video Diffusion for Prediction, Generation, and Interpolation\"\n- Harvey et al. (2022). \"Flexible Diffusion Modeling of Long Videos\"\n\n**Q7 (Inpainting)** We do not train for this in this study, but masked conditioning can be used for inpainting. However, we note that this task can also be tackled **without any additional training** in the case of diffusion models. As mentioned in Section 4, it is possible to guide the model with partial observations, which includes spatial and temporal \"inpainting\". We will provide more examples of guidance (e.g. inpainting, super resolution) in the camera-ready version.\n\n**Q6 (Sampling Strategy)** We apologize for the confusion. The Adams-Bashfort sampling method is **not novel**. It was proposed and extensively tested by Zhang et al. [74]. This method is also equivalent to the widespread \"linear multi-step\" method proposed by Katherine Crowson in their `k-diffusion` repository. We believe that this sampler is sufficiently established that we don't need to evaluate it against others samplers (notably DDPM, DDIM and Heun).\n\n**Q8 (Pixel-space Diffusion)** We do not train pixel-space diffusion baselines as they are too expensive (tens of thousand of GPU hours) to train and evaluate at the resolution and scale of our datasets. We agree that this is a limitation of our work and will discuss it explicitly in the manuscript. We note that our code (which we provided with the submission and will be released publicly) already supports training a pixel-space diffusion model. With enough resources, a third party could easily train these models.\n\n**L3 (Speed/Accuracy Tradeoff)** Indeed we are not comparing our models against numerical solvers at different space and/or time resolutions. We agree that this would be a valuable addition to this study. Unfortunately, we do not have the expertise to implement these solvers ourselves. We will discuss this limitation in the manuscript.\n\nWe would like to note, however, that neural networks have other advantages than being fast emulators. They enable other tasks which are impossible with traditional numerical solvers. Notably, neural networks are differentiable, which can be very useful in the case of inverse problems. Diffusion-based emulators can also be guided with partial/noisy observations. Perhaps most importantly, data-driven emulators do not require knowledge of the underlying physics/PDEs, which is extremely valuable in real-world settings.\n\nRegarding the \"knob\" to reduce speed and improve the accuracy, there isn't one currently. However, that is a very interesting research question. We believe this could be achieved with a tokenizer with a variable number of tokens such as FlexTok (Bachmann et al., 2025) as auto-encoder. Our methodology could then be applied on this \"flexible latent space\" and we would modify the computational cost by choosing the number of tokens.\n\n- Bachmann et al., 2025. \"FlexTok: Resampling Images into 1D Token Sequences of Flexible Length\"\n\n**L4 (Latent Representation)** As we explain in the discussion, a broader study of different latent representation strategies would be a valuable addition to this study. However, we believe that a comprehensive assessment of all latent representation methods is vastly outside of the scope of a modest academic conference paper and would require 10 to 100 times more time and compute. Therefore, in this work, we focus on the most widespread latent representation, that is the one learned by an auto-encoder.\n\n**Formatting** As per the paper formatting instructions: \"The authors may optionally choose to include some or all of the technical appendices in the same PDF [...]\"\n\nWe hope that our answers and additional experiments address your concerns. If so, we kindly ask you to reconsider your score."}, {"review_id": "uQgIDMXi4u", "reviewer": "Reviewer_v3JQ", "summary": "The authors propose the use of latent diffusion models as emulators for physical systems governed by partial differential equations (PDEs). The primary objective of the paper is to investigate whether compression impacts the emulation capacity. The main finding is that the emulation quality remains high even in high-compression regimes, even when the autoencoder provides poor reconstructions. The authors evaluate their methods across several canonical PDES and provide qualitative and quantitative results.", "questions": "- Why did you not include comparisons to more established neural solvers such as Fourier Neural Operators (FNO), U-Nets, or Physics-Informed Neural Networks (PINNs)? \n\n- It would be good to understand what physical properties are preserved in the latent space. For instance, are the latent variables learning specific quantities, such as velocity, pressure, etc., and are these preserved despite the poor reconstruction?\n\n- Given that the decoder can hallucinate perceptually plausible outputs, how can we be sure that the generated fields are physically meaningful? Have you considered incorporating physics-based constraints or projection steps post-decoding? This also raises the question of how this model performs under different boundary conditions. Do I need to retrain the model, or does it generalize?\n\n- Can you provide an intuitive explanation or a theoretical perspective of why the emulation remains strong despite poor reconstructions?\n\n- Latent diffusion models can be very slow to train. Can you comment on the generalization capabilities of your model and how it would hold up in practice against more traditional alternatives?\n\n- Your model generates statistically correct trajectories. In applications requiring deterministic repeatability (e.g. control), how would you address the stochastic nature of diffusion outputs?", "limitations": "See questions and weaknesses.", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n- I believe that the experiments are well-motivated and extensive. In this regard, the paper's main finding, that good emulation does not require good reconstruction, is important, as it is, at least in my opinion, unexpected. \n- The paper is well written and quite easy to understand, which is great. The numerical experiments/applications are interesting and validate the research question.\n### Weaknesses\n- I believe a major weakness of this approach is the reduced interpretability, as the method does not incorporate any explicit physical knowledge. This issue is crucial when dealing with safety-critical applications, for instance, in medical imaging, where certain applications may lead to physically invalid outputs. \n- This is compounded with some missing baselines, for instance, https://arxiv.org/abs/2305.07671 proposes a PINN hybrid approach it would be good to compare against the physics-informed models counterparts.", "was_revised": true, "final_justification": "I believe that the method is sound, and the authors in the rebuttal went beyond the original metrics to prove that the learned latents are meaningful. I recommend acceptance.", "rebuttal": "Thank you for the quality of your review. We greatly appreciate your interest in our findings. We are thankful for the opportunity to improve and clarify our manuscript.\n\n**W1 (Interpretability)** We believe that the lack of interpretability is a limitation of data-driven emulators, rather than a weakness of our work. Whether interpretability is necessary for emulators, is an open debate, which we believe is outside of the scope of this work. An example where interpretability is not necessary is weather forecasting, where speed and accuracy are much more relevant for consumers.\n\n**W2 & Q1 (Missing Baselines)** At the time of submission, no emulation results for the datasets of TheWell were available in the literature, other than the preliminary baselines from TheWell's paper [38]. Training our own competing baselines would be unreasonably expensive and lead to fairness concerns. Nevertheless, we agree that a more comprehensive set of baselines would help the reader assess the significance of our results within the context of data-driven emulators. Fortunately, in the past months, several studies have published results for TheWell datasets, which we compile here and will add to the manuscript.\n\n| Source | Method | Dataset | Parameters | lead time -> VRMSE |\n|:---|:---:|:---:|:---:|:---:|\n| Ohana et al. [38] | FNO | Euler | ~20M | 6:12 -> 1.13 |\n| Ohana et al. [38] | U-Net | Euler | ~20M | 6:12 -> 1.02 |\n| Ours | ViT | Euler | 860M | 1:20 -> 0.138 |\n| Ours | LDM $\\div_{80}$ | Euler | 240M + 220M | 1:20 -> 0.075 |\n| Ohana et al. [38] | FNO | RB | ~20M | 6:12 -> 10+ |\n| Ohana et al. [38] | U-Net | RB | ~20M | 6:12 -> 10+ |\n| Nguyen et al. (2025) | PhysiX | RB | 4.5B | 2:8 -> 1.067 |\n| Wu et al. (2025) | COAST | RB | ? | 5 -> 0.282 |\n| Mukhopadhyay et al. (2025) | ViT + CSM | RB | 100M | 10 -> 0.140 |\n| Ours | ViT | RB | 860M | 1:20 -> 0.210 |\n| Ours | LDM $\\div_{64}$ | RB | 240M + 220M | 1:20 -> 0.162 |\n| Ohana et al. [38] | FNO | TGC | ~20M | 6:12 -> 3.55 |\n| Ohana et al. [38] | U-Net | TGC | ~20M | 6:12 -> 7.14 |\n| Mukhopadhyay et al. (2025) | ViT + CKM | TGC | 100M | 10 -> 0.527 |\n| Ours | LDM $\\div_{48}$ | TGC | 720M + 220M | 1:10 -> 0.297 |\n\n> Table A. Average VRMSE results from different studies using TheWell datasets. Even though our latent diffusion models (LDMs) outperform most published baselines, we emphasize that we are not making any claims regarding the novelty of our method and do not position our models as state-of-the-art, notably due to the discrepancies in parameters count, training and evaluation.\n\nThank you for the PINN reference you provided, we will read it with attention. PINNs are very interesting and a developing area of research. However, we believe that it is unfair to compare PINNs with purely data-driven emulators, which do not require any knowledge of the physics.\n\n**Q2 & Q4 (Insights)** Indeed, our empirical findings raise many questions, which prompt the need for theoretical explanations. We believe that this is precisely the goal of empirical research, and hence a strength of our work. Nevertheless, we agree that the lack of explanation could leave the reader puzzled. We provide two non-exclusive hypotheses that could explain our findings:\n\n1. As explained in Section 5, many models, while accurate for short-term prediction, exhibit long-term instabilities as errors accumulate, pushing the predictions out of the training data distribution. Generative models, particularly diffusion models, partially address this problem [18, 19, 22–25] as they produce statistically plausible states, even when they diverge from the ground-truth solution.\n\n    However, distribution shift doesn't explain everything. At the first prediction step, before distribution shift can take effect, diffusion models are already slightly better than deterministic neural solvers. This could be explained by the iterative nature of DM sampling which allows the model to refine its prediction.\n\n2. For many physics, and especially fluids, dynamics are mainly explained by large-scale/low-frequency content, while small-scale/high-frequency content influences the (very) long-term evolution of the system. Learning to process small-scale content and predict its influence on long-term dynamics might require very large networks, as well as huge amounts of data and compute. When the latter are limited, it becomes more tractable to focus on larger scales. Therefore, by discarding some of the small-scale content, latent diffusion models are able to learn and reproduce the dynamics more easily than pixel-space models.\n\nValidating these hypotheses is an interesting avenue for future research. We would appreciate your feedback on these hypotheses and whether we should include them in the manuscript.\n\n**Q3 (Hallucinations)** There are two ways our models can \"hallucinate\": either the decoder introduces artifacts that are not plausible or the latent emulator generates implausible dynamics.\n\n1. Our decoder is not trained with a perceptual loss. We therefore expect the decoding to have a \"smoothing\" behavior on the fields, which we observe in practice (high-frequency cutoff). Due to this degradation of small-scale features, some physicality is indeed lost.\n\n    To address this issue, we considered training a generative decoder during this project. However, this would not increase the amount of information compressed in the latent space. The additional small-scale details would be hallucinated by the decoder, without contributing to emulation accuracy, which would be misleading for humans. We never implemented this idea.\n\n    We did not consider introducing physics-based objectives during the training of the auto-encoder, as we assume that the physics are unknown. It is however an interesting research direction.\n\n2. Our latent diffusion-based emulators are not perfect and often diverge from the ground-truth solution. In this light, it can be considered that individual trajectories are \"hallucinated\". However, our evaluation (using the spread-skill ratio) demonstrates that, as an ensemble, the trajectories are well calibrated, meaning that they generally cover plausible scenarios.\n\n**Q5 (Generalization)** You are perfectly right, we observed that (L)DMs need to be trained for a long time, and keep improving in the later training stages. Whether these models generate to out-of-distribution tasks is therefore very relevant. We propose to add an experiment where the models are used to emulate with parameters $\\theta$ that are not present in the training distribution. Our preliminary results suggest that the models generalize to new parameters $\\theta$ when these are close to or in between parameters seen during training. However, they fail for completely new parameters, including new types of boundary conditions.\n\n**Q6 (Repeatability)** Although diffusion models are generative models, there exist deterministic sampling strategies for them, which always lead to the same output when starting from the same initial noise. In fact, our sampling scheme (Adams-Bashforth, see Section 3.3) is one such strategy.\n\nWe believe that this rebuttal addresses most of your concerns and, therefore, kindly ask you to reconsider your score."}, {"review_id": "OTPgauSY9y", "reviewer": "Reviewer_mPpC", "summary": "This paper investigates the potential of *latent diffusion models* for 2-dimensional fluids emulations. For the aforementioned problem setting, the authors demonstrate 1) latent space models match or exceed the performance of pixel (raw data) space models and 2) diffusion significantly improves emulation performance compared to deterministic point-regression neural solvers and 3) latent modelling brings considerable speedup at inference time. They also investigate the reconstruction ability of autoencoders for various compression rates.", "questions": "In addition to the weaknesses already mentioned (which affect my rating), I have a few questions that will not affect my rating, but I hope that the authors can clarify to me.\n\n1. To what extent are such neural emulators actually useful for current applications? Can you provide an example of a concrete application that would benefit from the trade-off in accuracy for speed?\n2. In the case of low-resolution conditioned generation, would it be possible to use a simulator to predict a spatially-coarse-grained but accurate sequence of states and then upscale it using a diffusion model?\n3. How do neural emulators perform at \"OOD\" generalisation like on $\\theta$ or $i$ not seen during training time? Do they collapse? Time extrapolation seems like an important application but the paper does not even consider it.", "limitations": "Yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Note that I am familiar with diffusion but not physics emulation, so I will only assign low confidence to this review. I also welcome the authors to correct any mistakes/misunderstandings on my part that appear in this review.**\n\n## Strengths\n- The writing is clear, straightfoward and easy to follow/understand (for the most part). \n- The takeaways seem to have clear impact in the field the paper is targeting. Various design choices are explored, leading ultimately to the advantages of the proposed latent diffusion approach being clearly presented. (Although I am personally not familiar with this application).\n## Weaknesses\n1. The paper lacks a clear intuition as to why diffusion is superior for this application (although perhaps it lies in the literature). Could the authors provide such an intuition?\n2. This application seems very similar to video generation, especially considering the data is spatially 2D dimensional and the diffusion model is not trained with any \"physics-informed\" inductive biases. The authors would do better to link the performance of LDMs for video generation to this work. In fact, perhaps answers to why diffusion is effective for physics emulation can be found in the literature for image/video generation ([see this blog post](https://sander.ai/2024/09/02/spectral-autoregression.html]) and its accompanying references).\n3. Although the authors present the result that the compression level of the autoencoder doesn't seem to affect diffusion model performance much, they don't suggest any potential reasons. Could this be due to e.g. *emulation* VRMSE being dominated by lower frequency information (larger blocks of \"colour\") which arises from the diffusion model and not the autoencoder? I would encourage the authors to provide at least an informed suggestion to provide some additional insight to the reader.\n4. **Minor**:\n    1. Compression rate would be better visualised using a colour map that changes in intensity according to the compression value. Figures 5/7 are difficult to interpret currently.\n    2. The use of \"well calibrated\" in conjunction with \"uncertainty\" L249 is a bit confusing, since it refers typically to quite a specific concept in the uncertainty literature. I would suggest using different wording.\n    3. Various equations would benefit from some underbraces for readability e.g. indicating which latents are noised and which are clean in Eq. 7.\n    4. The use of eq 5. seems somewhat arbitrary (although the use of KL regularisation is similarly arbitrary in my experience). Was this based on eyeballing a histrogram of the element values of $z$? Did the authors ablate this choice? If I recall correctly DCAE doesn't use KL regularisation or this tanh-style squashing.\n    4. Figures 1 and 2 are a little unclear. It would be better to clearly separate the emulation timesteps to better illustrate how the masking is applied, as well as which frames serve as conditioning. For example, in Figure 1 it is unclear whether $z^{5:8}$ is conditioned on  $z^{1:4}$ or just $z^{4}$. My understanding is that it is the latter, although I wouldn't be surprised if the former produced better results.", "was_revised": true, "final_justification": "The authors clarified my concerns, however, as I am not familiar with the application (although I am familiar with the algorithms employed), I do not feel like I can give a firm recommendation. As such I will keep my score of weak accept.", "rebuttal": "Thank you for the pertinence of your review. Your feedback is very valuable, especially since your background gives us different perspectives on our work.\n\n**W1 & W3 (Insights)** Indeed, our empirical findings raise many questions, which prompt the need for theoretical explanations. We believe that this is precisely the goal of empirical research, and hence a strength of our work. Nevertheless, we agree that the lack of explanation could leave the reader puzzled. We provide two non-exclusive hypotheses that could explain our findings:\n\n1. As explained in Section 5, many models, while accurate for short-term prediction, exhibit long-term instabilities as errors accumulate, pushing the predictions out of the training data distribution. Generative models, particularly diffusion models, partially address this problem [18, 19, 22–25] as they produce statistically plausible states, even when they diverge from the ground-truth solution.\n\n    However, distribution shift doesn't explain everything. At the first prediction step, before distribution shift can take effect, diffusion models are already slightly better than deterministic neural solvers. This could be explained by the iterative nature of DM sampling which allows the model to refine its prediction.\n\n2. For many physics, and especially fluids, dynamics are mainly explained by large-scale/low-frequency content, while small-scale/high-frequency content influences the (very) long-term evolution of the system. Learning to process small-scale content and predict its influence on long-term dynamics might require very large networks, as well as huge amounts of data and compute. When the latter are limited, it becomes more tractable to focus on larger scales. Therefore, by discarding some of the small-scale content, latent diffusion models are able to learn and reproduce the dynamics more easily than pixel-space models.\n\nValidating these hypotheses is an interesting avenue for future research. We would appreciate your feedback on these hypotheses and whether we should include them in the manuscript.\n\n**W2 (Link to Video Generation)** There are definitely links with the image and video generation literatures. We already make connections in the discussion, especially regarding latent image generation which works much better than in pixel-space. Thank you for the blog post, we read it with attention. Note that we already refer to a blog post by Dieleman [56], although a different one. We will further look into the video diffusion model literature regarding the effects of compression.\n\nHowever, there is one very important difference between dynamical systems and videos. Dynamical systems are Markovian, meaning that knowing the current state is enough to predict the future (or the past). Therefore, there is no point in remembering the history, unlike for videos where it is very important.\n\n**W4.1 (Color map)** Thank you for your feedback. We will try to use a color map that changes in intensity.\n\n**W4.2 (Well Calibrated)** We are not sure to understand what you find confusing, but we are appreciate this opportunity to clarify our manuscript. Could you explain your perspective more precisely?\n\nFrom our perspective, the spread-skill ratio is a measure of calibration. Spread represents the uncertainty of the model. Skill represents the error of the model. For a well calibrated model, spread should be equal to skill (and the ratio be 1), meaning that the model should be as uncertain as it is wrong.\n\n**W4.3 (Equations Readability)** We will make sure to improve the readability of the equations in the camera-ready version.\n\n**W4.4 (Latent Saturation)** Yes, this is quite arbitrary. We tried a few saturating functions (which you can find in the code) in our preliminary experiments and all worked similarly. We used this one as it has a smooth derivative and saturates very slowly, which prevents vanishing gradients. $B = 5$ is chosen such that the values range between $-5$ and $5$, which is close to the range of a standard Gaussian. With these choices the mean and variance of the latents are close to 0 and 1 (empirically). Therefore, we do not need to re-scale the latents before training the diffusion model. Conversely, DCAE does not use this saturating function, but relies on rescaling the latents to train the latent diffusion model.\n\n**W4.5 (Figures)** Thank you again for your feedback. Indeed, these figures could be misleading. We will improve them in the camera-ready version, but cannot provide them in this rebuttal, as per NeurIPS guidelines.\n\nIn most of our experiments, the latent emulators are only conditioned on a single past frame. However thanks to our masking strategy, we are able to test whether conditioning on several past frames would help. We already do so in Table 7, where we vary the number of context frames from 1 to 3. We find that context length has little to no impact on emulation accuracy with our latent models.\n\nThis is likely due to dynamical systems being Markovian, as explained previously. In a Markov chain, knowing the current state is enough to predict the future. One could argue, however, that in the latent space, the system is not Markovian anymore as some information is lost. The Markov chain becomes a hidden Markov model (HMM). However, the fact that emulation accuracy is robust to compression and independent from context length, contradicts this argument.\n\n**Q1 (Applications)** There exists many applications for fast emulators. Common ones are weather forecasting and aerodynamic design (cars, airplanes, ...), for which traditional numerical solvers are extremely slow and expensive. Anything relying on simulations for real-time decision making too.\n\nWe would like to note, however, that neural networks have other advantages than being fast emulators. They enable other tasks which are impossible with traditional numerical solvers. Notably, neural networks are differentiable, which can be very useful in the case of inverse problems. Diffusion-based emulators can also be guided with partial/noisy observations. Perhaps most importantly, data-driven emulators do not require knowledge of the underlying physics/PDEs, which is extremely valuable in real-world settings.\n\n**Q2 (Upscaling)** This is called \"downscaling\" in numerical weather forecasting. This generally doesn't work well because for most fluid systems, numerical solvers fail at low resolution. Therefore \"spatially-coarse-grained but accurate\" numerical simulations typically do not exist. Besides, upscaling with a diffusion model could lead to \"good looking\" states, but the additional high-frequencies would not be linked to the past states.\n\n**Q3 (Generalization)** Whether our models generate to out-of-distribution tasks is very relevant. We propose to add an experiment where the models are used to emulate with parameters $\\theta$ that are not present in the training distribution. Our preliminary results suggest that the models generalize to new parameters $\\theta$ when these are close to or in between parameters seen during training. However, they fail for completely new parameters, including new types of boundary conditions. \n\nRegarding time extrapolation, unrolling the trajectory past the training length ($i > L$) works quite well qualitatively, although we are not able to assess it quantitatively as we don't have access to data past this length.\n\nWe hope that our answers address your concerns. If so, we kindly ask you to reconsider your score."}]

Author rebuttal:

Meta-review: The paper introduces an empirical study of the effect of the latent space compression of the efficiency of physical emulators. The main finding is that the emulation quality remains high even in high-compression regimes.

Initially, the reviewers appreciated the topic of the paper, clear writing and found the ideas presented in the paper very relevant to the research community. At the same time, several important concerns were brought up that included validation breadth, paper positioning and asked a fair amount of clarification question. With the rebuttal and post rebuttal discussion the authors were able to address almost all of the reviewer’s concerns. The only undressed concern is about the scope of the empirical study (validation breadth). Reviewer pmvt considers the validation of the observations with one generative model (LDM) too narrow and recommends to reject. Reviewers 52YY and v3JQ, agree with the argument of limited validation however, they argue that point out that the study is already very complete and of interest for NeurIPS community. AC would like to point out that the paper is well presented and reduces the scope of model exploration very early on — in the title, as such the AC agrees with the positive reviewers and thus recommends to accept. The authors are kindly asked to include the additions from the discussion and rebuttal in the camera ready (including the promised visualizations).
Decision: Accept (poster)