arxiv: 2604.17566 · v1 · submitted 2026-04-19 · 📡 eess.SY · cs.LG· cs.SY· physics.flu-dyn

Recognition: unknown

Target Parameterization in Diffusion Models for Nonlinear Spatiotemporal System Identification

Achraf El Messaoudi , Noureddine Khaous , Karim Cherifi

Authors on Pith no claims yet

Pith reviewed 2026-05-10 05:30 UTC · model grok-4.3

classification 📡 eess.SY cs.LGcs.SYphysics.flu-dyn

keywords diffusion modelsnonlinear system identificationturbulent flowtarget parameterizationspatiotemporal systemsrollout stabilityclean-state prediction

0 comments

The pith

Clean-state prediction in diffusion models improves long-term rollout stability for turbulent nonlinear systems over noise or velocity targets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how the target chosen during training of a diffusion model affects its ability to identify and forecast nonlinear systems that evolve in space and time. Using turbulent flow simulations as the test case and a straightforward patch-based transformer, it compares clean-state prediction against the more common noise-prediction and velocity-prediction objectives. Clean-state targets produce forecasts that remain stable over many steps and accumulate less error at long horizons, with the gap widening as each spatial patch contains more information. This design choice matters because turbulent regimes quickly amplify small prediction mistakes, limiting how far classical or standard diffusion approaches can be trusted for real engineering systems.

Core claim

In diffusion-based identification of nonlinear spatiotemporal systems, clean-state prediction as the modeling target yields consistently better rollout stability and lower long-horizon error than velocity- or noise-based objectives, and the benefit grows with increasing per-token dimensionality.

What carries the argument

Target parameterization (clean-state versus noise or velocity prediction) inside the diffusion training objective, which directly shapes how the model learns to reverse the forward noise process and thereby controls error accumulation across iterative rollouts on high-dimensional spatial fields.

Load-bearing premise

That turbulent-flow simulation with a simple patch-based transformer isolates the effect of target choice without confounding influences from architecture details or data specifics.

What would settle it

Demonstrating on any other nonlinear spatiotemporal system that clean-state prediction produces equal or higher long-horizon error than noise or velocity prediction would falsify the reported advantage.

Figures

Figures reproduced from arXiv: 2604.17566 by Achraf El Messaoudi, Karim Cherifi, Noureddine Khaous.

**Figure 1.** Figure 1: Geometric assumption. Under the manifold assumption, physical states x ∈ X concentrate near a lowdimensional subset M ⊂ X , while injected noise ϵ ∼ N (0, I) is isotropic in X . evaluating the identified surrogate in free-running simulation, which is the regime relevant for long-horizon prediction and downstream control-oriented use. A central difficulty in this setting is compounding error: small inaccu… view at source ↗

**Figure 2.** Figure 2: Method overview. A self-contained patch-token transformer maps the noised next-state zτ to a field-level prediction, conditioned on the history window xt−k+1:t, diffusion time, and parameters θ. A. Self-contained patch-token transformer backbone [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Sum of Global MSE across dimensions. The dashed [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 5.** Figure 5: Spectral fidelity of long rollouts. Temporal frequency spectrum of the downstream probe signal (log scale), comparing generated rollouts to the reference simulation. noise prediction. This advantage becomes more pronounced as per-token dimensionality increases, highlighting target parameterization as a key design choice for diffusion-based spatiotemporal modeling. Several directions merit further study. F… view at source ↗

read the original abstract

Machine learning is becoming increasingly important for nonlinear system identification, including dynamical systems with spatially distributed outputs. However, classical identification and forecasting approaches become markedly less reliable in turbulent-flow regimes, where the dynamics are high-dimensional, strongly nonlinear, and highly sensitive to compounding rollout errors. Diffusion-based models have recently shown improved robustness in this setting and offer probabilistic inference capabilities, but many current implementations inherit target parameterizations from image generation, most commonly noise or velocity prediction. In this work, we revisit this design choice in the context of nonlinear spatiotemporal system identification. We consider a simple, self-contained patch-based transformer that operates directly on physical fields and use turbulent flow simulation as a representative testbed. Our results show that clean-state prediction consistently improves rollout stability and reduces long-horizon error relative to velocity- and noise-based objectives, with the advantage becoming more pronounced as the per-token dimensionality increases. These findings identify target parameterization as a key modeling choice in diffusion-based identification of nonlinear systems with spatial outputs in turbulent regimes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Clean-state prediction gives better rollout stability than noise or velocity targets in their turbulent-flow experiments, but the single setup leaves room for confounding.

read the letter

The paper tests a straightforward design choice in diffusion models for identifying high-dimensional nonlinear systems. They fix a patch-based transformer and run it on turbulent flow simulation data, then compare three prediction targets: the clean state, velocity, and noise. The reported outcome is that clean-state prediction produces more stable long-horizon rollouts and lower error, with the margin increasing as per-token dimension grows. That is the concrete empirical observation they add. The work is useful because it takes an idea already used in image generation and checks it on a testbed where compounding errors matter, using a minimal architecture that keeps the comparison focused. The turbulent-flow setting is reasonable for the claim they want to make. The soft spot is isolation. The stress-test note is on point: when only one transformer and one dataset are used, differences in gradient behavior, normalization, or attention patterns could interact with the target choice rather than the parameterization itself driving the result. If the full paper shows explicit controls or ablations that hold the other factors fixed across objectives, the attribution strengthens; otherwise the advantage remains suggestive for this specific case. No load-bearing circularity or invented quantities appear. The paper is aimed at people building diffusion models for scientific simulation or system identification. A reader who already works with these models would get a practical data point on target choice. It is coherent enough on its own terms to deserve referee time, even if the experiments need expansion for broader claims.

Referee Report

1 major / 2 minor

Summary. The paper examines target parameterization choices in diffusion models applied to nonlinear spatiotemporal system identification, focusing on turbulent flow regimes where classical methods struggle with high dimensionality and error accumulation. Using a patch-based transformer operating directly on physical fields as a testbed, the authors compare clean-state, velocity, and noise prediction objectives and report that clean-state prediction yields consistently better rollout stability and lower long-horizon errors, with the gap widening at higher per-token dimensionality.

Significance. If the empirical isolation holds, the result identifies target parameterization as a practically important modeling decision for diffusion-based identification of nonlinear systems with spatial structure. It supplies a concrete, reproducible comparison on a challenging testbed and could inform more robust probabilistic forecasting architectures in control and dynamical systems applications.

major comments (1)

[Experimental Results / Setup] The central claim requires that only the prediction target varies while loss scaling, gradient magnitudes, normalization, conditioning, and optimization trajectories remain equivalent across the three objectives. The manuscript employs a single patch-based transformer on one turbulent-flow dataset; without explicit ablations or controls demonstrating invariance to architecture-data interactions (e.g., patch embedding sensitivity to dimensionality), the observed advantage cannot be confidently attributed to parameterization alone.

minor comments (2)

[Abstract] The abstract states that clean-state prediction improves rollout stability but supplies no quantitative metrics, error bars, or statistical tests; these details should appear in the main text and figures for verifiability.
[Methods] Notation for the three target parameterizations (clean state, velocity, noise) should be introduced with explicit equations early in the methods section to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment emphasizing the need for rigorous isolation of the target parameterization effect. We address the concern in detail below and will incorporate clarifications and discussion in the revised manuscript.

read point-by-point responses

Referee: [Experimental Results / Setup] The central claim requires that only the prediction target varies while loss scaling, gradient magnitudes, normalization, conditioning, and optimization trajectories remain equivalent across the three objectives. The manuscript employs a single patch-based transformer on one turbulent-flow dataset; without explicit ablations or controls demonstrating invariance to architecture-data interactions (e.g., patch embedding sensitivity to dimensionality), the observed advantage cannot be confidently attributed to parameterization alone.

Authors: We agree that the central claim requires careful controls to ensure differences arise from the prediction target. In our setup the patch-based transformer architecture, patch embedding, positional encodings, number of layers, attention heads, conditioning (timestep embedding), optimizer, learning-rate schedule, and batch size were held fixed; the sole change was the network regression target (clean state, velocity, or noise). Targets were normalized to comparable variance and we monitored gradient norms to confirm similar optimization behavior. We acknowledge, however, that the single-architecture, single-dataset design limits claims of invariance to architecture-data interactions. We will revise the manuscript to add an explicit subsection detailing these controls and a limitations paragraph discussing potential sensitivity of patch embeddings to per-token dimensionality. revision: partial

Circularity Check

0 steps flagged

No circularity in empirical comparison of diffusion target parameterizations

full rationale

The paper presents an empirical study comparing clean-state, velocity, and noise prediction objectives within a fixed patch-based transformer architecture on turbulent flow simulation data. All central claims rest on observed rollout stability and long-horizon error metrics obtained from independent simulation runs; no mathematical derivation, first-principles result, or uniqueness theorem is asserted that reduces by construction to its own inputs or to a fitted parameter renamed as a prediction. No self-citations are invoked as load-bearing justification for the target-parameterization choice, and the methodology does not involve ansatz smuggling, renaming of known empirical patterns, or self-definitional loops. The reported advantage of clean-state prediction is therefore an experimental outcome rather than a tautological restatement of the modeling assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the contribution is an empirical comparison of standard diffusion training objectives.

pith-pipeline@v0.9.0 · 5489 in / 980 out tokens · 40836 ms · 2026-05-10T05:30:53.734072+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 3 canonical work pages · 2 internal anchors

[1]

Deep Networks for System Identification: A Survey,

G. Pillonetto, A. Aravkin, D. Gedon, L. Ljung, A. H. Ribeiro, and T. B. Sch ¨on, “Deep Networks for System Identification: A Survey,” Automatica, vol. 171, p. 111907, 2025

2025
[2]

From System Models to Class Models: An In-Context Learning Paradigm,

M. Forgione, F. Pura, and D. Piga, “From System Models to Class Models: An In-Context Learning Paradigm,”IEEE Control Systems Letters, vol. 7, pp. 3513–3518, 2023

2023
[3]

Enhanced Trans- former Architecture for In-Context Learning of Dynamical Systems,

M. Rufolo, D. Piga, G. Maroni, and M. Forgione, “Enhanced Trans- former Architecture for In-Context Learning of Dynamical Systems,” inEuropean Control Conference (ECC), pp. 819–824, 2025

2025
[4]

Can Transformers Learn Optimal Filtering for Unknown Systems?

Z. Du, H. Balim, S. Oymak, and N. Ozay, “Can Transformers Learn Optimal Filtering for Unknown Systems?”IEEE Control Systems Letters, vol. 7, pp. 3525–3530, 2023

2023
[5]

Interpretable Spatial-Temporal Fusion Transformers: Multi-Output Prediction for Parametric Dynamical Systems with Time-Varying Inputs,

S. Sun, L. Feng, and P. Benner, “Interpretable Spatial-Temporal Fusion Transformers: Multi-Output Prediction for Parametric Dynamical Systems with Time-Varying Inputs,”arXiv preprint arXiv:2505.00473, 2025

work page arXiv 2025
[6]

G. I. Beintema,Modeling and Identification of Nonlinear Dynamical Systems Using Deep Neural Network Architectures, Ph.D. dissertation, Eindhoven University of Technology, 2024

2024
[7]

Benchmarking autoregressive conditional diffusion models for turbulent flow simulation,

G. Kohl, L.-W. Chen, and N. Thuerey, “Benchmarking autoregressive conditional diffusion models for turbulent flow simulation,”Neural Networks, vol. 199, art. 108641, July 2026

2026
[8]

Denoising Diffusion Probabilistic Models,

J. Ho, A. Jain, and P. Abbeel, “Denoising Diffusion Probabilistic Models,” inAdvances in Neural Information Processing Systems, 2020

2020
[9]

Improved Denoising Diffusion Probabilistic Models,

A. Nichol and P. Dhariwal, “Improved Denoising Diffusion Probabilistic Models,” inProceedings of the 38th International Conference on Machine Learning, 2021

2021
[10]

Elucidating the Design Space of Diffusion-Based Generative Models,

T. Karras, M. Aittala, T. Aila, and S. Laine, “Elucidating the Design Space of Diffusion-Based Generative Models,” inAdvances in Neural Information Processing Systems, 2022

2022
[11]

Scalable Diffusion Models with Transformers

W. Peebles and S. Xie, “Scalable Diffusion Models with Transformers,” arXiv preprint arXiv:2212.09748, 2022

work page internal anchor Pith review arXiv 2022
[12]

Back to Basics: Let Denoising Generative Models Denoise

T. Li and K. He, “Back to Basics: Let Denoising Generative Models Denoise,”arXiv preprint arXiv:2511.13720, 2025

work page internal anchor Pith review arXiv 2025