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REVIEW 2 major objections 5 minor 1 cited by

On Foundation Models for Dynamical Systems from Purely Synthetic Data

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A transformer trained only on synthetic smooth dynamics transfers to real cart-pole and Furuta states, and fine-tunes from little data.

desk verdict A plausible feasibility study of pretraining a transformer on purely synthetic RKHS dynamics, but the central pretraining-vs-capacity comparison is confounded and needs a matched-capacity baseline before the foundation-model claims hold. read the letter →

arxiv 2412.00395 v2 pith:ZVJF373N submitted 2024-11-30 cs.LG cs.ROstat.ML

Martin Ziegler , Andres Felipe Posada-Moreno , Friedrich Solowjow , Sebastian Trimpe This is my paper · ORCID
classification cs.LGcs.ROstat.ML
keywords foundationmodeldynamicalsystemssyntheticdatapretrainingreproducingkernelHilbertspacestatepredictiontransformerfine-tuningzero-shotgeneralization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether a foundation model for control can be built from purely synthetic data, since no Internet-scale dataset of physical dynamics exists. It claims yes: a decoder-only transformer pretrained on trajectories generated from random smooth functions sampled from a reproducing kernel Hilbert space learns a general predictor that transfers to unseen dynamical systems. The authors validate the claim on simulated and physical cart-pole setups and a physical Furuta pendulum, and show that fine-tuning with a small slice of target data beats specialist models trained from scratch. If the claim holds, pretraining for control tasks does not require collecting a giant real-world dataset; a well-chosen synthetic prior plus a transformer architecture provides the generalization, data efficiency, and run-to-run stability that foundation models promise elsewhere.

What carries the argument

The load-bearing mechanism is the synthetic pretraining prior: dynamics functions are drawn from a reproducing kernel Hilbert space with an RBF kernel, with random supporting points and coefficients, norm-scaled to a uniformly chosen target, discretized by Euler's method, and filtered by total variation so the input and output halves of each trajectory have comparable complexity and no single behavior dominates. The second mechanism is the decoder-only transformer, which embeds state-action pairs through residual blocks, applies causal multi-head self-attention, and predicts future states under random masking and output patching. It is trained to map a context window plus future-action placeholders to future states, so the same network can be evaluated zero-shot or fine-tuned on a target system.

What would settle it

Train the same pipeline and evaluate it zero-shot on a dynamical system with genuinely non-smooth dynamics, such as a bouncing ball or an oscillator with Coulomb friction, comparing prediction error to the reported cart-pole results; if errors are orders of magnitude larger and small-data fine-tuning cannot recover, the RKHS synthetic prior is not representative of the target class.

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Extended reading notes

Core claim

The core discovery is that the missing dataset problem for dynamical systems can be circumvented by sampling dynamics functions from an RKHS with an RBF kernel, scaling each function's norm, converting it to a continuous-time vector field, and discretizing with Euler's method. The resulting trajectories are filtered and binned by total variation to keep the pretraining distribution balanced. After pretraining, a 3.4-million-parameter transformer predicts future states of unseen systems in simulation and on hardware, and fine-tuning it with small data subsets improves accuracy beyond that of a linear regression baseline, a feedforward network, and a smaller transformer trained from scratch. The paper frames this as demonstrating generalization, data efficiency, and lower-variance performance across repeated runs.

Load-bearing premise

The central assumption is that random smooth RKHS functions, discretized with Euler's method and filtered by total variation, cover the behavior of real dynamical systems well enough that a transformer trained on them transfers to cart-pole and Furuta pendulums; if the prior misses the target class, the zero-shot and fine-tuning advantages disappear.

Editorial extensions

If this is right

  • A pretrained dynamics model can predict future states of a cart-pole system it has never seen, matching or beating linear regression and a feedforward net in low-data regimes in simulation.
  • Fine-tuning the pretrained model on as little as 2% of a target system's trajectories yields lower prediction error than training a specialist model from scratch on the same data.
  • The fine-tuned model shows less run-to-run variance across 20 training repetitions than the baselines, including on hardware cart-pole and Furuta pendulum data.
  • Zero-shot performance degrades when control actions are present, because pretraining used zero action, but fine-tuning closes most of that gap on hardware.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If this transfer holds, the same recipe could initialize state estimators and model-predictive controllers across many hardware systems, replacing per-system identification with a short fine-tuning step.
  • The zero-action pretraining prior is likely the main reason zero-shot performance drops on action-driven hardware; a direct testable extension is to include nonzero control inputs in the synthetic data generation.
  • Comparing the same pipeline with other synthetic function families, such as trigonometric or polynomial vector fields, would reveal whether smoothness alone or the specific RBF-kernel structure carries the transfer.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proposes training a transformer-based foundation model for state prediction in dynamical systems, using purely synthetic data generated by sampling dynamics functions from an RKHS, discretizing with Euler's method, and filtering by total variation. The resulting model is evaluated zero-shot and after fine-tuning on simulated cart-pole systems and on hardware cart-pole and Furuta pendulum data. The authors claim that this approach yields the typical foundation-model benefits: generalization across unseen systems, data efficiency in fine-tuning, and robustness across runs. The paper includes comparisons against linear regression, a feedforward network, and a smaller non-pretrained transformer.

Significance. If substantiated, the result would be a meaningful step toward foundation models for low-level control, addressing the lack of Internet-scale dynamical-systems data by using a synthetic pretraining distribution. The paper is clearly written and includes hardware validation, repeated runs, and a concrete data-generation pipeline. However, the central attribution of the reported gains to pretraining is weakened by the fact that the only transformer baseline has about 17 times fewer parameters than the proposed model, so capacity and pretraining are confounded. The paper also explicitly concedes that zero-shot performance is not competitive on action-driven hardware systems, which narrows the scope of the generalization claim. The code is not yet released, limiting reproducibility.

major comments (2)
  1. [§6.1, §6.3] The comparison between the fine-tuned model (Ft) and the smaller transformer (ST) cannot be used to isolate the effect of pretraining. Ft has 20 layers and approximately 3.4 million parameters, while ST has 8 layers and approximately 200k parameters; the paper states that ST is "not pretrained and thus shows the impact of pretraining," but any performance difference could instead be due to model capacity, optimization, or the larger pretraining corpus. This confound is load-bearing for the paper's central claims (P1)–(P3) because those claims are about the benefits of synthetic-data pretraining. The observation in §6.3 that the best ST run slightly outperforms Ft on the full hardware cart-pole dataset is consistent with capacity rather than pretraining driving part of the advantage. Please add a matched-capacity from-scratch transformer baseline (same architecture size, trained only on the task data), or otherwise control for capacity, before attributing the fine-tuning gains to pretraining.
  2. [§4, §6.3] The pretraining distribution is restricted to systems without control inputs (u ≡ 0 and zero placeholder actions in Eq. (4)), yet the problem formulation and the downstream evaluation include action-driven dynamics. The paper concedes in §6.3 that zero-shot performance on the hardware systems is "not competitive" because pretraining considered zero action. This is a substantive scope limitation: the claimed generalization across dynamical systems is demonstrated only for autonomous systems, while action-driven systems require fine-tuning. The paper should state this limitation prominently and temper the claim that the model generalizes across dynamical systems generally, or add experiments that include actions in the pretraining distribution.
minor comments (5)
  1. [Abstract] There is a typo in the abstract: "large-scale pretaining" should be "large-scale pretraining."
  2. [§1] In the sentence "models the capture the inherent 'dynamics' of the problem," "models the capture" should be "models that capture."
  3. [§3, §6.1] The paper defines robustness in §3 as consistency of MSE across runs, but the introduction lists robustness under distribution shifts, adversarial inputs, and noisy data. The evaluation only measures variance across initializations and data subsets; please clarify that the robustness claim is about run-to-run consistency, not the broader robustness properties listed in (P3).
  4. [§5] The indexing in the model description is slightly confusing: after giving the input sequence in Eq. (4), the text says the model generates {x̂_2,...,x̂_{c+m+1}} but only {x̂_{c+1},...,x̂_{c+m}} are used as valid predictions. Rewording or a small figure would help the reader verify the alignment of context and prediction windows.
  5. [§6] The paper states "The code will be released upon acceptance." Since the reproducibility of the experiments is part of the contribution, please consider releasing the code and data at submission time or providing a detailed appendix with hyperparameters and training details.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the pretrained model is evaluated on held-out systems and trajectories, fine-tuning uses task data before evaluation, and self-citations are background rather than load-bearing.

full rationale

No step in the derivation reduces to its own input. The pretraining distribution (Section 4) is a stated modeling choice—RKHS functions with RBF kernel, Euler discretization, and total-variation filtering—and is not fitted to the downstream cart-pole or Furuta data. Zero-shot evaluations (Pre) and fine-tuning evaluations (Ft) are run on task data not used for pretraining, and the paper reports that zero-shot is not competitive on action-driven hardware (Section 6.3: 'pretraining considered zero action (u ≡ 0)'), which is a negative empirical result inconsistent with a self-fulfilling construction. The main weaknesses are empirical-validity concerns rather than circularity. First, the sole transformer baseline ST has 8 layers / ~200k parameters versus 20 layers / ~3.4M for the proposed model (Section 6.1), so the Ft-vs-ST comparison cannot isolate pretraining; the paper even notes the best ST run slightly outperforms Ft on the full hardware cart-pole dataset (Section 6.3). This undermines attribution, not the independence of the evaluation. Second, whether RKHS-sampled functions cover real action-driven dynamics is an assumption about distributional coverage, not an equation-level reduction. Self-citations (Fiedler et al. 2021; Hose et al. 2024; Geist and Trimpe 2020) support background methods or hardware parameters and are not load-bearing for the central foundation-model claims. The footnote that code will be released upon acceptance is a reproducibility limitation, not evidence of circular reasoning. Hence no circularity score above 0 is warranted.

Assumptions & free parameters 9 free parameters · 5 assumptions · 0 invented entities

The central empirical claim rests on a pipeline of hand-picked design choices: RKHS kernel parameters, norm scaling, total variation filtering, augmentation ranges, and architecture hyperparameters. These are not fit to the target systems and are reported only partially, so the transfer result is conditional on the synthetic prior matching the test systems. No new physical entities are introduced.

free parameters (9)
  • RKHS kernel variance sigma^2 = not reported
    Scale of the RBF kernel used to sample dynamics functions in Section 4; controls function magnitudes and is chosen by hand.
  • RKHS kernel lengthscale l = not reported
    Determines smoothness of sampled dynamics functions; central to the synthetic prior.
  • Coefficient variance sigma_alpha^2 = not reported
    Variance of Gaussian coefficients alpha_i in function sampling.
  • State bounds xmin, xmax = not reported
    Support points and initial states sampled uniformly from this box; defines the state space of pretraining.
  • Target RKHS norm range [normmin, normmax] = not reported
    Functions are scaled to a target norm drawn uniformly from this range to vary behavior complexity.
  • Euler discretization step dt = not reported
    Converts continuous RKHS vector fields into discrete trajectories; affects stability and smoothness of data.
  • Total variation bounds TVmin, TVmax and delta = not reported
    Trajectories are filtered and binned by total variation to balance the pretraining distribution; these thresholds are manual.
  • Data augmentation ranges alpha, beta = not reported
    Random scaling and shifting applied to pretraining trajectories; augmentation strength chosen by hand.
  • Fine-tuning noise sigma^2 = not reported
    Gaussian noise added to state measurements during fine-tuning data augmentation.
assumptions (5)
  • domain assumption RKHS functions with an RBF kernel, after norm scaling and Euler discretization, are representative of the dynamics of real systems like cart-pole and Furuta pendulums.
    Invoked in Section 4 to justify the synthetic pretraining distribution; if this coverage fails, transfer to the test systems would not work.
  • ad hoc to paper Total variation filtering and binning do not remove the types of dynamical behavior needed for downstream transfer.
    Section 4 discards trajectories with low or imbalanced total variation and downsamples overrepresented bins; this shapes the pretraining distribution in a way that is not independently justified.
  • domain assumption Zero-action pretraining is a sufficient base for models that later see actions, provided fine-tuning is applied.
    Section 6.3 acknowledges zero-shot Pre is not competitive on hardware with actions; the main results for action systems rely on fine-tuning.
  • domain assumption State prediction over a fixed horizon is the right surrogate for control-relevant model quality.
    Section 3 equates the state prediction objective with what is needed in model-based control and estimation.
  • domain assumption The TimesFM decoder-only architecture, including residual blocks, patching, and masking, is an appropriate inductive bias for multi-step dynamics prediction.
    The architecture is taken from prior work (Das et al., 2024) without modification-specific justification for dynamics.

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Cite this review

Pith. "Pith review of On Foundation Models for Dynamical Systems from Purely Synthetic Data." pith.science (2026). https://pith.science/paper/ZVJF373N

@misc{pith2026241200395,
  author       = {Pith},
  title        = {Pith review of: On Foundation Models for Dynamical Systems from Purely Synthetic Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZVJF373N}},
  note         = {Machine review of arXiv:2412.00395}
}
read the original abstract

Foundation models have demonstrated remarkable generalization, data efficiency, and robustness properties across various domains. In this paper, we explore the feasibility of foundation models for applications in the control domain. The success of these models is enabled by large-scale pretaining on Internet-scale datasets. These are available in fields like natural language processing and computer vision, but do not exist for dynamical systems. We address this challenge by pretraining a transformer-based foundation model exclusively on synthetic data and propose to sample dynamics functions from a reproducing kernel Hilbert space. Our pretrained model generalizes for prediction tasks across different dynamical systems, which we validate in simulation and hardware experiments, including cart-pole and Furuta pendulum setups. Additionally, the model can be fine-tuned effectively to new systems to increase performance even further. Our results demonstrate the feasibility of foundation models for dynamical systems that outperform specialist models in terms of generalization, data efficiency, and robustness.

Figures

Figures reproduced from arXiv: 2412.00395 by the authors.

Figure 1
Figure 1. Proposed approach for a foundation model that predicts future states of dynamical sys [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Representative functions sampled from an RKHS (left) in a two-dimensional state ( [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Architecture of the decoder-only transformer model. The model projects inputs into an [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Custom cart-pole (image taken from (Hose et al., 2024)) and Quanser Qube Servo 2 Furuta systems. Datasets We use both simulated and real-world datasets for train￾ing and evaluation. The simulated datasets are: (i) a cart-pole simu￾lation with fixed parameters and const…
Figure 5
Figure 5. Figure 5: Simulation results for cart-pole with fixed parameters and constant action (left) and ran [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Hardware results for the cart-pole (left) and Furuta pendulum (middle). We compare the [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

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