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arxiv: 2606.05045 · v1 · pith:5GR6VUM4new · submitted 2026-06-03 · 🧮 math.DS · cs.LG

Learning Control-Affine Reduced-Order Models via Autoencoders

Ali Mjalled , Martin M\"onnigmann This is my paper

Pith reviewed 2026-06-28 03:49 UTC · model grok-4.3

classification 🧮 math.DS cs.LG
keywords autoencodersreduced-order modelscontrol-affine systemssystem identificationfeedback linearizationlatent space dynamicsnonlinear control
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The pith

Autoencoders can learn reduced-order models that keep the original system's control-affine structure.

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

The paper develops a method to find low-dimensional models for control by training an autoencoder jointly with a control-affine state-space model. This simultaneous training transforms high-dimensional states and inputs into a latent space where the dynamics stay control-affine. A sympathetic reader would care because this allows control techniques like feedback linearization to be applied to complex high-dimensional systems using the simpler model. The method also includes a sequence-based version that uses past states and inputs for better accuracy. Numerical tests show improved prediction and control performance compared to models that assume linear dynamics in the latent space.

Core claim

The framework identifies control-affine reduced-order models by using autoencoders to map high-dimensional states and inputs to reduced latent representations, with the autoencoder and state-space model trained simultaneously to preserve the control-affine form. This is extended to sequence-based models that process histories. The approach is motivated by the ability to apply feedback linearization to the resulting models, and is demonstrated on two numerical examples where it outperforms a baseline with linear latent dynamics.

What carries the argument

The joint training of the autoencoder and the control-affine state-space model to discover a latent space preserving the affine input effect.

If this is right

  • Feedback linearization can be applied directly to the learned reduced-order models.
  • The sequence-based extension improves prediction accuracy without losing the control-affine property.
  • The models achieve higher prediction accuracy on test data than baselines with linear state-space dynamics in latent space.
  • The framework provides guidelines for efficient use in control applications.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This method may generalize to other nonlinear control structures if similar latent representations can be found.
  • It could enable model-based control for systems like fluid flows or large mechanical systems where full-order models are too complex.
  • Further work might test whether the learned models maintain stability properties from the original system.

Load-bearing premise

A latent representation exists where the input effect on the state remains linear after the nonlinear transformation by the autoencoder.

What would settle it

If applying feedback linearization to the learned reduced model fails to stabilize or track trajectories on the original high-dimensional system, while the full-order model succeeds.

Figures

Figures reproduced from arXiv: 2606.05045 by Ali Mjalled, Martin M\"onnigmann.

Figure 1
Figure 1. Figure 1: Single-step inference of the control-affine ROM. The encoders compress the high-dimensional state and input [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Input/Output feedback linearization block diagram. The linearized model is controlled using a PID controller [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of ground-truth temperature profiles and ROM-predicted outputs at four representative time steps [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the latent dynamics predicted by the control-affine model and the corresponding ground-truth [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Latent space reference tracking using PID control in combination with feedback linearization. The reference [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of decoded closed-loop control inputs and reference signals (top row), and corresponding [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of (top) ground-truth frames, (middle) ROM-predicted outputs, and (bottom) absolute error at [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of the predicted latent dynamics using the control-affine model and the ground-truth encoded [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the (dashed lines) predicted autonomous latent trajectories using the control-affine model and [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Latent space reference tracking using P control in combination with feedback linearization (upper). A [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
read the original abstract

We present in this paper a framework for the identification of control-affine reduced-order models (ROMs). The proposed method utilizes autoencoders (AEs) to transform the high-dimensional states, and potentially the high-dimensional inputs, into reduced latent ones suitable for control-affine state-space dynamics. This is achieved by simultaneous training of the AE and the state-space model. In addition, we extend the discrete ROM formulation to a sequence-based model, which processes state and input histories to improve prediction accuracy while preserving the control-affine structure. We motivate our framework by applying feedback linearization to the derived models, and we present guidelines for its efficient use. The proposed framework is assessed on two numerical examples and its performance is compared to a baseline model, where the AE identifies a latent space with linear state-space dynamics. The assessment involves evaluating the prediction accuracy of the ROM on test data and its effectiveness in controlling the system to a desired state or trajectory.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a framework to identify control-affine reduced-order models by jointly training autoencoders that compress high-dimensional states (and optionally inputs) into a latent space together with a control-affine state-space model in that space. It further extends the formulation to a sequence-based model that ingests state and input histories while preserving the affine-in-u structure, motivates the approach via feedback linearization, supplies usage guidelines, and validates the method on two numerical examples by comparing prediction accuracy and closed-loop control performance against a baseline that learns linear latent dynamics.

Significance. If the empirical results hold, the framework offers a data-driven route to reduced models that remain compatible with standard nonlinear control tools such as feedback linearization. The joint training and sequence extension are pragmatic engineering contributions; the fact that control-affine structure is automatically inherited from any differentiable encoder (via the chain rule) means the central modeling choice is imposed rather than discovered, so the value lies in the training procedure and the numerical demonstrations rather than in a new theoretical guarantee.

major comments (2)
  1. [Numerical examples section] § on numerical examples (the two test cases): the abstract and reader summary indicate that prediction accuracy and control performance are evaluated, yet no quantitative metrics, error bars, or statistical comparisons are supplied in the provided description; without these, it is impossible to verify whether the control-affine ROM meaningfully outperforms the linear baseline or reliably supports feedback linearization.
  2. [Method / dynamics derivation] The claim that the learned latent dynamics remain control-affine and usable for linearization is load-bearing; because the structure follows immediately from the chain rule for any differentiable encoder E (˙z = DE(x)·(f(x)+g(x)u)), the paper should explicitly state this preservation result (perhaps as a short proposition) rather than present it only as an empirical outcome of joint training.
minor comments (2)
  1. Clarify whether the input encoder is permitted to be nonlinear or is constrained to remain linear in the latent input; the abstract mentions “potentially the high-dimensional inputs” but does not specify the restriction needed to keep the latent dynamics control-affine.
  2. Add a short statement on reproducibility (e.g., whether code or trained models will be released) to strengthen the empirical claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive recommendation of minor revision and for the constructive comments on the numerical examples and theoretical presentation. We address each point below and will incorporate the suggested changes.

read point-by-point responses

  1. Referee: [Numerical examples section] § on numerical examples (the two test cases): the abstract and reader summary indicate that prediction accuracy and control performance are evaluated, yet no quantitative metrics, error bars, or statistical comparisons are supplied in the provided description; without these, it is impossible to verify whether the control-affine ROM meaningfully outperforms the linear baseline or reliably supports feedback linearization.

    Authors: We agree that explicit quantitative metrics, error bars, and statistical comparisons are necessary to substantiate the performance claims. The numerical examples section reports prediction errors and closed-loop metrics via tables and figures comparing the proposed method to the linear baseline, but we will revise to include error bars from repeated training runs, standard deviations, and additional statistical tests to allow direct verification of the improvements. revision: yes

  2. Referee: [Method / dynamics derivation] The claim that the learned latent dynamics remain control-affine and usable for linearization is load-bearing; because the structure follows immediately from the chain rule for any differentiable encoder E (˙z = DE(x)·(f(x)+g(x)u)), the paper should explicitly state this preservation result (perhaps as a short proposition) rather than present it only as an empirical outcome of joint training.

    Authors: We agree that an explicit statement of the preservation result improves rigor. While the control-affine structure in latent space follows directly from the chain rule applied to any differentiable encoder (as noted in the derivation), we will add a short proposition in the method section to state and briefly prove this property, clarifying that it holds independently of the joint training. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper imposes control-affine structure on the latent dynamics as an explicit modeling choice and trains the autoencoder jointly with the ROM parameters to match data. Preservation of the affine-in-u form under differentiable encoding follows directly from the chain rule and is not claimed as a derived prediction. No self-citations, fitted inputs renamed as predictions, or ansatzes smuggled via prior work appear in the derivation. The method is validated empirically on two examples against a linear baseline; the central claim remains an independent empirical procedure rather than a tautology.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The framework rests on the domain assumption that a latent space with control-affine dynamics can be learned from data and that this structure is preserved under the chosen autoencoder architecture. No free parameters or invented entities are named in the abstract.

free parameters (1)
  • latent dimension
    Dimension of the reduced space must be chosen; not specified in abstract.
axioms (1)
  • domain assumption High-dimensional nonlinear dynamics admit a latent representation in which the input matrix remains state-independent (control-affine form).
    Invoked when the method states that the AE produces states suitable for control-affine state-space dynamics.

pith-pipeline@v0.9.1-grok · 5692 in / 1382 out tokens · 44287 ms · 2026-06-28T03:49:35.528126+00:00 · methodology

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