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arxiv: 2607.00669 · v1 · pith:MSSGFOERnew · submitted 2026-07-01 · 🧮 math.NA · cs.NA· stat.ML

Convolutional Symmetric AutoEncoders: enhancing latent stability via differential geometry

G. Li Causi , N.Tonicello , L. Magri , G. Rozza This is my paper

Pith reviewed 2026-07-02 07:59 UTC · model grok-4.3

classification 🧮 math.NA cs.NAstat.ML
keywords convolutional autoencodersreduced order modelingsymmetric autoencodersparametric PDEsmanifold learninglatent space stability
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The pith

Symmetric convolutional autoencoders preserve manifold parametrization to deliver more accurate latent trajectories in reduced-order PDE models.

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

This paper extends the idea of representation-consistent autoencoders to convolutional layers by introducing symmetric Convolutional AutoEncoders. These are designed to embody the primary properties of manifold parametrization mappings that standard autoencoders often miss despite minimizing reconstruction error. Tested within a reduced-order modeling framework on the Linear Advection, Viscous Burger, and Kuramoto Sivashinsky equations, the symmetric approach produces more accurate latent trajectories, lower reconstruction errors, and greater robustness than classical CAEs.

Core claim

The authors claim that a novel class of symmetric Convolutional AutoEncoders, by embodying the primary properties of manifold parametrization mappings, when integrated into a ROM framework, demonstrates significantly improved predictive capabilities, with numerical results on three one-dimensional PDE test cases showing consistently more accurate latent trajectories, lower reconstruction errors, and enhanced model robustness.

What carries the argument

The symmetric Convolutional AutoEncoder architecture that extends representation consistency from fully connected layers to convolutional layers to preserve essential manifold parametrization properties.

If this is right

  • ROMs using symmetric CAEs achieve higher accuracy in latent space predictions for parametric PDEs.
  • The symmetric design leads to lower reconstruction errors compared to standard CAEs.
  • Model robustness is enhanced through the symmetric approach across the tested equations.

Where Pith is reading between the lines

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

  • The method could extend to multi-dimensional problems or other network architectures for broader applicability in scientific computing.
  • Integration with other ROM techniques might further stabilize predictions for slowly decaying Kolmogorov n-width problems.

Load-bearing premise

Extending representation consistency from fully connected to convolutional layers will automatically preserve the essential manifold parametrization properties required for stable and accurate ROMs.

What would settle it

Numerical experiments on the same test cases where the symmetric CAE does not show lower reconstruction errors or more accurate latent trajectories than the classical CAE.

Figures

Figures reproduced from arXiv: 2607.00669 by G. Li Causi, G. Rozza, L. Magri, N.Tonicello.

Figure 1
Figure 1. Figure 1: The architecture consists of a nonlinear encoder (E) for dimensionality re￾duction and a symmetric decoder (D) for physical field reconstruction. Here, σ denotes a Bi-lipschitz nonlinear activation function applied element-wise between subsequent encoder layers, while σ −1 represents its inverse, utilized within the decoder layers to maintain the structural symmetry and bi-orthogonality of the mapping. wit… view at source ↗
Figure 2
Figure 2. Figure 2: Rational activation function σα, its inverse σ −1 α , and Tanh activation func￾tion. The dashed gray line denotes the bisector of the first and third quadrants. 4. Data-driven approach Having introduced the enhanced version of convolutional autoencoders, which are distinguished by their advantageous parameterization properties, we can now employ them for manifold learning tasks to obtain a reduced represen… view at source ↗
Figure 3
Figure 3. Figure 3: Sketch of the LF net. encoding mapping z = E(uh(t; µ)). The ground-truth latent velocity z˙ is derived by computing the differential of the encoder with respect to the high-fidelity state uh, evaluated at uh(t; µ), and applying to the high-fidelity dynamics fh as follows: z˙(z, t; µ) = dE (uh(t; µ)) · fh (t, uh(t; µ); µ). (14) The LF-net is trained by minimizing a Mean Squared Error (MSE) loss function, wh… view at source ↗
Figure 4
Figure 4. Figure 4: Solution of Linear Advection equation (left), Viscous Burgers (middle) and Kuramoto-Sivanshisky equation (right). 5.1. Linear Advection. The one-dimensional Linear Advection equation describes the transport of a passive scalar field within a spatial domain of size L over a temporal horizon T. The governing equations, assuming periodic boundary conditions, are defined as:    ∂u ∂t + v ∂u ∂x = 0, x ∈ [0… view at source ↗
Figure 5
Figure 5. Figure 5: Cumulative sum of the squared singular values of the snapshot matrix X = [u1,u2, . . . ,uNT ] ∈ R Nh×Nt the performance of the ROMs, we define the relative L2 reconstruction error as: ϵu(t) = ∥u(t) − u˜(t)∥L2 ∥u(0)∥L2 (22) where u˜(t) is the solution predicted by the ROM. For the POD-ROM, the reconstruction is given by u˜(t) = Φc(t). As illustrated in [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Error ϵu defined in Equation 22 with an increasing number of modes l, as function of time t. −1 0 1 z ˆ 1 −1.0 −0.5 0.0 0.5 1.0 ˆz 2 −1 0 1 z ˆ 1 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Normalized latent trajectory produced by 10 classic ROMs (left) and 10 symmetric ROMs (right) To facilitate a direct comparison between models trained independently, which may produce topolog￾ically equivalent but geometrically distinct latent representations, we map all latent trajectories onto a unit circle. For a high-dimensional trajectory uh(t, µ), first we project it onto the latent space (z(t, µ) = … view at source ↗
Figure 8
Figure 8. Figure 8: Statistical evolution of the high-dimensional reconstruction error (top) and latent space error (bottom) for the classic and symmetric ROMs. Results are computed over a 10-period time evolution across the test set. Solid lines represent the mean error, while the shaded regions indicate the standard deviation. generated by the symmetric ROMs tend to stay closer to the unit circle, where they should belong. … view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the latent space trajectories (l = 2) for the parametric Vis￾cous Burgers’ equation using Classic CAE (left) and Symmetric CAE (right). Solid lines represent the projection of the high-fidelity solution through the encoder, z = E(uh), while dashed lines denote the trajectories predicted by the ROM. The trajectories are colored according to the viscosity ν, with the thicker lines highlighting … view at source ↗
Figure 10
Figure 10. Figure 10: First line: spatio-temporal solutions generated by the FOM (left), the clas￾sical ROM (center), and the symmetric ROM (right), using a viscosity value ν selected from the training set. Second line: point-wise error as functions of the spatial and tem￾poral coordinates for classical ROM (center) and symmetric ROM (right) [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: First line: spatio-temporal solutions generated by the FOM (left), the clas￾sical ROM (center), and the symmetric ROM (right), using a viscosity value ν selected from the test set. Second line: point-wise error as functions of the spatial and temporal coordinates for classical ROM (center) and symmetric ROM (right). The performance of the classical and symmetric ROMs is evaluated via the mean absolute err… view at source ↗
Figure 12
Figure 12. Figure 12: Error ϵ abs u defined as the time mean of Equation 26, for ROM based on standard CAE (left) and symmetric CAE (right). 6.3. Kuramoto-Sivashinsky Equation. The application of ROMs to chaotic systems, such as the Kuramoto-Sivashinsky equation, represents a significant challenge due to the inherent unpredictability of the dynamics. In chaotic regimes, trajectories exhibit a sensitive dependence on initial co… view at source ↗
Figure 13
Figure 13. Figure 13: Reference solution of the KS equation (left), prediction of the ROM model based on classical CAE (center) and point-wise absolute error (right) [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Reference solution of the KS equation (left), prediction of the ROM model based on symmetric CAE (center) and point-wise absolute error (right). models—utilizing both the classical CAE ( [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Lyapunov Spectrum predicted by the classical and symmetric ROMs and the reference spectrum obtained from the high fidelity simulation. Another useful analysis concerns the Lyapunov exponents, , which measure the sensitivity of a solution to initial conditions by quantifying the average rate of growth or decay of a small perturbation along a given direction. For a rigorous theoretical treatment and numeric… view at source ↗
Figure 16
Figure 16. Figure 16: Probability Density Function (PDF) of the kinetic energy. characterization of the Kuramoto-Sivashinsky system, we compute the averaged kinetic energy spectrum utilizing Welch’s method. This analysis allows for a precise evaluation of how the reduced-order models distribute energy across different spatial scales. The symmetric ROM demonstrates a superior capability in capturing the energy content across th… view at source ↗
Figure 17
Figure 17. Figure 17: Kinetic energy spectrum against wave number for the FOM, classical and symmetric ROM systems. literature has introduced augmented neural network types, specifically autoencoders with fully connected layers, that address the representation consistency property. This paper extends this property to hybrid convolutional autoencoders. In this context, “hybrid” refers to an encoder composed of convolutional lay… view at source ↗
read the original abstract

Autoencoders (AEs) have emerged as powerful tools for non-linear dimensionality reduction, often surpassing traditional linear methods such as Proper Orthogonal Decomposition (POD) in scenarios characterized by slowly decaying Kolmogorov $n$-widths. In the realm of Reduced-Order Modelling (ROM), these models are increasingly utilized to learn low-dimensional representations of solution manifolds associated with parametric Partial Differential Equations (PDEs). However, the high expressivity of AEs presents a challenge: although trained networks typically minimize reconstruction error, they often struggle to capture the essential properties necessary for building accurate and robust ROMs. Recent works by arXiv:2307.15288v2 and arXiv:2506.11641v1 have tackled this challenge in fully connected AEs by proposing representation-consistent architectures, which preserve some of the properties belonging to POD. This study builds upon that concept by extending representation consistency for convolutional layers. We introduce a novel class of symmetric Convolutional AutoEncoders (CAEs) designed to embody the primary properties of manifold parametrization mappings. When integrated into a ROM framework, this architecture demonstrates significantly improved predictive capabilities. Specifically, we compared the performance of the ROMs based on classical and symmetric CAEs on three one dimensional academic test cases, namely the Linear Advection, the Viscous Burger and the Kuramoto Sivashinsky equation. Numerical results demonstrate that our proposed symmetric approach consistently yields more accurate latent trajectories, lower reconstruction errors, and enhanced model robustness.

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 / 1 minor

Summary. The paper proposes symmetric Convolutional AutoEncoders (CAEs) that extend representation consistency from fully connected layers (as in prior self-cited works) to convolutional layers, with the aim of preserving primary properties of manifold parametrization mappings for use in Reduced-Order Modeling (ROM) of parametric PDEs. It evaluates the approach against classical CAEs on three 1D test cases (Linear Advection, Viscous Burgers, Kuramoto-Sivashinsky), claiming more accurate latent trajectories, lower reconstruction errors, and improved robustness.

Significance. If the convolutional symmetry is shown to preserve the required manifold parametrization properties and the performance gains are reproducible with proper controls, the work could strengthen nonlinear ROM techniques for problems where POD fails due to slowly decaying Kolmogorov n-widths. The differential-geometry motivation for latent stability is a coherent direction, but the absence of explicit verification for the convolutional case limits the current impact.

major comments (2)
  1. [Abstract] Abstract (architecture design paragraph): the central claim that the symmetric CAE 'embodies the primary properties of manifold parametrization mappings' for convolutional layers is not supported by any derivation, invariance check, or comparison showing that the convolutional symmetry satisfies the same consistency conditions with POD-like mappings as the fully connected case in the cited priors (arXiv:2307.15288v2, arXiv:2506.11641v1). Without this step the reported ROM gains cannot be attributed to the claimed geometric preservation.
  2. [Numerical results] Numerical results (comparison on three test cases): the abstract asserts that the symmetric approach 'consistently yields more accurate latent trajectories, lower reconstruction errors, and enhanced model robustness,' yet supplies no quantitative tables, error metrics, error bars, training details, or ablation studies. This prevents verification of the empirical claim and leaves open the possibility that gains arise from unstated architectural differences rather than the symmetry.
minor comments (1)
  1. [Abstract] The abstract references three specific 1D PDE test cases but does not indicate whether the reported improvements hold under variations in network depth, filter sizes, or training hyperparameters.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our contributions. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract (architecture design paragraph): the central claim that the symmetric CAE 'embodies the primary properties of manifold parametrization mappings' for convolutional layers is not supported by any derivation, invariance check, or comparison showing that the convolutional symmetry satisfies the same consistency conditions with POD-like mappings as the fully connected case in the cited priors (arXiv:2307.15288v2, arXiv:2506.11641v1). Without this step the reported ROM gains cannot be attributed to the claimed geometric preservation.

    Authors: We agree that an explicit derivation and verification step for the convolutional extension would strengthen the link to the manifold parametrization properties established in the fully connected priors. The symmetric convolutional architecture is defined by enforcing weight symmetry across the encoder-decoder pair in a manner that mirrors the representation-consistency construction used for dense layers, thereby inheriting the same invariance under reparametrization. In the revised manuscript we will add a dedicated subsection that derives the consistency conditions for the convolutional case, includes an invariance check with respect to POD-like mappings, and provides a direct comparison to the cited works. revision: yes

  2. Referee: [Numerical results] Numerical results (comparison on three test cases): the abstract asserts that the symmetric approach 'consistently yields more accurate latent trajectories, lower reconstruction errors, and enhanced model robustness,' yet supplies no quantitative tables, error metrics, error bars, training details, or ablation studies. This prevents verification of the empirical claim and leaves open the possibility that gains arise from unstated architectural differences rather than the symmetry.

    Authors: The manuscript presents comparative results on the three test cases through figures, but we acknowledge that tabulated quantitative metrics, error bars, training details, and ablation studies are not provided. In the revision we will add tables reporting reconstruction and latent-trajectory errors (with standard deviations over multiple random seeds), full hyperparameter specifications, and ablation experiments that isolate the effect of the symmetry constraint while keeping all other architectural choices identical. revision: yes

Circularity Check

1 steps flagged

Representation consistency for CAEs justified primarily via self-citation to prior fully-connected AE works

specific steps
  1. self citation load bearing [Abstract]
    "Recent works by arXiv:2307.15288v2 and arXiv:2506.11641v1 have tackled this challenge in fully connected AEs by proposing representation-consistent architectures, which preserve some of the properties belonging to POD. This study builds upon that concept by extending representation consistency for convolutional layers. We introduce a novel class of symmetric Convolutional AutoEncoders (CAEs) designed to embody the primary properties of manifold parametrization mappings."

    The assertion that the new CAE class embodies the required manifold parametrization properties is justified solely by building upon the two self-cited prior works; the abstract provides no separate derivation or verification that the convolutional symmetry satisfies the same consistency conditions shown for fully-connected layers in those citations.

full rationale

The paper's core architectural claim—that the proposed symmetric CAEs 'embody the primary properties of manifold parametrization mappings'—is introduced by direct reference to two prior arXiv preprints on fully-connected representation-consistent AEs. No independent derivation, invariance proof, or explicit check that convolutional symmetry inherits the same POD-like properties appears in the provided abstract or skeptic summary. Numerical gains on the three PDE cases are presented as validation, but the load-bearing geometric premise reduces to the cited prior results. This qualifies as moderate self-citation load-bearing without reducing the entire result to a fit or tautology, hence score 4 rather than 0 or 6+.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

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discussion (0)

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