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REVIEW 4 major objections 8 minor 2 cited by

Loss Terms and Operator Forms of Koopman Autoencoders

T0 review · 4 major / 8 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper argues that Koopman autoencoder training has robust default choices: full accuracy loss, reconstruction or consistency loss, tridiagonal operator form, and unitary operator loss.

desk verdict Broad empirical comparison of Koopman autoencoder design choices, but single-run experiments and table mismatches undercut the 'most robust' claims. read the letter →

arxiv 2412.04578 v1 pith:5CQ2C7HN submitted 2024-12-05 cs.LG physics.comp-ph

Dustin Enyeart , Guang Lin This is my paper · ORCID
classification cs.LGphysics.comp-ph
keywords Koopmanautoencoderoperatorlearninglossfunctionformtridiagonalunitarydifferentialequationsneuralnetwork
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

Koopman autoencoders are a popular way to learn the time evolution of differential equations, but the literature disagrees on how to build the loss function and what form the learned linear operator should take. This paper runs a systematic comparison of those choices across eight ODE and PDE benchmarks and tries to identify which options are robustly best. It introduces several new loss terms—discounted accuracy, unitary, determinant, and absolute max—and also compares dense, tridiagonal, and Jordan operator forms. The paper's recommendation is a concrete default configuration: full accuracy loss, reconstruction loss (with consistency loss as a safe alternative), the tridiagonal operator form, and the unitary loss. If these defaults survive full-length training, practitioners gain a strong baseline that removes much of the guesswork in Koopman autoencoder design.

What carries the argument

The load-bearing objects are the loss terms themselves and the structural form of the Koopman operator. The accuracy terms compare predicted and target states; the encoding terms constrain the encoder–decoder pair; the operator terms push the learned matrix toward unitarity. The operator forms are the dense matrix, the tridiagonal matrix (motivated by the fact that a unitary operator is tridiagonal in some basis), and the Jordan form built from $2\times 2$ blocks $\left(\begin{smallmatrix}a & b \\ -b & a\end{smallmatrix}\right)$. The recommended unitary loss $\lVert K K^{T} - I \rVert^{2}$ directly enforces that the operator be orthogonal, and the tridiagonal mask cuts parameters while preserving expressiveness. These choices together determine the inductive bias of the whole architecture.

What would settle it

Re-run the recommended configuration (full accuracy + reconstruction + unitary loss + tridiagonal operator) against the main alternatives on the pendulum and Lorenz benchmarks with many random seeds and full training; if the recommended configuration is not consistently among the best, or if another configuration consistently beats it, the robustness claim is unsupported.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that training choices for Koopman autoencoders have robust winners. Across the mean effects, best-combination tables, and direct comparisons, the full accuracy loss—the plain mean squared error between predictions and targets—is the most dependable accuracy term; the reconstruction loss and the consistency loss are the most dependable encoding terms; the tridiagonal form of the operator is the most dependable operator form; and the unitary loss, which penalizes $\lVert K K^{T} - I \rVert^{2}$, is the most dependable operator loss. The paper also reports that the norm loss is consistently weak, that the metric loss is usually weak, and that auxiliary losses such as the absolute max loss or energy conservation do not consistently help. Novel loss terms introduced here—discounted accuracy loss, unitary loss, determinant loss, and absolute max loss—are documented so that later work can test them independently.

Load-bearing premise

The conclusions rest on short grid searches with a single run per configuration; if those runs are noisy or do not extrapolate to full training, the 'most robust' rankings could change.

Editorial extensions

If this is right

  • A default recipe now exists: full accuracy loss, reconstruction loss, unitary loss, and a tridiagonal operator, so new Koopman autoencoder projects can start there instead of grid-searching.
  • The metric loss and norm loss can be deprioritized in most settings; the paper finds them consistently less robust than their alternatives.
  • The consistency loss can substitute for reconstruction loss when a problem makes reconstruction expensive, since the two perform similarly overall.
  • The unitary and determinant losses, both introduced here, are the strongest operator regularizers in the comparisons.

Reading between the lines

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

  • An implication not tested here: the recommended configuration is a natural default for long-horizon rollout, since a nearly orthogonal tridiagonal operator should limit error growth, though the paper's experiments are short.
  • Because the grid searches used single runs, the defaults are probably conservative: a practitioner with a specific equation and enough compute may find that the Jordan form or a different operator loss wins for that equation after full convergence.
  • The same comparison could be extended to Koopman-based control, where the operator's spectrum and boundedness matter for controller design; a unitary-plus-tridiagonal inductive bias may interact favorably with stability constraints.
  • The novel determinant loss could become practical at scale if combined with the tridiagonal form, since the determinant of a tridiagonal matrix is computable in linear time; the paper notes this but does not scale the test.
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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

4 major / 8 minor

Summary. The paper presents a comparative empirical study of loss terms and operator parameterizations for Koopman autoencoders. It defines several accuracy, encoding, and operator loss terms (full, max, and discounted accuracy; reconstruction, consistency, and metric; isometry, norm, unitary, and determinant; plus auxiliary absolute-max and energy-conservation losses) and three operator forms (dense, tridiagonal, and a block-diagonal "Jordan" form), claims novelty for the discounted accuracy, unitary, determinant, and absolute-max losses, and runs grid searches on eight ODE/PDE benchmarks. Based on short grid searches and longer per-equation comparisons, it recommends the full accuracy loss, the reconstruction loss, and the unitary loss with a tridiagonal operator.

Significance. If the conclusions were supported, the paper would provide useful practitioner guidance and fill a gap in the Koopman-autoencoder literature, where loss terms and operator forms are often chosen ad hoc. The systematic enumeration of choices across several equations, the public code, and the clearly stated final recommendation are strengths. However, the empirical basis is currently too fragile: the headline recommendations rest on single-run, short-horizon experiments with no variance estimates, and several table/caption inconsistencies make it difficult to verify the reported comparisons. The central claims are therefore not yet established.

major comments (4)
  1. [Section 5] The headline recommendations in Section 7 rest on grid searches whose "robust trends" are inferred from single runs at 20 to 50 epochs with no repeated seeds, no standard deviations, and no statistical tests. Section 5 itself states that the experiments are "grid searches that cover a large number of combinations for only a small number of epochs" and that the objective is to "find robust trends," but robustness cannot be established from one realization per configuration. Concretely, Section 5.1 reports that full and discounted accuracy losses "performed about the same," yet no measure of variance is given, and Section 5.4 says that "not having as mask performed much better" while "seven of the best ten combinations used a mask," showing that mean-effect and top-k readings can conflict. Without repeated seeds, error bars, or an interaction analysis, the "most robust" conclusions in Sections 5.7 and 7 are not supported.
  2. [Section 6, Tables 14 and 16] The table captions and the surrounding text do not match. Table 14 is introduced in the text as "the results for the equation for the pendulum" but is labeled "Results for the Lorenz system" and contains Lorenz data; Table 16 is introduced as "the results for the equation for the fluid attractor equation" but is labeled "Results for Burger's equation." These mismatches are not cosmetic: the Section 6.2 conclusion that the tridiagonal form and unitary loss are "most robust" is reached by reading these tables across equations, so the reader cannot tell which equation produced which rank. The tables must be relabeled and every row verified against the experimental log.
  3. [Section 6.2] The conclusion that the unitary loss term is the most robust operator loss term is not supported by the five tables it is based on. In Tables 13-17, the unitary loss gives the best error in only two cases (Tables 13 and 17); in Table 14 the best entry uses no operator loss, in Table 15 it uses the determinant loss, and in Table 16 it uses no operator loss. The text acknowledges that "the best combination varied from equation to equation" but then concludes robustness without defining any quantitative aggregation rule, such as mean rank, median error, or win rate. A qualitative reading of five single-run tables does not establish that the unitary loss is the most robust option.
  4. [Sections 5.1-5.6] Several reported grid sizes and dimensions are inconsistent with the stated option sets, which prevents reproduction of the search. Section 5.5 says "the embedding dimensions 256 was considered," but Table 9 reports entries at dimension 128 and Figure 9's caption says the embedding dimension is 64. In addition, the stated numbers of combinations do not match the stated restriction that the determinant loss is only considered with a mask: for example, Section 5.1 reports 216 combinations, but 3 accuracy terms x 3 embedding terms x 4 operator terms x 2 mask choices x 3 dimensions gives 189 after excluding determinant-without-mask; Section 5.2 reports 288, but the analogous calculation gives 252. Section 5.3 contains two competing descriptions of the grid. These inconsistencies make it impossible to know what was actually run and should be corrected or explained.
minor comments (8)
  1. [Sections 1 and 3] The sentence "the loss functions and operator forms are introduction in Section 3" is repeated, and "introduction" should be "introduced."
  2. [Section 2] The text contains "to to learn" and the Introduction says "This chapter" even though the manuscript is an arXiv paper; both should be corrected.
  3. [Section 3.2] In the definition of the reconstruction loss, the text says "where E is the encoder, R is the encoder"; the second identifier should be "decoder."
  4. [Section 3.3] What is called the "Jordan form" is a real block-diagonal canonical form with 2x2 rotation-scaling blocks, not the usual Jordan form; if this nonstandard terminology is intended, it should be defined and justified explicitly.
  5. [Section 5.3] The experimental description is internally contradictory: the first paragraph lists embedding dimensions 8, 16, 32, 64 and discount factors 1.0, 0.975, 0.95, while the next paragraph restates the setup with only dimensions 32 and 64 and no discount factors; this looks like leftover text from an earlier draft and should be removed.
  6. [Table 7] The heat-equation top-ten table reports errors rounded to one significant digit (all rows are 1e-6), which obscures the ranking; more digits are needed.
  7. [Section 6.1, Table 16] The sentence "The most accurate combination was the tridiagonal form without an unitary loss" should be reworded, since the table's best entry is "tridiagonal none" and the intended meaning appears to be "without an operator loss term."
  8. [Abstract] The abstract says "systemic study"; this should be "systematic study."

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the recommendations are empirical grid-search comparisons against externally generated test data, with no fitted-parameter predictions and no load-bearing self-citations.

full rationale

This paper makes no derivation that reduces to its own inputs. The central claims in Sections 5.7, 6.2, and 7 ('The full accuracy loss term is the most robust accuracy loss term... the tridiagonal form of the operator is the most robust operator form... the unitary loss term is the most robust operator loss term') are empirical conclusions drawn from grid searches over loss-term combinations, evaluated on numerically integrated test data for multiple ODEs and PDEs. The full accuracy loss is used both as a training term and as the test metric, but this is standard practice: the evaluation metric is not a parameter fitted from the data, and using the same metric for training and testing does not make the comparison definitionally forced. The paper introduces novel loss terms (discounted accuracy loss, unitary loss, determinant loss, absolute max loss) by defining their formulas, not by deriving them from the conclusions. There are no self-citations by the authors that carry a load-bearing argument; the references cited for existing loss terms and mathematical facts are external and independent. No uniqueness theorem is invoked, and no ansatz is smuggled in via self-citation. The observed inconsistencies in table captions (e.g., Table 14 introduced as pendulum but containing Lorenz data) concern data-pipeline quality and statistical support, not circularity. Accordingly, no circular step can be quoted with a specific equation-to-equation reduction, and the honest finding is no significant circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The recommendations rest on the unitary Koopman assumption, the tridiagonal sufficiency assumption, and the reliability of short grid-search runs. No new physical entities are introduced; the ledger is dominated by unstated hyperparameters and methodological assumptions.

free parameters (3)
  • discount factor lambda = 1.0, 0.975, 0.95 (values searched for Lorenz; otherwise unspecified)
    Hyperparameter for the discounted accuracy loss, chosen by hand or grid search; the paper does not justify it beyond reinforcement-learning inspiration.
  • embedding dimension = 8 to 1024 depending on the system
    Latent space size is searched per system and fixed for each grid; it is a configuration choice that affects all comparisons.
  • relative weights for combined loss terms = not reported
    The paper never specifies how accuracy, encoding, and operator losses are combined (weighted sum with weights equal to 1, or some other weighting). This choice affects every comparison and is not documented.
assumptions (4)
  • domain assumption The Koopman operator for classical mechanics is unitary.
    Section 3.3 motivates the operator losses and the tridiagonal form from this premise.
  • domain assumption A unitary operator is tridiagonal in some basis.
    Section 3.3 uses this claim to justify the tridiagonal mask on the operator.
  • ad hoc to paper Restricting the operator to 2x2 Jordan blocks is sufficiently expressive.
    Section 3.3 states that at worse it will express redundant information, but gives no proof or capacity analysis.
  • ad hoc to paper Short grid searches without repeated seeds reveal robust trends.
    The whole Section 5 methodology depends on this; the paper does not validate it with repeated runs or statistical tests.

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Cite this review

Pith. "Pith review of Loss Terms and Operator Forms of Koopman Autoencoders." pith.science (2026). https://pith.science/paper/5CQ2C7HN

@misc{pith2026241204578,
  author       = {Pith},
  title        = {Pith review of: Loss Terms and Operator Forms of Koopman Autoencoders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5CQ2C7HN}},
  note         = {Machine review of arXiv:2412.04578}
}
read the original abstract

Koopman autoencoders are a prevalent architecture in operator learning. But, the loss functions and the form of the operator vary significantly in the literature. This paper presents a fair and systemic study of these options. Furthermore, it introduces novel loss terms.

Figures

Figures reproduced from arXiv: 2412.04578 by the authors.

Figure 1
Figure 1. Discretization of the Koopman formulation into a numerical scheme: [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The mean effect of the different options of the numerical experiment [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Direct comparison of loss terms for the simple harmonic motion. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: The mean effect of the different options of the numerical experiment [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: The mean effect of the different options of the numerical experiment [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: Direct comparison of loss terms for the Lorenz system. On the left, [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: The mean effect of the different options of the numerical experiment [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: The mean effect of the different options of the numerical experiment [PITH_FULL_IMAGE:figures/full_fig_p029_8.png]
Figure 9
Figure 9. Figure 9: Direct comparison of loss terms for the wave equation. Each plot is [PITH_FULL_IMAGE:figures/full_fig_p031_9.png]
Figure 10
Figure 10. Figure 10: The mean effect of the different options of the numerical experiment [PITH_FULL_IMAGE:figures/full_fig_p033_10.png]
Figure 11
Figure 11. Figure 11: Direct comparison of loss terms for Burger’s equation. Each plot is [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]
Figure 12
Figure 12. Figure 12: Progress learning the equation for the pendulum: The model uses [PITH_FULL_IMAGE:figures/full_fig_p039_12.png]
Figure 13
Figure 13. Figure 13: Comparison of two operator loss terms for the fluid attractor equa [PITH_FULL_IMAGE:figures/full_fig_p041_13.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Adversarial Autoencoders in Operator Learning

    cs.LG 2024-12 conditional novelty 4.0 of 10

    Adding adversarial autoencoder training improves DeepONet and Koopman autoencoder accuracy by 4% to 27% on five small-data benchmarks.

  2. Some Best Practices in Operator Learning

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    Across six operator-learning tasks, GELU was the best activation, dropout always hurt, stochastic weight averaging helped only for small learning rates, and a learning rate finder was reported but not shown.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.