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arxiv: 2604.21097 · v2 · pith:L7OVMBI4new · submitted 2026-04-22 · 📊 stat.ML · cs.LG

Learning to Emulate Chaos: Adversarial Optimal Transport Regularization

Gabriel Melo , Leonardo Santiago , Peter Y. Lu This is my paper

Pith reviewed 2026-07-05 01:30 UTC · model glm-5.2

classification 📊 stat.ML cs.LG
keywords chaotic dynamicsoptimal transportadversarial traininginvariant measureemulatorWasserstein distancenoise robustnessSinkhorn divergence
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The pith

Adversarial Optimal Transport Trains Chaos Emulators from One Noisy Trajectory

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

This paper proposes a method for training neural network emulators of chaotic dynamical systems (such as weather models, turbulent flows, and plasma physics simulations) that does not require long-term trajectory accuracy, which is theoretically impossible for chaotic systems due to sensitivity to initial conditions. The standard approach of minimizing one-step prediction error (MSE) is shown to degrade catastrophically under noise, because the emulator's Jacobian amplifies perturbations exponentially at a rate governed by the maximal Lyapunov exponent. The authors instead regularize the MSE loss with an optimal transport (OT) cost that measures the discrepancy between the distribution of summary statistics of the true system's invariant measure and that of the emulator's induced measure. The key innovation is that the summary statistics are not handcrafted by domain experts but are learned adversarially: a summary map is trained to maximize the OT discrepancy between true and emulated trajectories, while the emulator is simultaneously trained to minimize it. This min-max game automatically discovers the most discriminative statistics, removing the need for expert knowledge or multi-environment datasets. The authors provide two tractable instantiations — a WGAN-style dual formulation for the 1-Wasserstein distance and a Sinkhorn divergence formulation for the p-Wasserstein distance — and prove that the OT regularizer is bounded by the one-step prediction error (Proposition 5.1), that MSE diverges exponentially under noise while the Wasserstein distance stabilizes to a finite value determined only by measurement noise and model error (Corollary 5.9 and Theorem 5.14), and that the adversarial summary objective inherits this noise robustness (Corollary 5.16). Experiments on Lorenz-96, Kuramoto-Sivashinsky, and Kolmogorov flow demonstrate improved long-term statistical fidelity, better spectral distance, and more accurate Lyapunov exponents compared to both MSE-only training and fixed-statistics OT baselines.

Core claim

The central mechanism is an adversarial min-max objective over an emulator g and a summary map f. The emulator minimizes a combined loss of one-step MSE and a Wasserstein distance between the pushforward measures f#μ (true data) and f#ĝ (emulator output), while the summary map maximizes that same Wasserstein distance. This forces f to find the statistics where the emulator most badly fails, and forces g to fix exactly those failures. The theoretical analysis establishes a clean separation: MSE captures short-horizon trajectory fidelity but diverges exponentially under noise at long horizons due to chaotic amplification through the Jacobian, while the Wasserstein term captures long-horizon统计f

What carries the argument

The adversarial optimal transport objective in Eq. (4), instantiated as either a WGAN-style dual (Eq. 6) or a Sinkhorn divergence (Eq. 8). The noise robustness theorem (Theorem 5.14) relies on the assumption that the emulator is exponentially mixing in Wp, meaning the emulator's induced dynamics forget initial condition perturbations at a geometric rate, so the Wasserstein distance between the true invariant measure and the emulator's noisy rollout measure converges to a finite limit independent of the initial condition noise.

If this is right

  • If the approach generalizes beyond the tested benchmarks, it could enable data-efficient surrogate models for climate simulation, turbulence modeling, and plasma physics that are trained from a single observational run rather than requiring large multi-scenario datasets.
  • The adversarial summary learning mechanism could be applied to other distribution-matching problems where the choice of test statistics is non-obvious, such as simulation-based inference or generative model evaluation in scientific domains.
  • The theoretical separation between trajectory-level losses (which diverge under chaos) and distributional losses (which remain bounded) provides a principled basis for designing training curricula that shift weight from MSE to OT as the rollout horizon exceeds the system's mixing time.
  • The ability to recover accurate Lyapunov exponents (Table 3) suggests the learned emulators preserve not just static statistics but dynamical invariants, which could make them useful for bifurcation analysis and stability studies of the underlying physical systems.

Where Pith is reading between the lines

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

  • The exponential mixing assumption on the learned emulator (Definition 5.12) is the load-bearing condition for noise robustness. If one could establish that the adversarial training procedure itself encourages or guarantees mixing — for instance, because matching the invariant measure under a Lipschitz summary map constrains the emulator's spectral properties — this would close the gap between theo
  • The displacement covariance result (Proposition A.1) suggests that the learned linear summary maps are implicitly performing a form of principal component analysis on the optimal transport displacement field, which connects the adversarial summary learning to spectral analysis of the transport plan.
  • The framework could potentially be extended to non-ergodic or transient systems by replacing the global invariant measure with a time-windowed empirical measure, though the theoretical guarantees on noise forgetting would need to be reformulated for non-stationary dynamics.
  • The observation that Sinkhorn training requires early stopping to prevent degenerate collapse of the summary map (Appendix C.1) suggests that the entropic regularization parameter and the adversarial training dynamics interact in ways that are not fully captured by the population-level theory, pointing to a need for finite-sample analysis of the min-max game.

Load-bearing premise

The noise robustness theorem assumes the trained emulator is exponentially mixing — meaning its dynamics forget initial perturbations at a geometric rate — but there is no proof that the training procedure produces an emulator with this property; it is empirically observed on the tested benchmarks but could fail for unstable or poorly trained models. The paper also acknowledges that the analysis does not guarantee the existence of a saddle point for the full min-max objective

What would settle it

If the exponential mixing assumption fails for a trained emulator — for example, if the emulator develops a spurious slow manifold or periodic orbit that does not forget initial perturbations — then the Wasserstein noise forgetting bound (Theorem 5.14) would not hold and the OT regularizer could become noise-dominated just like MSE. Empirically, this would manifest as the OT-regularized emulator degrading at long rollout horizons under noise, similar to the MSE-only baseline.

Figures

Figures reproduced from arXiv: 2604.21097 by Gabriel Melo, Leonardo Santiago, Peter Y. Lu.

Figure 1
Figure 1. Figure 1: Adversarial optimal transport regularization for emulating chaotic dynamics. (a) Emulator training via one-step prediction loss with OT regularization. (b) Adversarial learning of summary statistics that maximize the discrepancy between real and generated trajectory distributions while minimizing the full loss. 4. Our Approach: Adversarial Optimal Transport Regularization Motivated by the fact that chaotic… view at source ↗
Figure 2
Figure 2. Figure 2: KS full roll-out evaluation (clean data). Our WGAN￾style emulator most faithfully replicates the diagonal wave patterns and spatial structure of the ground truth (numerical simulation) across the full evaluation rollout. Lorenz-63 Attractor Geometry Comparison Baseline (No OT) WGAN (Learnable) σ = 0.1 20 15 10 5 0 5 10 15 20 x 20 10 0 10 20 y 5 10 15 20 25 30 35 40 45 z Ground truth Emulator 20 15 10 5 0 5… view at source ↗
Figure 3
Figure 3. Figure 3: L63 emulator geometry at increasing noise level σ. At σ = 0.10, the MSE baseline (No OT) underestimates the spatial extent of the attractor; at σ = 0.15, it collapses to a limit cycle, losing the bilobal structure entirely. WGAN maintains coverage of both lobes at both noise levels, directly illustrating how distributional regularization prevents attractor collapse under noise. More results on L63 are prov… view at source ↗
Figure 4
Figure 4. Figure 4: Lipschitz bounds during training for the WGAN summary map f on L96, under three regularization settings. Upper bounds (dashed) are computed as Q ℓ ∥Wℓ∥2; lower bounds (solid) are estimated from the mean Jacobian spectral norm over a batch subset. Prescribed thresholds Lmax = 4 and Lmax = 10 are shown as horizontal dash-dotted lines. Practical trade-off. Explicit regularization via per-step Jaco￾bian comput… view at source ↗
Figure 5
Figure 5. Figure 5: shows the Lorenz–63 attractor colored by the learned summary value s = fφ(u) across the three canonical projections (x, y), (x, z), and (y, z), alongside the projection induced by the dominant eigenvector of the displacement covariance CT (Proposition A.1). 2 1 0 1 2 x 3 2 1 0 1 2 3 y (x, y) 2 1 0 1 2 x 3 2 1 0 1 2 3 z (x, z) 3 2 1 0 1 2 3 y 3 2 1 0 1 2 3 z (y, z) 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 f_… view at source ↗
Figure 6
Figure 6. Figure 6: shows long-run state-space visitation histograms, comparing the ground truth system against the emulator trained with the adversarial OT objective. Both histograms are computed from a single long trajectory after transient removal. 2 1 0 1 2 0.0 0.1 0.2 0.3 0.4 0.5 Density x(t) Predicted True 3 2 1 0 1 2 3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Density y(t) Predicted True 3 2 1 0 1 2 3 0.0 0.1 0.2 0.3 0.4 0.5 Density… view at source ↗
Figure 7
Figure 7. Figure 7: compares the distribution of the learned summary statistic s extracted from ground truth trajectory segments against emulator-generated segments. 3 2 1 0 1 2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Density f(u(t)) Predicted True Summary Histogram Comparison [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Space-time plots of u(x, t) for the Lorenz–96 system (d = 60), comparing ground truth numerical simulations (left) against emulator rollouts (right) over 1,500 timesteps. Each row corresponds to a different method. As expected for chaotic systems, pointwise trajectory agreement is not maintained beyond the Lyapunov time; the relevant comparison is the statistical structure of the attractor over long horizo… view at source ↗
Figure 9
Figure 9. Figure 9: Space-time plots of u(x, t) for the Kuramoto–Sivashinsky equation (d = 256), comparing ground truth numerical simulations (left) against emulator rollouts (right) over 1,000 timesteps. Each row corresponds to a different method. Trajectory-level correspondence is not expected beyond the Lyapunov time due to chaos; the relevant comparison is the statistical structure of the attractor rather than pointwise a… view at source ↗
Figure 10
Figure 10. Figure 10: Vorticity rollouts for 2D Kolmogorov flow (Re = 104 , α = 0.1) at selected timesteps. Each row shows autoregressive predictions from a different model alongside the ground truth (top row). For quantitative analysis, see Tables 2. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
read the original abstract

Chaos arises in many complex dynamical systems, from weather to power grids, but is difficult to accurately model with data-driven methods such as machine learning emulators. While emulators are promising tools for accelerating simulations and solving inverse problems, they still struggle to learn chaotic dynamics, where sensitivity to initial conditions renders exact long-term forecasts infeasible, especially given noisy data. Recent work instead trains emulators to match the statistical properties of chaotic attractors, but these approaches often rely on handcrafted summary statistics or large, diverse multi-environment datasets. In this work, we propose a family of adversarial optimal transport objectives that can jointly learn high-quality summary statistics and a physically consistent emulator from a single noisy trajectory. We theoretically analyze and experimentally validate a Sinkhorn divergence formulation (2-Wasserstein) and a WGAN-style dual formulation (1-Wasserstein) of our approach. Numerical experiments across a variety of chaotic systems, including ones with high-dimensional spatiotemporal chaos, show that emulators trained using our proposed objectives have significantly improved long-term statistical fidelity.

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

4 major / 8 minor

Summary. This paper proposes adversarial optimal transport (OT) regularization for training emulators of chaotic dynamical systems. The core idea is to jointly learn an emulator g and a summary map f via a min-max objective (Eq. 4), where the emulator minimizes a combination of MSE and an OT cost between summary distributions of real and generated trajectories, while the summary map maximizes this OT cost to discover informative statistics. Two tractable instantiations are developed: a WGAN-style dual formulation (1-Wasserstein) and a Sinkhorn divergence formulation (2-Wasserstein). Theoretical analysis (Section 5) shows the OT regularizer is bounded by one-step prediction error (Prop. 5.1) and that the Wasserstein objective is robust to initial-condition noise under an exponential mixing assumption on the emulator (Thm. 5.14), unlike MSE which diverges exponentially (Cor. 5.9). Experiments on Lorenz-96, Kuramoto-Sivashinsky, and Kolmogorov flow demonstrate improved long-term statistical fidelity over MSE-only and fixed-statistics OT baselines.

Significance. The paper addresses a practically important problem: training data-driven emulators of chaotic systems from noisy data, where standard trajectory-matching losses are known to fail. The idea of adversarially learning summary statistics via OT, rather than relying on handcrafted features or multi-environment datasets, is a natural and well-motivated contribution that builds on and extends the framework of Jiang et al. (2023). The theoretical results in Section 5.1 are clean: Prop. 5.1 and Cor. 5.2 provide parameter-free bounds linking the summary-space Wasserstein cost to prediction error, and Prop. A.1 gives an exact characterization for the linear summary case. The noise-robustness analysis (Thm. 5.14 vs. Cor. 5.9) provides a clear conceptual distinction between distributional and trajectory-based objectives. The experimental scope across three systems of increasing dimensionality, including a 2D turbulent flow, is commendable. The paper is transparent about the limitations of its theoretical guarantees, explicitly noting at the end of Section 5.1 that the analysis does not ensure saddle-point existence.

major comments (4)
  1. Theorem 5.14, the central theoretical contribution on noise robustness, assumes the emulator g is exponentially mixing in W_p with rate rho in (0,1) (Definition 5.12). This is an assumption about the learned emulator's dynamics, not the true system. The paper does not prove that the trained emulator satisfies this property, nor does it directly verify it empirically. Table 3 shows that the WGAN emulator's LLE (2.336) matches the ground truth (2.334) on L96, but LLE measures local exponential divergence along a single trajectory, which is necessary but not sufficient for exponential mixing in W_p. A model could have the correct LLE but wrong global recurrence structure (e.g., converging to a periodic orbit with the correct local expansion rate). This gap is load-bearing because the noise-forgetting guarantee (Eq. 14) — the key advantage over MSE — holds only under this assumption. The bar
  2. should be either (a) a direct empirical verification of exponential mixing (e.g., estimating the rate of convergence of W_p((g∘k)#η, ν) as k increases for various initial measures η), or (b) a more careful discussion of what evidence the LLE match provides and what it does not, with an explicit acknowledgment that the theorem's applicability to the trained models remains unverified. As written, the connection between the theorem and the experiments is weaker than the presentation suggests.
  3. The Sinkhorn variant requires early stopping at 350/400 out of 500 training steps (Appendix C.1, Table 4) to avoid degenerate collapse of the summary map. Table 4 shows that without early stopping, Jacobian spectral norms collapse near zero with rare large spikes (max 5.762), indicating the adversarial game does not converge to a meaningful equilibrium. This is a significant practical concern: it suggests the min-max objective (Eq. 8) may not have a stable saddle point for the Sinkhorn formulation, and the reported results depend on a hand-tuned stopping criterion. The paper should discuss this more prominently (not only in the appendix) and clarify whether the WGAN variant, which does not appear to require early stopping, is the more robust recommendation. The sensitivity of results to the early stopping step should be reported.
  4. Table 2: For L96 single-trajectory, the Sinkhorn variant achieves L1 hist. error of 0.222 (noisy) and 0.082 (clean), while WGAN achieves 0.151 (noisy) and 0.115 (clean). Notably, Sinkhorn is better on clean but worse on noisy for L1 hist., and the reverse pattern appears for spectral distance (Sinkhorn 0.137 vs. WGAN 0.145 on noisy). This inconsistency makes it difficult to assess which formulation is preferred. More importantly, on KS single-trajectory noisy, the Sinkhorn variant (L1 hist. 0.435) barely outperforms the No OT baseline (0.454), and its clean L1 hist. (0.190) is better than Fixed OT (0.310) but the noisy performance is not. The paper should discuss these mixed results explicitly rather than stating that learnable OT methods 'consistently outperform' baselines.
minor comments (8)
  1. Section 5.1, end of paragraph: the statement that the analysis 'does not by itself ensure tractability or existence of a saddle point for the full min–max problem' is important and should perhaps be elevated to the main text of Section 4 or the Introduction, rather than appearing only in the theoretical analysis section, since it affects the interpretation of the entire approach.
  2. Eq. (16): the horizon-dependent weighting scheme with w_k → 0 for k >> tau_mix is proposed but the practical choice w_1 = 1, w_{k>1} = 0 means only one-step MSE is used. This is a very conservative choice; it would be useful to discuss whether intermediate choices were tried and what effect they had.
  3. Table 2: the Kolmogorov Flow row labels the baseline as 'No OT ((baseline)' with an extra parenthesis.
  4. Section 6, Evaluation protocol: the noise level sigma=0.3 is used for training and noisy evaluation, but the relationship between sigma and the noise model in Section 5.2 (sigma_1, sigma_2) is not made explicit. Are sigma_1 = sigma_2 = 0.3 * std in the experiments?
  5. Appendix C.1, Table 4: the WGAN row reports Jacobian spectral norms (min 1.290, max 2.126) that are consistently higher than Sinkhorn with early stopping (min 0.250, max 0.716). It would help to discuss whether this difference in Lipschitz regime affects the comparison between the two formulations.
  6. Proposition A.1: the equality condition requires the Brenier map T* to be separable in the eigenbasis of C_T. The paper notes this 'provides some intuition' but the L63 visualizations in Figure 5 are used to support the practical relevance. It would be worth stating more explicitly how often this separability condition is expected to hold in practice.
  7. The paper uses 'Wasserstein noise forgetting' (Theorem 5.14) as a key term. This is evocative but could be confused with the standard notion of 'forgetting' in sequential learning. Consider clarifying that this refers to the decay of the initial-condition perturbation in the Wasserstein distance under the emulator's dynamics.
  8. References: the Peyré (2025) reference is to an arXiv preprint on 'Optimal transport for machine learners'; this appears to be a survey/lecture note and should be cited as such.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. All three major comments identify genuine gaps that we will address in revision. On Comment 1 (exponential mixing assumption), we agree the connection between Theorem 5.14 and the experiments is weaker than presented and will add both an empirical mixing-rate diagnostic and a careful discussion of what LLE does and does not certify. On Comment 2 (Sinkhorn early stopping), we agree this is a significant practical limitation that deserves prominent discussion in the main text, and we will promote it from the appendix and clarify that WGAN is the more robust recommendation. On Comment 3 (mixed results), we agree the word 'consistently' overstates the findings and will revise the discussion to address the mixed and system-dependent patterns honestly.

read point-by-point responses

  1. Referee: Theorem 5.14 assumes the emulator g is exponentially mixing in W_p (Definition 5.12), but the paper does not prove the trained emulator satisfies this property, nor does it directly verify it empirically. LLE matching is necessary but not sufficient for exponential mixing. The referee requests either (a) direct empirical verification of exponential mixing or (b) more careful discussion of what LLE does and does not show, with explicit acknowledgment that the theorem's applicability remains unverified.

    Authors: The referee is correct on both counts. LLE measures local exponential divergence along a single trajectory and is indeed necessary but not sufficient for exponential mixing in W_p: a model could have the correct local expansion rate but wrong global recurrence structure. We will address this in two ways. First, we will add an empirical mixing-rate diagnostic: for each trained emulator, we estimate W_p((g∘k)_#η, ν) as k increases for several initial measures η (e.g., point masses, uniform on a grid, and the noisy observed measure), where ν is the empirical invariant measure estimated from a long emulator rollout. If the emulator is exponentially mixing, this quantity should decay geometrically; we will report the estimated rate ρ and compare it across methods. Second, we will revise the discussion around Table 3 to explicitly state that LLE matching provides evidence of correct local chaotic behavior but does not certify the global mixing property required by Theorem 5.14, and that the theorem's applicability to the trained models remains an assumption verified only empirically (not proven). The current presentation overstates the connection between the theorem and the experiments, and we will correct this. revision: yes

  2. Referee: The Sinkhorn variant requires early stopping at 350/400 out of 500 training steps (Appendix C.1, Table 4) to avoid degenerate collapse of the summary map. Without early stopping, Jacobian spectral norms collapse near zero with rare large spikes, indicating the adversarial game does not converge to a meaningful equilibrium. This is a significant practical concern that should be discussed more prominently, not only in the appendix. The paper should clarify whether WGAN is the more robust recommendation and report sensitivity to the early stopping step.

    Authors: We agree that this is a significant practical limitation that deserves prominent discussion in the main text, not only in the appendix. The Jacobian spectral norm statistics in Table 4 clearly show that the Sinkhorn min-max game can degenerate without early stopping, and this does suggest potential non-existence of a stable saddle point for the Sinkhorn formulation in our setting. We will make three changes: (1) promote the early-stopping discussion and Table 4 (or a summary thereof) from Appendix C.1 to the main text, likely in Section 6 or 7; (2) explicitly state that the WGAN variant, which does not require early stopping in any of our experiments, is the more robust practical recommendation; and (3) report sensitivity to the early stopping step by including results at multiple stopping points (e.g., 300, 350, 400, 450, 500) for at least one system. We will also note that the Sinkhorn instability is consistent with known challenges in adversarial training of learnable-cost OT objectives, and that the joint hinge regularization proposed in Appendix C.2 offers a principled (if more expensive) alternative for stabilizing the summary map. revision: yes

  3. Referee: Table 2 shows mixed results: for L96 single-trajectory, Sinkhorn is better on clean but worse on noisy for L1 hist., and the reverse pattern appears for spectral distance. On KS single-trajectory noisy, Sinkhorn (L1 hist. 0.435) barely outperforms No OT (0.454). The paper should discuss these mixed results explicitly rather than stating that learnable OT methods 'consistently outperform' baselines.

    Authors: The referee is correct that the results are more mixed than our current language suggests. The specific examples cited are accurate: on L96 single-trajectory, Sinkhorn and WGAN trade off between noisy and clean settings and between L1 histogram error and spectral distance; on KS noisy, Sinkhorn's L1 hist. error (0.435) is only marginally better than No OT (0.454), while WGAN achieves 0.339. The word 'consistently' overstates the pattern. We will revise the discussion in Section 7 to present a more nuanced and honest assessment: (1) both learnable OT methods substantially outperform No OT on spectral distance across all systems and conditions, which is the clearest and most consistent advantage; (2) on L1 histogram error, the advantage is system- and condition-dependent, with WGAN generally stronger on noisy data and Sinkhorn sometimes stronger on clean data; (3) on KS noisy, Sinkhorn's advantage over No OT on L1 hist. is marginal, and we will state this directly. We will also discuss the possibility that the Sinkhorn formulation's instability (Comment 2) contributes to its weaker noisy-data performance relative to WGAN. The revised text will replace 'consistently outperform' with a system-by-system and metric-by-metric summary. revision: yes

Circularity Check

0 steps flagged

No significant circularity: theoretical results are parameter-free derivations from stated assumptions; evaluation uses independent fixed metrics.

full rationale

The paper's theoretical results (Proposition 5.1, Theorem 5.14, Corollary 5.9) are standard mathematical derivations from explicitly stated assumptions (Lipschitz summary maps, exponential mixing, Gaussian noise). None of these assumptions embed the target conclusion as a hidden input. The key bound in Proposition 5.1 (W_{d_S^p}(f#µ, f#µ̂) ≤ L_f^p E[d_U(Φ(u), g(u))^p]) follows from constructing a feasible coupling and applying the Lipschitz property—a direct, non-circular derivation. Theorem 5.14's noise-forgetting result follows from the triangle inequality plus the exponential mixing assumption (Definition 5.12); while that assumption is strong and unverified for the learned emulator (a correctness concern, not a circularity concern), it does not encode the theorem's conclusion. The evaluation metrics (L1 histogram error, spectral distance) use fixed handcrafted summary statistics (Table 1) that are independent of the learned summaries, explicitly stated: 'The histogram error is always computed using fixed and system-specific summary statistics given by S(u) listed in Table 1, regardless of the summary representation used during training, ensuring a fair comparison.' The paper builds on Jiang et al. (2023) for architecture and data generation, but these are external citations to different authors. The one self-citation to Jiang et al. (2025) is for the UNet architecture used in the Kolmogorov flow experiments, which is not load-bearing for the theoretical claims. No step in the derivation chain reduces to its inputs by construction. The score of 1 reflects the minor, non-load-bearing nature of the architectural self-citation and the absence of any circular reasoning in the theoretical or experimental methodology.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 1 invented entities

The axiom ledger captures the key assumptions and parameters. The exponential mixing assumption on the emulator is the most ad hoc, as it is not verified and is load-bearing for the main theoretical result.

free parameters (5)
  • λ (OT regularization weight) = 3 (L96), grid-searched (KS)
    Controls the balance between MSE and OT terms; chosen empirically per system.
  • ε (Sinkhorn entropic regularization) = 0.02 (L96, KS), 0.05 (L63)
    Controls the entropic regularization in the Sinkhorn divergence; chosen empirically.
  • p (Wasserstein order) = 2 (Sinkhorn), 1 (WGAN)
    Fixed by the choice of formulation.
  • d (summary dimension) = 3 (default), 1 (ablation)
    Output dimension of the summary map; chosen to match the fixed baseline's 3 handcrafted features.
  • Early stopping step (Sinkhorn) = 350/400 of 500
    Practical hack to prevent degenerate Jacobian collapse; not theoretically motivated.
axioms (5)
  • domain assumption The true dynamics Φ admits an invariant and ergodic probability measure μ on a compact attractor U.
    Invoked in §2.1 to justify learning from a single trajectory via ergodicity (Eq. 1). Standard for chaotic systems with strange attractors.
  • domain assumption The summary map f is L_f-Lipschitz.
    Required for Propositions 5.1, 5.3 and Corollary 5.16. Enforced in practice via weight clipping (Appendix D.1) but not explicitly guaranteed; empirically verified in Tables 4-5.
  • ad hoc to paper The emulator g is exponentially mixing in Wp with rate ρ∈(0,1).
    Definition 5.12, required for Theorem 5.14 (the key noise-forgetting result). This is assumed about the *learned* emulator, not just the true system. No proof that trained emulators satisfy this.
  • domain assumption Noise is additive isotropic Gaussian, independent of state.
    §5.2 noise model. The paper argues (Remark 5.17) that qualitative conclusions hold for other noise models, but all quantitative bounds depend on this assumption.
  • domain assumption The summary hypothesis class F contains a scaling map f(u)=L_f u when U⊆S⊆R^n.
    Required for Corollary 5.4 (tightness of the Wasserstein bound). Holds for the MLP architecture used but is a structural assumption on the hypothesis class.
invented entities (1)
  • Adversarial summary map f: U → S independent evidence
    purpose: Learns to maximize the OT discrepancy between true and emulator-induced distributions in summary space, automatically discovering informative statistics.
    The summary map is a neural network trained adversarially; its effectiveness is validated empirically on three benchmark systems (Table 2) and its Lipschitz properties are measured (Tables 4-5). The learned summaries are compared against transport-optimal linear directions (Proposition A.1, Figure 5) providing some independent theoretical grounding.

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    How Well Do WGANs Estimate the Wasserstein Metric?

    URL https://openreview.net/forum? id=c8P9NQVtmnO. Li, Z., Liu-Schiaffini, M., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., and Anandku- mar, A. Learning chaotic dynamics in dissipative systems. InAdvances in Neural Information Processing Systems, volume 35, pp. 16768–16781, 2022. Lu, L., Jin, P., Pang, G., Zhang, Z., and Karni...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1038/s42256-021-00302-5 2022

  5. [5]

    Our learned MLP summary map is: ut 7→MLP(u t 1, ut 2)∈R 2×d

    (∂tut 2,adv t 2, u t 2) ∈R 2×3. Our learned MLP summary map is: ut 7→MLP(u t 1, ut 2)∈R 2×d. In both cases, across the full trajectory you collectT×2vectors inR d as a point cloud. Overview of OT regularization strategies.We compare the fixed-statistics OT baseline introduced by Jiang et al. (2023) against two learnable OT-based regularizers. All three ap...

    work page 2023