Single-shot prediction of parametric partial differential equations

Aditya G. Nair; Khalid Rafiq; Wenjing Liao

REVIEW 4 major objections 7 minor 73 references

Single-shot prediction of parametric partial differential equations

T0 review · 4 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Flexi-VAE claims that one feed-forward evaluation of a latent propagator can replace iterative time-stepping for parametric PDE forecasting, offering constant-time cost and accurate long-horizon forecasts across unseen parameters.

desk verdict A useful single-shot latent-propagation architecture with a clean empirical core, but the theory is oversold and the comparisons need equal-budget controls. read the letter →

arxiv 2505.09063 v1 pith:2OSEJGHL submitted 2025-05-14 cs.LG cs.NAmath.NA

Khalid Rafiq , Wenjing Liao , Aditya G. Nair This is my paper · ORCID

classification cs.LGcs.NAmath.NA

keywords single-shotforecastingparametricPDEsvariationalautoencoderlatentspacepropagationmodelreductionBurgersequationadvection-diffusiongeometry

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

Flexi-VAE aims to establish that a variational autoencoder with a feed-forward latent propagator can forecast parametric PDEs in a single shot: encode the current solution field, advance the low-dimensional code by any requested horizon $\tau$ in one evaluation conditioned on the parameters, and decode the result. The paper reports accurate long-horizon forecasts for the 1D viscous Burgers equation and the 2D advection-diffusion equation, including Reynolds numbers outside the training range, with constant runtime and up to 50x CPU and 90x GPU speedups over an AE-LSTM baseline. It also claims that the direct-concatenation propagator yields disentangled latent coordinates tied to physical quantities, and that propagated latent states occupy more stable regions of the decoder than directly encoded states. If these claims hold, PDE surrogate modeling changes character: a single function evaluation replaces sequential rollout, making long-horizon parametric studies and real-time control problems substantially cheaper. The supporting theory proves existence of such networks with an error depending on the intrinsic manifold dimension, though it is framed on a patched latent space rather than the single fixed latent space used in the experiments.

What carries the argument

The key object is the parametric latent propagator $P_{\theta_p}(\tilde{z}(t,\zeta); \tau, \zeta)$, a feed-forward network that advances the latent code from time $t$ to time $t+\tau$ in one pass, conditioned on the forecast horizon and the PDE parameters. Two instantiations are studied: the Direct Concatenation Propagator (DCP), which appends $\zeta$ and $\tau$ directly to the latent vector, and the Positional Encoding Propagator (PEP), which adds sinusoidal embeddings of $\zeta$ and $\tau$ in a higher-dimensional space before projecting back down. The propagator is trained jointly with the encoder and a shared decoder by a loss combining reconstruction error, KL divergence, and a physical-space propagated reconstruction error; that last term is what aligns latent evolution with true physical state reconstruction. The accompanying representation theorem guarantees that, for solutions lying on a compact smooth $m$-dimensional manifold, networks of size scaling as $\epsilon^{-m}\log\epsilon^{-1}$ can realize the whole pipeline with $O(\epsilon)$ prediction error, making the intrinsic dimension rather than the ambient state dimension the controlling complexity.

What would settle it

Train Flexi-VAE on one part of the Re range and evaluate on a region where the MLE intrinsic-dimension estimate (with the same $k$-nearest-neighbor settings) rises above the fixed latent dimension; if reconstruction remains accurate but forecast error diverges there, the global-chart assumption is the cause. A second decisive check is to run the trained propagator at horizons far beyond training, for example $\tau/\Delta t > 450$ on Burgers, and compare against the closed-form solution: the single-shot claim predicts roughly horizon-independent error, whereas a drift growing with $\tau$ would indicate error accumulation and falsify the mechanism.

Watch

Extended reading notes

Core claim

The paper's central claim is that the composition $\hat{u}(x,t+\tau,\zeta) = D_{\theta_d}(P_{\theta_p}(E_{\theta_e}(u(x,t,\zeta)), \tau, \zeta))$ approximates the true parametric evolution operator $F(u(x,t,\zeta),\tau,\zeta)$ well enough to serve as a surrogate model, and that this is achieved in a single feed-forward evaluation rather than by stepping through intermediate times. The propagator loss is evaluated in physical space, so the propagated latent code $\hat{z}$ is not required to equal the encoded code $\tilde{z}$ of the true future state; the paper shows these can be distinct yet decode to nearly identical fields, and that the propagated code lies in a flatter, better-conditioned region of the decoder as measured by the Jacobian $\partial D_{\theta_d}/\partial z$ and $\det(J^\top J)$. Between the two propagator designs, the Direct Concatenation Propagator is claimed to generalize better, be more data-efficient, and produce disentangled coordinates tied to physical quantities such as peak position and shock sharpness. Theorem 1 asserts that ReLU encoder, propagator, and decoder networks exist with prediction error bounded by $\epsilon$ and network size scaling as $\epsilon^{-m}\log \epsilon^{-1}$, where $m$ is the intrinsic manifold dimension; the theorem is proved on a patched latent space of dimension $C_M(m+1)$, while the experiments use a single global latent space of dimension $m=2$ or $m=3$.

Load-bearing premise

The load-bearing premise is that the full family of PDE solutions across all parameters and horizons can be represented in one fixed low-dimensional coordinate system of dimension 2 or 3, with the decoder staying stable along propagated coordinate paths; the existing theory is proved on a patched space of dimension $C_M(m+1)$, not on a single global chart.

Editorial extensions

If this is right

The wall-clock cost of a forecast becomes independent of the horizon, since one forward pass produces any requested $\tau$; the reported constant runtime replaces the linear-in-$\tau$ cost of autoregressive LSTM rollouts.
Long-horizon parametric studies become feasible in real-time settings, with measured speedups of up to 50x on CPU and 90x on GPU against the AE-LSTM baseline.
Unseen parameter values can be forecast without retraining when the solution family is captured by the fixed latent coordinate system, as shown on extrapolated Reynolds numbers for both equations.
The DCP variant supplies a physically interpretable latent space in which individual coordinates track identifiable features such as peak position and shock sharpness.
Propagated latent trajectories are more robust to decoder perturbations than directly encoded trajectories, supporting stable long-horizon decoding.

Reading between the lines

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

A direct stress test is to push the training parameter range until the estimated intrinsic dimension exceeds the fixed latent dimension; at that point the single-chart assumption predicts visible forecast failure, and the location of the failure would pinpoint where the global-coordinate premise breaks.
The geometric stability result suggests a training-time regularizer the authors did not implement: adding a penalty on the decoder Jacobian norm or on $\det(J^\top J)$ at propagated latents could actively steer the propagator toward flat regions and possibly improve extrapolation.
Because the proved theorem needs $C_M$ charts, a charted or mixture-of-experts latent space is the natural generalization if a single global chart cannot coordinatize larger PDE families; this would test whether the speed and simplicity of a single-chart propagator survive a principled patching procedure.
If the approach scales to higher-dimensional flows, its main contribution may be computational rather than accuracy-based: changing the complexity class of online tasks such as PDE-constrained control, where a constant-time surrogate can be evaluated many times per control step.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing to know: the empirical core of Flexi-VAE is plausible and worth engaging. The architecture—a VAE with a shared decoder and a feedforward parameter-and-horizon-conditioned latent propagator—delivers constant-time forecasts on Burgers and advection-diffusion, with DCP beating PEP in their runs, interpretable latent coordinates (peak position and sharpness), and 50-90x speedups over AE-LSTM. They ship code and a demo, which is more than most.

What's actually new: the DCP/PEP comparison and the decoder-Jacobian analysis are the main contributions, plus the specific design of conditioning latent propagation on (ζ, τ) in a single feedforward pass. The observation that propagated latents land in flatter decoder regions is interesting, though it is an empirical observation, not a representation-theoretic proof of DCP's superiority as the abstract suggests.

The soft spots are real but mostly addressable. The biggest is the theory-apparatus mismatch. Theorem 1 is proved for a patched latent space of dimension C_M(m+1), not for the fixed 2D or 3D latent codes the model actually uses. It doesn't justify a single global chart of dimension m. And as your stress-test note says, the proof applies Lemma 2 to P*, but P* is only shown to be Lipschitz, not C^k; Lemma 2 requires C^k. So Eq. (6) isn't established as written. This doesn't sink the empirical story—the experiments are direct regression on future states—but the paper overclaims by citing Theorem 1 as motivation for m=2.

Second, the comparisons are not clean. PEP was trained on 20k tuples, DCP on 80k; batch sizes and epochs differ. There are no error bars or repeated seeds, and the central MSE heatmaps are presented without uncertainty. The advection-diffusion section only tests DCP, so the DCP-vs-PEP claim is limited to Burgers. Stronger baselines like FNO or DeepONet are missing; AE-LSTM alone is a thin bar. The data-efficiency power law, MSE~p^{-0.57}, is fit through few points and shouldn't be over-read. And 'extensibility to higher-dimensional systems' is stated, not shown.

Citation pattern looks fine; they build on the relevant latent-dynamics and manifold literature without inventing prior work. The main thing is that the theory and the experiments are not as tightly coupled as the narrative claims.

Who this is for: people building latent-dynamics surrogates for parametric PDEs. It's a practical design with a useful inductive bias. It deserves a serious referee—the central idea is clear and the evaluation, while flawed, is substantial. My recommendation: send it to review, ask for equal-budget comparisons, uncertainty estimates, a corrected or heavily softened theorem, and at least one operator-learning baseline. Conditional accept rather than reject.

Referee Report

4 major / 7 minor

Summary. The paper introduces Flexi-VAE, a variational-autoencoder framework for single-shot (non-autoregressive) forecasting of parametric PDEs. Given a state u(x,t,ζ), an encoder produces a low-dimensional latent code z̃, a feedforward propagator P_θ(z̃; τ, ζ) advances it to a horizon τ, and a shared decoder returns the forecasted field. Two propagator designs are compared: DCP, which concatenates (ζ, τ) to the latent code, and PEP, which uses sinusoidal embeddings in a higher-dimensional space. Experiments on the 1D viscous Burgers equation (Re ∈ [400,2400]) and the 2D advection–diffusion equation report accurate interpolation and extrapolation, latent-space interpretability for DCP, and 50×/90× CPU/GPU speedups over an AE-LSTM baseline. A representation theorem (Theorem 1) is stated and proved in Appendix A to justify the encoder-decoder-propagator architecture on a low-dimensional manifold.

Significance. The single-shot, parameter-conditioned latent propagator is a practically relevant idea for PDE surrogate modeling, and the reported constant-time inference with large wall-clock speedups over an autoregressive baseline is valuable if reproducible. The paper ships code and an interactive demo, which supports reproducibility. The empirical study covers two canonical benchmarks and includes interpolation/extrapolation splits, which is reasonable for a first demonstration. The main weakness is that the theoretical result does not actually cover the implemented architecture: Theorem 1 constructs a patched C_M(m+1)-dimensional latent representation, while the experiments use a single m-dimensional latent space; the propagator in the theorem takes only the latent code as input, not (z, τ, ζ); and the proof applies a C^k approximation lemma to a target that is only shown Lipschitz. These gaps do not by themselves invalidate the empirical findings, but they mean the paper's representation-theoretic claims currently outrun the proof.

major comments (4)

[Theorem 1 (§2.2, Appendix A.2)] The theorem's propagator network is declared to belong to FNN(C_M(m+1), C_M(m+1), ...), i.e., its input is only the latent code z, yet the architecture in Eq. (3) and Algorithm 1 feeds (z, ζ, τ) to the propagator. Because P*(z, τ, ζ) = E*∘F(D*(z), τ, ζ) depends on τ and ζ, a network whose input dimension is C_M(m+1) cannot satisfy the stated uniform bound 'for all t, τ, ζ' in Eq. (24). The statement and proof must either include the parameter and horizon dimensions in the propagator's input class or restrict the theorem to a single fixed (τ, ζ); the latter would remove the parametric character of the claim.
[Appendix A.2, propagator approximation] The bound in Eq. (24) is derived by applying Lemma 2 to P*(·, τ, ζ), but Lemma 2 requires the target function to be C^k (in the application, C^1). The manuscript only proves that P* is Lipschitz, with Lipschitz constant bounded by Lip E* Lip F Lip D*; no C^1 regularity of F is assumed. Consequently Eq. (24) is not established and the error estimate in Theorem 1 does not follow as written. The proof can be repaired by assuming F is C^1 (or C^k) and showing P* inherits this regularity, or by invoking a Lipschitz approximation result; without such a change the theorem's guarantee is unproven.
[Theorem 1 vs. §3.1.1 and §3.2.1] The implemented Flexi-VAE uses a single latent space of dimension m (m=2 for Burgers, m=3 for advection-diffusion), but the oracle construction in Theorem 1 operates in R^{C_M(m+1)} using a C_M-chart atlas with partition-of-unity weights. The theorem never asserts that one global chart of dimension m exists, and the MLE intrinsic-dimension estimates quoted in §3.1.1 and §3.2.1 do not imply global coordinatizability. Thus the sentence in §3.1.1 that the latent dimension is 'motivated by the theoretical guarantee in Theorem 1' overstates what the theorem provides; the representational premise of the fixed low-dimensional latent space is not covered by the theory.
[Abstract and §3.1.4] The abstract states that DCP's superior long-term generalization is demonstrated 'through representation-theoretic analysis.' Theorem 1 is an existence result for a generic encoder-decoder-propagator triple; it does not compare DCP with PEP or analyze parameter concatenation. The DCP-versus-PEP conclusions are entirely empirical (Figs. 6, 9, 10). This phrasing should be corrected to avoid claiming theoretical support for an architectural comparison the theory does not address.

minor comments (7)

[Eq. (6), Theorem 1] The displayed inequality is missing a closing parenthesis after Eθe(u(x,t,ζ)) and after Dθd∘Pθp, making the statement hard to parse; the proof in Appendix A.2 makes the intended expression clear.
[Appendix A.2, Step 2] The proof uses d where the intrinsic dimension m is meant, e.g., 'E* : M → R^{C_M(d+1)}' and '(z_j)_{1:d}'; this conflicts with the theorem's notation and is confusing given that d is also used for the PEP embedding dimension.
[Eq. (2), Tables 1–2, Algorithm 1] The propagation-loss weight is denoted η in Eq. (2) and Algorithm 1 but γ in Tables 1 and 2; please unify the notation.
[§3.1.5, Figs. 11–13] The claim that propagated latents generically occupy lower-sensitivity decoder regions rests on a single pair of latent vectors. Please compute statistics over many (Re, τ) samples, or temper the claim to a case study.
[§3.1.3] The test Reynolds numbers 'Re∈{550,1025,...,2450}' are listed with an irregular step and include 1025, which lies inside the training set; please clarify the intended test set.
[Fig. 9, §3.1.4] The power-law fit with slope −0.57 is over the short range 7K–80K tuples; please report the number of fit points, the fit uncertainty, and use error bars.
[Throughout] Minor typos include 'Propogator' in Eq. (2), 'diffferent' in §3.1, 'Activiation' in §2.2, and 'the goal is of this paper' in §2.1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Flexi-VAE forecasts are supervised directly on future states, and Theorem 1 is a constructive approximation result rather than an input-output tautology.

full rationale

The paper's central prediction pipeline does not reduce to a fitted value or to a self-citation. The propagated reconstruction loss in Eq. (2) is a direct supervised regression on future physical states, and the reported evaluation is carried out on interpolation and extrapolation splits of the joint (Re, tau) space, so the test errors are not forced by construction. The latent dimension choice is motivated by an independent MLE intrinsic-dimension estimate (“the results consistently yield intrinsic dimension estimate of 1.97, supporting the use of a 2D latent space”), not by fitting the forecast targets. Theorem 1 is also not circular: the oracle encoder/decoder pair E* and D* is explicitly constructed in Appendix A.2 (“we still present its construction for completeness”), and the approximation bound in Eq. (6) follows from standard neural-network approximation lemmas via a triangle inequality. The fact that the proof invokes prior manifold-approximation results by co-author Liao ([8], [37], [38]) is a self-citation, but it is not load-bearing in the sense of an unverified uniqueness theorem; the relevant construction is reproduced in the appendix, and the empirical claims stand independently of those citations. The known gap between Theorem 1's patched C_M(m+1)-dimensional representation and the implemented fixed m-dimensional latent space is a correctness/assumption risk, not a circularity, because the theorem is not used to define the trained output. No equation or fitted parameter is renamed as a prediction, and no central claim is equivalent to its input by definition.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central empirical claims rest mainly on supervised data from exact analytical solutions and on a set of tuned hyperparameters. The theory adds an existence result under a manifold-plus-smoothness assumption but does not justify the actual 2D or 3D global latent representation, and one step of the proof requires regularity not established. No new physical entities are postulated.

free parameters (7)

Latent dimension m = 2 (Burgers), 3 (advection-diffusion)
Set from MLE intrinsic-dimension estimates (1.97 and 2.2) and treated as model capacity; the central forecasting claim depends on this compression.
KL divergence weight beta = 4e-5 (PEP), 1.2e-5 (DCP Burgers), 1.152e-5 (2D DCP)
Bayesian-optimized regularization weight in Eq. (2); balances reconstruction and latent structure.
Propagation loss weight eta = 0.60 (PEP), 1.70 (DCP Burgers), 1.15 (2D DCP)
Bayesian-optimized weight on the propagated reconstruction loss; controls forecast fidelity.
Embedding dimension d for PEP = 64
Hand-chosen or optimized dimension for sinusoidal parameter embeddings in the PEP propagator.
Training dataset size = 20,000 tuples (PEP), 80,000 tuples (DCP)
Different data budgets for the compared propagators; this confounds the reported DCP-better conclusion.
Learning rate and batch size = alpha = 8e-4 / 7e-4 / 7e-4; batch = 256 / 64 / 64
Bayesian-optimized training hyperparameters reported in Tables 1 and 2.
Empirical scaling exponent for DCP data efficiency = -0.57
Least-squares slope of validation MSE versus dataset size in the 7K to 80K range; reported as an empirical scaling law.

assumptions (6)

domain assumption The set of PDE solutions lies on a compact m-dimensional Riemannian manifold isometrically embedded in R^n with reach tau_M > 0.
Invoked in Theorem 1 and Sec. 2.2; if the solution family is not intrinsically low-dimensional, the fixed low-rank latent propagator has no theoretical basis.
domain assumption The solution map F(u(t), tau, zeta) = u(t+tau, zeta) is Lipschitz in u uniformly in tau and zeta.
Footnote 3, used to prove Lipschitzness of the oracle propagator; not verified empirically.
ad hoc to paper The oracle latent propagator P* is regular enough (C^k) for the neural-network approximation lemmas used in the proof.
Appendix A applies the C^k approximation result of Yarotsky to P*, but only Lipschitz continuity is proved for P*, leaving a regularity gap.
domain assumption The closed-form Cole-Hopf solution Eq. (11) and the Gaussian advection-diffusion solution Eq. (14) correctly reproduce the PDEs with the stated boundary and initial conditions.
All supervised labels are generated from these formulas; if the formulas are wrong, every reported error is meaningless.
ad hoc to paper A single latent space of dimension m, rather than the C_M(m+1) chart space, can coordinate the solution manifold.
Implemented m=2 and m=3 while Theorem 1 uses a chart count C_M; the MLE estimate is suggestive but the theorem does not imply global single-chart representability.
standard math The neural-network approximation results of Yarotsky and Chen et al. for functions on low-dimensional manifolds are valid.
Used throughout the proof of Theorem 1 in Appendix A.

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Cite this review

Pith. "Pith review of Single-shot prediction of parametric partial differential equations." pith.science (2026). https://pith.science/paper/2OSEJGHL

@misc{pith2026250509063,
  author       = {Pith},
  title        = {Pith review of: Single-shot prediction of parametric partial differential equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2OSEJGHL}},
  note         = {Machine review of arXiv:2505.09063}
}

read the original abstract

We introduce Flexi-VAE, a data-driven framework for efficient single-shot forecasting of nonlinear parametric partial differential equations (PDEs), eliminating the need for iterative time-stepping while maintaining high accuracy and stability. Flexi-VAE incorporates a neural propagator that advances latent representations forward in time, aligning latent evolution with physical state reconstruction in a variational autoencoder setting. We evaluate two propagation strategies, the Direct Concatenation Propagator (DCP) and the Positional Encoding Propagator (PEP), and demonstrate, through representation-theoretic analysis, that DCP offers superior long-term generalization by fostering disentangled and physically meaningful latent spaces. Geometric diagnostics, including Jacobian spectral analysis, reveal that propagated latent states reside in regions of lower decoder sensitivity and more stable local geometry than those derived via direct encoding, enhancing robustness for long-horizon predictions. We validate Flexi-VAE on canonical PDE benchmarks, the 1D viscous Burgers equation and the 2D advection-diffusion equation, achieving accurate forecasts across wide parametric ranges. The model delivers over 50x CPU and 90x GPU speedups compared to autoencoder-LSTM baselines for large temporal shifts. These results position Flexi-VAE as a scalable and interpretable surrogate modeling tool for accelerating high-fidelity simulations in computational fluid dynamics (CFD) and other parametric PDE-driven applications, with extensibility to higher-dimensional and more complex systems.

Figures

Figures reproduced from arXiv: 2505.09063 by the authors.

**Figure 2.** To ensure that the propagator Pθp effectively integrates both temporal evolution and parametric dependencies, we explore two distinct parameter-informed strategies. The first, the positional encoding propagator (PEP), encodes system parameters and temporal offsets into a structured high-dimensional embedding inspired by transformer architecture[67]. This is achieved by mapping parameters ζ and τ into an embedding … view at source ↗

**Figure 2.** Schematic representation of the Flexi-VAE architecture. The encoder compresses the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

Figures from the paper (15 more)

**Figure 3.** Figure 3: Comparison of two parameter-informed latent propagation strategies in the Flexi-VAE [PITH_FULL_IMAGE:figures/full_fig_p008_3.png]

**Figure 4.** Figure 4: Overview of the Flexi-VAE architecture. The encoder [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Forecasting performance of the DCP based Flexi-VAE model on the 1D viscous Burgers’ [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: Mean Squared Error (MSE) distribution across 30,000 randomly sampled future state [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: Temporal evolution analysis for left and right extrapolation Reynolds numbers. Pre [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Prediction time comparison for Flexi-VAE and AE-LSTM across a range of temporal [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: Mean squared error (MSE) on the Reynolds number extrapolation region as a function [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 10.** Figure 10: Latent space structure for PEP (left) and DCP (right) across training sizes. Each point [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

**Figure 11.** Figure 11: High-dimensional reconstructions of u(x, t, ζ) obtained by decoding (left) z˜ and (right) zˆ, each colorcoded by the magnitude of the Jacobian |∂Dθd /∂z1| at that point. Despite yielding nearly identical outputs, the decoder exhibits sharper sensitivity at z˜, particu…

**Figure 12.** Figure 12: Effect of perturbations applied independently along each latent coordinate for the direct [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]

**Figure 13.** Figure 13: From left to right: (Left) Singular value spectra Σ(J(z˜)) and Σ(J(zˆ)), illustrating the principal local scaling behavior of the decoder at direct and propagated representations. (Middle) Frobenius norms ∥J(z˜)∥F and ∥J(zˆ)∥F , providing an aggregate measure of decod…

**Figure 14.** Figure 14: Schematic representation of the advection-diffusion process. The scalar field is trans [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]

**Figure 15.** Figure 15: Model generalization across temporal and parametric regimes. The bottom-left panel [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]

**Figure 16.** Figure 16: Workflow about the evolutionary operator, encoder, decoder and latent propagator. [PITH_FULL_IMAGE:figures/full_fig_p027_16.png]

**Figure 17.** Figure 17: Data splitting strategy across parameter [PITH_FULL_IMAGE:figures/full_fig_p029_17.png]

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