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PDE-JEPA: Predictive Representation Learning of Latent Dynamics Modeling for Parametric PDEs

Zhentao Tan, Jianrong Zhang, Ruijie Quan, Yi Yang

arXiv:2609.34715v1cs.AI

TL;DR

Parametric PDE representation learning has largely emphasized reconstructing observed fields, while the suitability of learned states for evolution and extrapolation remains a central question. PDE-JEPA combines masked-latent pretraining, geometry alignment, and a physics-structured predictor; across nine benchmarks, it reports strong in-distribution performance and substantial gains under unseen governing conditions.

  • Problem

    Reconstruction-focused representation learning leaves open how to obtain latent states suited to accurate parametric PDE prediction and extrapolation across governing conditions.

    Source: §1 · p. 1

  • Method

    PDE-JEPA uses masked-latent pretraining, aligns latent trajectory geometry with physical evolution, and separates shared evolution from parameter-dependent responses.

    Source: §1 · p. 3, §3.2 · p. 4

  • Results

    Across nine PDE benchmarks, PDE-JEPA achieves strong in-distribution performance and substantially improves extrapolation to unseen governing conditions.

    Source: §6 · p. 9

  • Takeaways & Limitations

    Predictive pretraining provides physically informative representations, but trajectory alignment and structured prediction accompany accurate evolution and extrapolation.

    Source: §1 · p. 2, §6 · p. 9

    The stated scope is time-dependent physical systems governed by a family of PDEs.

    Source: §3.1 · p. 4

Abstract

from arXiv · show

Physical trajectories contain more than snapshots of a system: they also reveal how its states evolve under governing conditions. However, representation learning for parametric partial differential equations (PDEs) has largely relied on reconstruction-based objectives that emphasize recovering observed physical fields. In this paper, we investigate predictive representation pretraining as an alternative to reconstruction-based learning. We find that predictive representations preserve rich physical information, yet this advantage alone does not ensure accurate field evolution. Based on these observations, we introduce PDE-JEPA for parametric PDE dynamics. Specifically, we first train an encoder using a masked-latent prediction to capture the underlying regularities of PDE dynamics. To explicitly adapt the pretrained representation toward a more dynamics-aligned state space, we then introduce a geometry projector that aligns latent trajectory geometry with the evolution geometry of physical fields. Finally, building on this geometry-aligned latent space, we further develop a physics-structured latent predictor that decomposes the dynamics into parameter-independent evolution and parameter-dependent response components. Extensive experiments on nine widely used PDE benchmarks demonstrate that our framework outperforms existing state-of-the-art methods by an average of 33.4\% in-distribution, while achieving an average improvement of 51.4\% when extrapolating to unseen governing parameters. The project page is available \href{https://tanpig-x.github.io/PDE-JEPA/}{here}.

1 INTRODUCTION

Parametric PDE forecasting must capture evolution across changing governing conditions, yet reconstruction-focused representations may not expose physically relevant information or support reliable temporal evolution. PDE-JEPA builds on predictive pretraining with geometry alignment and structured latent prediction, addressing the gap between informative representations and accurate rollout.

1 INTRODUCTION

  • Parametric PDE models must capture temporal evolution while generalizing across variations in coefficients, forcing terms, and boundary conditions.Requiring accurate solutions across these conditions motivates learning-based forecasting as an alternative to computationally costly classical numerical methods.Source: §1 · p. 1
  • Reconstruction fidelity alone does not establish whether physical information is accessible or whether the latent space supports evolution under shifted governing conditions.The paper asks what makes a latent representation suitable for accurate prediction and extrapolation.Source: §1 · pp. 1–2
  • JEPA yields stronger frozen-feature probes, yet vanilla JEPA has higher in-distribution rollout errors than the reconstruction baseline on Wave-2D and Vorticity.The probes assess recovery of instantaneous fields and inference of governing conditions.Source: §1 · p. 2
  • PDE-JEPA achieves the lowest rollout error across most of nine in-distribution benchmarks and all five benchmarks evaluated under out-of-distribution governing conditions.The benchmarks span transport, diffusion, reaction–diffusion, wave propagation, and fluid dynamics.Source: §1 · p. 3
  • PAG projects latent trajectory geometry toward physical-field evolution geometry while using an anchor loss to preserve pretrained representations.The lightweight module is intended to improve rollout accuracy.Source: §1 · p. 3
  • PSP separates parameter-independent evolution from parameter-dependent responses to improve extrapolation to unseen governing conditions.This decomposition reflects how governing parameters modulate dynamical components in parametric PDEs.Source: §1 · p. 3

2 RELATED WORK

Related work spans neural operators for parametric forecasting, reconstruction-oriented latent dynamics, and predictive representation learning for physical systems. PDE-JEPA studies how latent states should be structured for evolution and extrapolation across governing conditions.

2.1 PARAMETRIC PDE SOLVERS

  • Neural-operator research addresses parametric generalization through parameter conditioning, multi-environment adaptation, in-context learning, and large-scale multiphysics pretraining.The paper focuses on parametric forecasting through latent-state dynamics rather than covering parametric PDE solvers broadly.Source: §2.1 · p. 3

2.2 LATENT LEARNING FOR PDE DYNAMICS

  • Latent PDE models commonly learn states through reconstruction, while PDE-JEPA uses masked latent prediction and then aligns latent trajectory geometry for downstream dynamics.Prior latent approaches also add losses for dynamical predictability or autoregress over quantized or continuous representations.Source: §2.2 · pp. 3–4

2.3 PREDICTIVE LEARNING FOR PHYSICAL SYSTEMS

  • Prior predictive physical representations probe governing factors, while this work asks how predictive latent states should be adapted for long-horizon evolution and parameter extrapolation.Related work also explores physical invariance, masked reconstruction, aerodynamic-field JEPA, and explicit shaping of latent trajectory geometry.Source: §2.3 · p. 4

3 PRELIMINARIES AND PROBLEM SETUP

Parametric PDE dynamics treats changing governing conditions as a family of related evolutions, while JEPA offers a latent-prediction alternative to reconstructing observations.

3.1 PARAMETRIC PDE DYNAMICS

  • A fixed governing environment ξ defines one dynamical system, while changing coefficients, forcing, or boundary conditions yields related but distinct evolutions.Source: §3.1 · p. 4

3.2 JOINT-EMBEDDING PREDICTIVE LEARNING

  • JEPAs predict masked content in latent space rather than reconstructing observations, using encoded visible context to predict target representations.Source: §3.2 · p. 4
  • Prior V-JEPA work reports temporally consistent representations (Mur-Labadia et al., 2026), while physical-system studies find that JEPA features encode governing factors (Qu et al., 2026).Source: §3.2 · p. 4
  • Figure 2 presents a two-stage framework: a geometry projector aligns frozen JEPA latents, then a structured predictor evolves them through an ODE solver.Source: Figure 2 · p. 5

4 METHOD

PDE-JEPA keeps predictive pretraining as its base, then aligns latent trajectory geometry and structures governing-parameter effects for continuous-time evolution.

4.1 PREDICTIVE PHYSICAL STATES

  • The encoder maps frame-wise physical states to spatial latent tokens and remains frozen; ξ is withheld during pretraining to encourage parameter-agnostic representations.Source: §4.1 · p. 5

4.2 PHYSICS-ALIGNED LATENT GEOMETRY

  • Latent turns exceed physical trajectories (Vorticity: 60.5° vs 31.3°); the projector aligns multi-lag directions while an identity anchor preserves pretrained information.Source: §4.2 · pp. 5–6
  • An auxiliary causal predictor estimates Δq_t to keep aligned coordinates dynamically predictable; the geometry-stage projector is then frozen before final dynamics training.Source: §4.2 · p. 6

4.3 PHYSICS-STRUCTURED LATENT PREDICTOR

  • The latent vector field separates shared state evolution from parameter-specific state responses weighted by normalized governing parameters, as a physics-inspired bias rather than exact PDE operator recovery.Source: §4.3 · p. 6
  • An ODE solver integrates the learned field between observations using fixed-step RK4; recursive integration yields long-horizon forecasts.Source: §4.3 · p. 7

5 EXPERIMENTS

Across nine parametric PDE benchmarks, PDE-JEPA performs strongly on in-distribution rollout and improves geometry alignment; the experiments also assess extrapolation and parameter-response consistency.

5.1 EXPERIMENTS SETUP

  • The experiments cover nine PDE benchmarks spanning transport, diffusion, waves, reaction–diffusion, and fluid dynamics, with comparisons across several solver paradigms.Accuracy is evaluated using relative L2 error over the full rollout trajectory, where lower values indicate better long-horizon forecasting.Source: §5.1 · pp. 7–8

5.2 IN-DISTRIBUTION GENERALIZATION

  • PDE-JEPA achieves the lowest relative L2 on eight of nine in-distribution benchmarks and ranks second on Advection (0.0074 vs. 0.0068 for CoDA).The reported gains are particularly large on Burgers, Heat, Wave-B, Wave-2D, and Vorticity.Source: §5.2 · p. 8

5.3 OUT-OF-DISTRIBUTION EXTRAPOLATION

  • Table 4 reports relative L2 rollout error across PDE benchmarks under out-of-distribution governing conditions, with lower values indicating better performance.The supplied passages do not report the table's individual benchmark values.Source: Table 4 · p. 8

5.4 EFFECT OF PAG

  • Adding PAG reduces Vorticity rollout error from 0.086 to 0.040 on ID data and from 0.491 to 0.397 on OOD data.These correspond to 53.4% and 19.1% improvements, respectively.Source: §5.4 · p. 8, Table 2 · p. 8
  • Alignment reduces geometry errors, especially on Vorticity (78%/77%/84%) and Wave-2D (71%/68%/74%) across three trajectory measures.The measures assess local turning angles, second-order temporal variation, and directional consistency; they are computed directly on the original and aligned representations without a dynamics predictor.Source: Table 3 · p. 8, §5.4 · p. 9

5.5 PARAMETER-RESPONSE MECHANISM FOR PSP

  • Table 5 examines parameter-response consistency under matched initial conditions.The supplied passage identifies the table's scope but gives no specific results.Source: Table 5 · p. 9

6 CONCLUSION

The conclusion emphasizes that predictive pretraining provides physically informative representations, while trajectory alignment further improves long-horizon evolution. PDE-JEPA combines aligned latent geometry with a structured predictor and reports strong in-distribution performance and improved extrapolation across nine PDE benchmarks.

6 CONCLUSION

  • Across nine PDE benchmarks, PDE-JEPA achieves strong in-distribution performance and substantially improves extrapolation to unseen governing conditions.Further analyses report better physical trajectory alignment and more consistent parameter-conditioned responses.Source: §6 · p. 9
  • PDE-JEPA combines physics-aligned latent geometry with a structured predictor that separates shared evolution from parameter-dependent responses.The paper describes predictive JEPA pretraining as physically informative, with explicit trajectory alignment further improving long-horizon evolution.Source: §6 · p. 9
  • The conclusion argues that generalizable physical forecasting depends not only on the evolution model, but also on state-space geometry suited to evolution and extrapolation.Source: §6 · p. 9

A DATASET DETAILS

The appendix documents nine PDE datasets, their parameter settings, and the trajectory configurations used for in-distribution and out-of-distribution evaluation.

A.1 ADVECTION

  • Advection uses periodic transport with speed β sampled from [0, 4], 1,200 training environments, and trajectories saved as 140-frame segments on a 256-point grid.The retained frames cover approximately t ∈ [44.18, 100].Source: §A.1 · pp. 17–18

A.2 BURGERS EQUATION; A.3 HEAT EQUATION

  • Burgers and Heat use periodic intervals of length 16, with environment-specific forcing; Heat varies diffusivity β over log-uniform [10^-3, 5].Burgers uses WENO reconstruction and adaptive Dormand–Prince integration; Heat uses finite-difference diffusion and the same integrator.Source: §A.2 · p. 18, §A.3 · p. 19

A.4 WAVE-B: ONE-DIMENSIONAL WAVES WITH VARYING BOUNDARIES

  • Wave-B varies endpoint boundary types across DD, DN, ND, and NN environments, while its train, validation, and test splits all include every boundary combination.The 256-point nonuniform Chebyshev grid is reordered into forward time, then downsampled to 25-frame trajectories.Source: §A.4 · p. 20

A.5 COMBINED EQUATION

  • Combined Equation trains on 1,200 parameter triplets in α ∈ [0, 1], β ∈ [0, 0.4], and γ ∈ [0, 1], with OOD trajectories extending α beyond its training range.The OOD split varies α over [1.0682, 1.7581] while β and γ remain within their in-distribution ranges.Source: §A.5 · pp. 20–21

A.6 VORTICITY: TWO-DIMENSIONAL INCOMPRESSIBLE FLOW

  • Vorticity varies viscosity ν over [10^-3, 10^-2] in-distribution and supplies OOD viscosities from [10^-5, 10^-4] for unforced periodic flow.Trajectories are subsampled to 128 × 128 resolution with 30 frames.Source: §A.6 · pp. 21–22

A.7 HETERNS

  • HeterNS uses viscosity and forcing frequency as environment parameters, with 15 training configurations and separate viscosity-interpolation, viscosity-extrapolation, and forcing OOD settings.The three OOD settings comprise 900, 510, and 90 trajectories, respectively.Source: §A.7 · pp. 22–23

A.8 WAVE-2D: DAMPED TWO-DIMENSIONAL WAVES

  • Wave-2D training covers c ∈ [100, 500] and k ∈ [0, 50], while its OOD release contains 110 strict extrapolation trajectories and 10 boundary-support trajectories.Each parameter pair has ten initial conditions.Source: §A.8 · pp. 23–24

A.9 GRAY–SCOTT EQUATION

  • Gray–Scott fixes diffusion coefficients and evaluates OOD reaction parameters F ∈ [0.045, 0.0467] and k ∈ [0.0570, 0.0590] against the stated in-distribution ranges.The system uses periodic boundaries on a 32 × 32 spatial grid.Source: §A.9 · p. 24

B ARCHITECTURE DETAILS

PDE-JEPA learns representations and dynamics through staged latent-space training, then reconstructs physical fields from autoregressive latent rollouts.

B.1 PRETRAINING STAGE

  • Masked-latent JEPA pretraining predicts masked target representations from visible context tokens, using a momentum-updated target encoder, and retains the trained encoder for later stages.The prediction objective aligns predicted target representations with target-encoder representations.Source: §B.1 · p. 25

B.2 GEOMETRY ALIGNMENT STAGE

  • A geometry projector adapts predictive latents because their trajectory geometry is not explicitly constrained to match physical evolution geometry.The geometry loss compares dimension-independent directional similarities across temporal offsets, while an anchor objective keeps projected states near pretrained latents; the encoder remains fixed.Source: §B.2 · pp. 25–27
  • An auxiliary causal predictor estimates the next latent displacement from projected latent history and physical parameters, while the projector itself receives no parameter input.The projector and auxiliary predictor are optimized jointly with the pretrained encoder frozen.Source: §B.2 · p. 27

B.3 PREDICTOR LEARNING STAGE

  • The physics-structured latent predictor decomposes transitions into parameter-independent evolution and parameter-dependent response components.Its branches preserve spatial-token structure, and the parameter-dependent branch captures changes in latent evolution induced by governing parameters.Source: §B.3 · pp. 27–28
  • Long-horizon prediction recursively feeds predicted latent states into the next transition, evolving dynamics in latent space and decoding physical predictions only when needed.This avoids repeatedly mapping autoregressive predictions between physical and latent domains.Source: §B.3 · p. 28

B.4 DECODER LEARNING STAGE

  • Decoder training freezes the encoder, projector, and predictor, and reconstructs physical fields frame-wise without modeling temporal evolution in the decoder.Training constructs decoder inputs from the initial physical state and rolled-out latent states; future ground-truth states serve only as reconstruction targets, exposing the decoder to rollout deviations.Source: §B.4 · pp. 28–29

C.1.1 PHYSICS-STRUCTURED LATENT PREDICTOR IMPLEMENTATIONS

  • PDE-specific predictor implementations can separate latent vector-field components by governing term, including transport, diffusion, and dispersion for Combined Equation and viscosity and forcing for HeterNS.Component fields share latent-state spatial and channel dimensions, and their coefficients are normalized parameter-dependent weights.Source: §C.1.1 · pp. 30–31

C.1.2 LATENT TIME INTEGRATION.

  • Continuous latent vector fields advance each observation interval with classical RK4 using four substeps; the stages use AdamW with dataset-specific settings.The implementation specifies AdamW betas of (0.9, 0.999), except structured predictors, which use (0.9, 0.95).Source: §C.1 · p. 30, §C.1.2 · p. 31

C.2 BASELINE IMPLEMENTATIONS

The appendix describes the baseline implementations and protocols, then outlines representation analyses that distinguish physical information, latent organization, and temporal evolvability.

C.2 BASELINE IMPLEMENTATIONS

  • The evaluation compares PDE-JEPA with operator, transformer, autoencoder, and representation-learning baselines, using a common evaluation trajectory, metric, and rollout protocol for reimplemented baselines.Some reported results are adopted directly from Zebra, while the remaining baselines are implemented and evaluated under the paper’s setting.Source: §C.2 · p. 33
  • Baseline implementations span autoregressive FNO, parameter-conditioned GEPS and UniSolver, and pretrained or task-fine-tuned models including DPOT-S and Poseidon-T.Other described baselines include LE-PDE, LNS, MAE-PDE, and ENMA.Source: §C.2 · pp. 33–36

D ADDITIONAL EXPERIMENTS

  • The appendix labels JEPA as Vanilla, the model with Physics-Aligned Latent Geometry as Geo, and the full PAG-plus-PSP model as Physics, making Vanilla→Geo and Geo→Physics the respective component comparisons.These labels isolate PAG and the additional PSP contribution, respectively.Source: D ADDITIONAL EXPERIMENTS · p. 39

D.1 REPRESENTATION ANALYSES

  • Representation analyses on Vorticity ask what physical information is encoded, how it is organized, and whether it is easy to evolve, using parameter and local-state probes, latent-organization measures, and rollout accuracy.The analyses compare JEPA with LNS and VideoMAE.Source: §D.1 · p. 39, §D.1.1 · p. 39

D.1.1 MODELS AND EVALUATION PROTOCOL

  • The latent-organization protocol analyzes 2,400 training, 120 external-ID, and 120 far-OOD trajectories using model-specific standardized PCA16 spaces, with separate internal-holdout, external-ID, and far-OOD viscosity-readout evaluations.PCA axes are fitted independently per model and are not assumed to be shared across JEPA, LNS, and VideoMAE.Source: §D.1.1 · p. 40

D.1.2 PARAMETER INFORMATION ANALYSES

These analyses examine how parameter and physical-state information in representations relates to trajectory geometry, temporal predictability, and rollout. They report that physics aligns more closely with projected numerical trajectory geometry, while the structured predictor improves physical-field forecasting despite limits in extrapolation and latent-space error.

D.1.2 PARAMETER INFORMATION ANALYSES

  • For Vorticity viscosity readout, JEPA’s normalized test MSE falls from 0.042597 at 10% supervision to 0.009310 at 100%, where it reduces error by 55.3% relative to LNS and 50.9% relative to VideoMAE.The advantage over VideoMAE grows from 7.9% to 47.5% to 50.9% across the 10%, 50%, and 100% budgets.Source: §D.1.2 · p. 40, Table 19 · p. 40
  • Despite strong ID readout, all three linear probes have negative R2 on far-OOD viscosities; JEPA has the smallest standardized RMSE at 4.0933, while kNN-5 R2 is 0.7820 for JEPA, 0.6780 for LNS, and 0.7586 for VideoMAE.LNS’s external-ID R2 drops to 0.1422 despite a Spearman correlation of 0.9605, consistent with a calibration offset; VideoMAE’s external-ID R2 is 0.9194.Source: §D.1.2 · p. 41, Figure 13 · p. 41

D.1.3 VARIANCE CONCENTRATION AND LATENT ORGANIZATION

  • LNS concentrates variance in two principal components (97.11%), while JEPA and VideoMAE distribute it more broadly; representation similarity does not directly rank physical usefulness.Entropy effective ranks are 1.31 for LNS, 2.77 for JEPA, and 8.92 for VideoMAE; linear CKA is 0.8018 for JEPA–LNS, 0.1361 for JEPA–VideoMAE, and 0.1964 for LNS–VideoMAE, even though VideoMAE has external-ID viscosity R2 of 0.9194.Source: §D.1.3 · p. 41, Table 21 · p. 42, Table 22 · p. 43

D.1.4 TEMPORAL AND SPATIAL ORGANIZATION

  • LNS has straighter, more monotonically progressing latent trajectories than JEPA, while VideoMAE’s pooled geometry depends on clip-relative position; local spatial structure persists across representations.Path efficiency is 0.7581 for LNS, 0.4901 for JEPA, and 0.1070 for VideoMAE; VideoMAE’s relative tubelet position explains 94.5% of pooled PCA16 variance. On shared 16 × 16 grids, ID neighbor–distant cosine lifts are 0.3809 for LNS and 0.2335 for JEPA.Source: §D.1.4 · pp. 42–43, Figure 15 · p. 43, Table 23 · p. 43

D.1.5 EXTENSION TO OTHER PDES

  • At full supervision, JEPA reduces parameter-readout test MSE by 21.5% on Wave2D, 59.1% on Burgers, and 13.8% on Heat, with bootstrap gains conclusive for Wave2D and Burgers but not Heat.JEPA has lower test MSE across all label budgets on these three PDEs; the Heat confidence interval crosses zero.Source: §D.1.5 · pp. 43–44, Table 24 · p. 44, Figure 18 · p. 45
  • On a shared 8 × 8 grid, JEPA has stronger ID/OOD local-state readout than LNS on CFD and Wave2D, and its leading components show coherent, physically aligned spatiotemporal patterns across PDEs.JEPA’s CFD R2 is 0.9921/0.9492 and Wave2D R2 is 0.8773/0.8633 for ID/OOD, versus 0.9578/0.8818 and 0.8258/0.7254 for LNS; MAE-PDE has weaker controlled local-state readout (R2 ≈0.10 and 0.37).Source: §D.1.5 · p. 45, Table 25 · p. 45, Figure 19 · p. 46

D.1.6 DIRECT RECONSTRUCTION VERSUS AUTOREGRESSIVE ROLLOUT

  • Reconstruction fidelity and temporal predictability are distinct: JEPA’s rollout-to-reconstruction ratio ranges from 8.12 to 12.31, while Zebra has much larger gaps on several benchmarks despite smaller reconstruction errors.The ratio does not rank rollout accuracy: Zebra has a larger ratio but lower absolute rollout error on Wave2D, whereas JEPA has lower rollout error on Burgers, Heat, Wave-B, and Vorticity.Source: §D.1.6 · p. 47, Table 26 · p. 47

D.2.2 PARAMETER-DEPENDENT DYNAMICS

  • On Vorticity and Wave2D, Physics improves parameter-response direction and magnitude calibration over Geo, although Wave2D’s late-horizon latent response direction declines.On Vorticity, field-response cosine rises from 0.5161 for Vanilla to 0.6259 for Geo and 0.7079 for Physics; on Wave2D, Physics raises field-response cosine from 0.9582 to 0.9861 and moves the magnitude ratio from 0.9786 to 0.9959. On Wave2D at steps 22–29, latent cosine decreases from 0.8900 to 0.8789.Source: §D.2.2 · p. 48, Table 27 · p. 48, Figure 23 · p. 49

D.2.3 LATENT TRAJECTORY GEOMETRY

  • Physics better matches projected cross-parameter trajectory geometry than Geo on Vorticity and reproduces Wave2D’s near-orthogonal parameter trajectories.On Vorticity, cross-parameter separation is 0.2743 for Physics, 0.2694 for projected truth, and 0.2995 for Geo; velocity-direction similarity is 0.2296, 0.2124, and 0.1804, respectively. On Wave2D, velocity-direction similarity is −0.0014 for Physics, −0.0010 for projected truth, and −0.0026 for Geo, so direction similarity should be judged against numerical dynamics rather than as a standalone measure.Source: §D.2.3 · p. 50, Figure 25 · p. 50
  • Physics matches projected numerical speed and curvature more closely on Wave2D, but is slower and smoother than the reference on Vorticity, where it still achieves better response direction and lower field rollout error.Thus, smoothness alone does not establish that latent dynamics are more physical; agreement with projected numerical trajectories is the relevant criterion.Source: §D.2.3 · p. 51, Table 29 · p. 51, Figure 28 · p. 54

D.2.4 FIELD AND LATENT ROLLOUT ACCURACY

  • 15.6% and 24.3% Vorticity field-error reductions on ID validation and OOD accompany Wave2D OOD and far-OOD NRMSE drops from 0.3039 to 0.1521 and 0.2652 to 0.1405.These averages cover steps 1–29; Wave2D's far-OOD latent error instead rises from 0.2212 to 0.2366, while rollout differences persist across the horizon.Source: Table 30 · p. 51, §D.2.4 · p. 51, Figure 29 · p. 54
  • Wave2D final-step visualizations show smaller OOD field errors for Physics despite a late-horizon limitation in the latent metrics.The final-step fields compare parameter changes for the same initial condition.Source: §D.2.4 · p. 54, Figure 30 · p. 54, Figure 31 · p. 55

D.2.5 PHYSICS SUPERVISION REDUCES ERROR PROPAGATION

Physics-D reduces long-horizon perturbation amplification under Bridge and far-OOD shifts despite slightly worse one-step fitting, while ID amplification changes little. Frozen geometry projection also improves trajectory-geometry metrics on three one-dimensional PDEs, but lower geometry error does not guarantee better forecasting.

D.2.5 PHYSICS SUPERVISION REDUCES ERROR PROPAGATION

  • Despite slightly higher teacher-forced error on all splits, Physics-D lowers far-OOD final-step latent error from 0.169647 to 0.159624 and whole-rollout error from 0.096307 to 0.090602.ID rollout error slightly increases, while Bridge follows the far-OOD improvement trend with a smaller margin.Source: §D.2.5 · pp. 55–56, Table 31 · p. 56
  • Across 480 initial states, log gain correlates about 0.81 with final-step latent error for both predictors, but the association weakens after controlling for viscosity.The authors conclude that reduced error propagation contributes to extrapolation gains, but explains only part of rollout behavior.Source: §D.2.5 · pp. 55–56
  • At K = 29, Physics-D cuts median gain from 8.8056 to 4.9087 on Bridge and 15.5361 to 6.4286 on far-OOD, with little ID change.Bridge uses ν ∈ [10^-4, 10^-3], between the training and far-OOD viscosity ranges.Source: §D.2.5 · p. 55, Table 31 · p. 56, Figure 32 · p. 56

D.2.6 EXTENSION TO 1D PDES

  • Projection lowers turning-angle MAE from 46.0253 to 23.3122° on Burgers, 78.1210 to 23.0318° on Heat, and 13.3181 to 5.7452° on Wave-B.Weighted multi-lag geometry error also decreases on all three datasets; the comparison uses the same held-out trajectories before and after projection.Source: Table 32 · p. 56, §D.2.6 · p. 56
  • On Wave-B, projection brings mean latent turning angle from 111.8439° to 118.4202°, near the physical 118.7228°, showing alignment can increase or decrease curvature.This coordinate-level geometry improvement does not establish lower forecast error.Source: §D.2.6 · p. 56

D.3.1 FULL OOD RESULTS

The method achieves the best OOD rollout performance across all evaluated PDE benchmarks, with especially large reported gains on Combined and Wave2D.

D.3.1 FULL OOD RESULTS

  • The method achieves the best OOD rollout performance across all evaluated PDE benchmarks; error falls from 0.038 to 0.008 on Combined and 0.610 to 0.157 on Wave2D.On Vorticity, error improves over Zebra from 0.320 to 0.288; substantial gains are also reported on HeterNS and Gray–Scott.Source: Table 33 · p. 57, §D.3.1 · p. 57

D.3.2 GEOMETRY ALIGNMENT MODULE

Geometry alignment reduces latent-trajectory geometry discrepancy, while the anchor constraint helps retain useful representation information. Its benefit is not representation-agnostic: alignment improves JEPA rollout accuracy but slightly degrades LNS accuracy.

D.3.2 GEOMETRY ALIGNMENT MODULE

  • Across four benchmarks, geometry loss generally falls and stabilizes while anchor loss rises then settles, with noisier geometry optimization on Gray–Scott.The joint loss behavior suggests alignment proceeds without unconstrained deformation of the pretrained representation.Source: §D.3.2 · p. 57, Figure 33 · p. 58
  • Removing the anchor further lowers turning-angle error but worsens rollout: Vorticity ID/OOD error rises from 0.040/0.397 to 0.047/0.423, and Wave-2D from 0.143/0.321 to 0.166/0.354.Local-state probing deteriorates across evaluated datasets, alongside higher parameter-probing error.Source: §D.3.2 · pp. 57–58, Table 34 · p. 58
  • Geometry alignment improves JEPA angle discrepancy from 29.1° to 6.30° and ID rollout error from 0.086 to 0.040, but worsens LNS rollout error from 0.059 to 0.062.More aggressive unanchored deformation of LNS lowers angle error to 5.94° but increases rollout error to 0.071.Source: §D.3.2 · p. 58

D.3.3 PREDICTOR MODULE

This appendix evaluates latent predictor design through matched Gray–Scott comparisons, factorization and physical-tangent ablations, integration tests, and qualitative forecasts across PDE settings.

D.3.3 PREDICTOR MODULE

  • 0.275269 to 0.253280 final-step error and 0.171373 to 0.162613 whole-rollout error show a further, more modest gain from eight to twenty-four structured blocks.The comparison holds optimizer-step budget constant.Source: §D.3.3 · p. 59, Figure 34 · p. 59
  • 0.630593 to 0.275269 final-step pooled latent error and 0.457266 to 0.171373 whole-rollout error favor the structured over the direct Gray–Scott predictor.The matched models have equal parameter counts and training budgets, with essentially identical teacher errors (0.041156 vs. 0.041170).Source: Table 36 · p. 59, §D.3.3 · p. 59, Figure 34 · p. 59
  • 0.1100 to 0.0848 raw whole-rollout latent error and 0.1009 to 0.0769 field RelL2 accompany M0-to-M1 factorization; M2 further lowers raw whole-rollout error to 0.0200.JVP supervision improves aggregate trajectory metrics, while M1 retains lower phase and speed errors than M2.Source: Table 37 · p. 60, §D.3.3 · p. 60
  • 0.1914 to 0.159669 final-step error and 0.0936 to 0.077184 whole-rollout error accompany Euler-to-RK4 integration; RK8 adds little forward-prediction improvement.The passage reports that RK8 further reduces round-trip error, while forward-prediction metrics essentially saturate beyond RK4.Source: Table 38 · p. 60, §D.3.3 · p. 60

D.4.1 ADVECTION

  • Advection, Burgers, and Heat figures show ground truth, forecasts, errors, and spatial profiles across two β values; Burgers and Heat also specify fixed α and γ.Burgers fixes α = 1 and γ = 0; Heat fixes α = 0 and γ = 0.Source: §D.4.1 · p. 61, §D.4.2 · p. 61, §D.4.3 · p. 61

D.4.4 WAVE-B

  • Wave-B forecasts are shown under two boundary conditions, Combined Equation forecasts under representative ID and OOD parameters, and Wave-2D forecasts under ID and OOD (c, k).The figures include ground truth, forecasts, and absolute errors; Wave-B and Combined Equation also show representative spatial profiles over time.Source: §D.4.4 · p. 62, §D.4.5 · p. 62, §D.4.6 · p. 62

D.4.7 VORTICITY

  • Vorticity figures contrast ID and OOD viscosity, Gray–Scott contrasts ID and OOD reaction parameters, and HeterNS contrasts ID, OOD forcing, and OOD viscosity.These figures show ground truth, forecasts, and absolute errors across representative rollout frames.Source: §D.4.7 · p. 63, §D.4.8 · p. 63, §D.4.9 · p. 63

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