Advectra: Asymmetric Latent Transport for Non-Stationary Physics

Nodens Koren; Thomas Hofmann; Georgios Kissas

cs.LGOct 4, 2026

Advectra: Asymmetric Latent Transport for Non-Stationary Physics

Authors: Nodens Koren, Thomas Hofmann, Georgios Kissas

Organizations: Department of Computer Science ETH Zürich Zürich, Switzerland · Swiss Data Science Center ETH Zürich Zürich, Switzerland

Abstract

Many latent neural operators represent input and output fields in a stationary latent chart. In particular, common latent routing mechanisms use fixed or shared assignment weights for feature projection and reconstruction, limiting their ability to model transport-dominated systems where coherent structures move relative to fixed coordinate frames. We propose Advectra, a transport-aware latent operator that introduces a regularized kinematic coordinate map to decouple source and target coordinate systems. This yields an approximately co-moving latent reference frame and enables asymmetric feature aggregation and reconstruction. Combined with a geometry-aware ordering mechanism for state-space models, Advectra captures advective dynamics while maintaining stable global interactions. Advectra achieves the best performance among evaluated geometry-constrained and form-free baselines on advection-dominated benchmarks, including passive scalar transport in Navier--Stokes flows and Rayleigh--Taylor instability, while demonstrating strong generalization on real-world engineering tasks. These results highlight the benefit of explicit moving-frame structure in neural operators for non-stationary physics.

Figures & tables

Figure 1 : Overview of the Advectra operator.

Model	AirfRANS				ShapeNet-Car
Model	Volume $\downarrow$	Surf $\downarrow$	$C_{L}$ $\downarrow$	$\rho_{L}$ $\uparrow$	Volume $\downarrow$	Surf $\downarrow$	$C_{D}$ $\downarrow$	$\rho_{D}$ $\uparrow$
GraphSAGE [ 24 ]	0.0087	0.0184	0.1476	0.9964	0.0461	0.1050	0.0270	0.9695
PointNet [ 25 ]	0.0253	0.0996	0.1973	0.9919	0.0494	0.1104	0.0298	0.9583
Graph U-Net [ 26 ]	0.0076	0.0144	0.1677	0.9949	0.0471	0.1102	0.0226	0.9725
MeshGraphNet [ 6 ]	0.0214	0.0387	0.2252	0.9945	0.0354	0.0781	0.0168	0.9840
GNO [ 8 ]	0.0269	0.0405	0.2016	0.9938	0.0383	0.0815	0.0172	0.9834

Table 2 : Quantitative results on industrial benchmarks AirfRANS and ShapeNet-Car. We report relative

L_{2}

error (

\downarrow

) and Spearman’s rank correlation (

\uparrow

). Top-2 results in each column are highlighted: best (orange) and second (blue).

Configuration	NS-Tracer-PwC		ShapeNet-Car
Configuration	$\mathbf{v}$ $\downarrow$	tracer $\downarrow$	Volume $\downarrow$	Surf $\downarrow$	$C_{D}$ $\downarrow$	$\rho_{D}$ $\uparrow$
Advectra (Full)	0.1209	0.1607	0.0149	0.0716	0.0094	0.9941
Independent Warps	0.1548	0.2183	0.0174	0.0732	0.0101	0.9931
w/o Ordering	0.1446	0.1840	0.0169	0.0766	0.0113	0.9923
w/o Transport	0.1858	0.2435	0.0159	0.0742	0.0108	0.9928
w/o Both	0.2001	0.2525	0.0169	0.0797	0.0129	0.9918

Table 3 : Ablation study of Advectra. We evaluate Geometry-Aware Ordering, Latent Transport, and an independent-warp variant that uses separate source/target displacement maps instead of the shared transported update. Top-2 results per column are highlighted: best (orange) and second (blue).

Figure 2 : Visualizing the contrast between Eulerian and Lagrangian perspectives in Navier-Stokes flow. The Eulerian view treats the domain as a fixed grid, measuring fluid velocity at specific, stationary locations (cyan streamlines). In contrast, the Lagrangian view follows the individual trajectories of material parcels as they move through space. Here, a dense block of tracers (colored by initial position) acts as a set of “moving observers.” As $t$ increases from $0$ to $9$ , these tracers are advected by the unsteady flow; their severe stretching and folding demonstrates how the Lagrangian frame of reference deforms over time, physically reconfiguring the spatial relationships between different regions of the fluid.

Model	AirCraft
Model	$C_{p}$ $\downarrow$	$\rho$ $\downarrow$	$U$ $\downarrow$	$V$ $\downarrow$	$W$ $\downarrow$	Pressure $\downarrow$
GraphSAGE [ 24 ]	0.1442	0.0803	0.0148	0.2154	0.2395	0.0923
PointNet [ 25 ]	0.1472	0.0783	0.0147	0.2104	0.2528	0.0937
Graph U-Net [ 26 ]	0.1475	0.0784	0.0152	0.2225	0.2477	0.0947
MeshGraphNet [ 6 ]	0.1372	0.0682	0.0164	0.1835	0.2175	0.0893
GNO [ 8 ]	0.1556	0.0785	0.0167	0.2012	0.2318	0.1071

Table 4 : Quantitative results on the AirCraft benchmark (3D external aerodynamics). We compare against geometric baselines and neural operators. We report relative

L_{2}

error (

\downarrow

). Top-2 results in each column are highlighted: best (orange) and second (blue).

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Model	COM $\downarrow$	Mass $\downarrow$	Raw MAE $\downarrow$	Ratio $R$ $\downarrow$
Advectra	0.0043	0.67%	0.0285	1.0099
Advectra (No Transport)	0.0071	0.93%	0.0419	1.0656
LaMO	0.0072	1.10%	0.0397	1.0973

Appendix

Table 5 : Transport diagnostics on NS-Tracer-PwC.

Panel of Figure 3 — Figure 3 : Comparison of Raw Error versus Center-of-Mass (COM) Aligned Error across three randomly selected test trajectories. By translating the predicted tracer distribution by the displacement vector $d=\mu_{\mathrm{gt}}-\mu_{\mathrm{pred}}$ before computing pointwise error, this diagnostic separates bulk COM misalignment from residual structural mismatch. Higher COM-aligned error indicates deformation, diffusion, or local misalignment that cannot be corrected by a global translation, while lower COM-aligned error indicates better preservation of tracer structure under this diagnostic.

Panel of Figure 4 — Figure 4 : 1D horizontal cross-sections of the tracer concentration taken at the spatial peak for three randomly selected test trajectories. Advectra produces profiles that more closely match the ground truth in peak amplitude and spatial alignment. In contrast, the baseline models exhibit lower peak amplitudes, broader spatial spreads, and shifted centers, indicating stronger diffusion and advection lag.

Panel of Figure 5 — Figure 5 : Qualitative comparison of predicted tracer transport across three randomly selected test trajectories. Each panel contrasts the Ground Truth state at $t_{out}$ against the predictions generated by the proposed Advectra architecture, the ablated Advectra (No Transport) variant, and the LaMO baseline. The visualizations illustrate the differing spatial distributions and structural deformations produced by each operator learning formulation.

Figure 6 : Distribution of consecutive centroid distances for randomly selected test samples. Geometry-aware ordering reduces both the average and median jump distance, indicating improved locality. At the same time, it preserves a small number of larger jumps, corresponding to structured long-range transitions necessary to capture global geometry.

Panel of Figure 7 — Figure 7 : Latent sequence paths mapped to physical space for three randomly selected ShapeNet-Car test geometries. Without ordering, the sequence exhibits frequent long-range jumps between spatially discontinuous regions (e.g., front and rear of the geometry). With geometry-aware ordering, the paths become smoother and spatially coherent, traversing the domain through predominantly local transitions with only occasional long-range connections.

Panel of Figure 8 — Figure 8 : Aggregation field visualization for three randomly selected ShapeNet-Car test geometries. Without ordering, the field exhibits long-range, crossing interactions spanning large portions of the domain, indicating spatially incoherent aggregation. With geometry-aware ordering, the field decomposes into localized, spatially coherent clusters with bounded extent, reflecting predominantly local interactions.

Sorting Interval	ShapeNet-Car
	Volume $\downarrow$	Surf $\downarrow$
25	0.0158	0.0736
50	0.0150	0.0719
100	0.0149	0.0716
200	0.0153	0.0716
400	0.0156	0.0718

Appendix

Table 6 : Effect of ordering frequency on the ShapeNet-Car dataset. We report relative

L_{2}

errors on volume and surface fields. Geometry-aware ordering significantly improves performance compared to no ordering, while results remain stable across a wide range of sorting intervals (i.e., refresh intervals of the nearest-neighbor latent ordering).

Ordering Strategy	ShapeNet-Car
	Volume $\downarrow$	Surf $\downarrow$
Greedy nearest-neighbor	0.0149	0.0716
Random-walk traversal	0.0159	0.0734
MST traversal	0.0166	0.0743
No sorting	0.0169	0.0766

Appendix

Table 7 : Effect of latent ordering strategy on the ShapeNet-Car dataset. We report relative

L_{2}

errors on volume and surface fields. Greedy nearest-neighbor ordering performs best among the tested traversal strategies.

Model	OOD Reynolds		OOD Angles
Model	$C_{L}$ $\downarrow$	$\rho_{L}$ $\uparrow$	$C_{L}$ $\downarrow$	$\rho_{L}$ $\uparrow$
Simple MLP	0.6205	0.9578	0.4128	0.9572
GraphSAGE [ 24 ]	0.4333	0.9707	0.2538	0.9894
PointNet [ 25 ]	0.3836	0.9806	0.4425	0.9784
Graph U-Net [ 26 ]	0.4664	0.9645	0.3756	0.9816
MeshGraphNet [ 6 ]	1.7718	0.7631	0.6525	0.8927

Appendix

Table 8 : Generalization performance on Out-of-Distribution (OOD) tasks for the AirfRANS dataset. We evaluate extrapolation to unseen Reynolds numbers and angles of attack.

\downarrow

indicates lower is better;

\uparrow

indicates higher is better (Spearman’s rank correlation). Top-2 results in each column are highlighted: best (orange) and second (blue).

Model	Time (s/iter) $\downarrow$	GFLOPs / batch $\downarrow$
Transolver	0.1111	266.09
LaMO	0.1150	266.69
Advectra (Ours)	0.1209	258.06

Appendix

Table 9 : Runtime comparison on the Airfoil benchmark. We report average time per iteration and FLOPs per batch.

Figure 9 : Relative

L_{2}

error on the Airfoil benchmark as a function of token count. All methods improve with increasing token count and eventually saturate. Advectra reaches its saturated regime at a lower token count in this sweep, while the stronger

128

-slice setting benefits Transolver and LaMO.

	Training Configuration						Model Configuration
Benchmark	Loss	Epochs	Init LR	Optimizer	Batch	Scheduler	$L$	$H_{r}$	$D$	$G$
Airfoil	Relative L2	500	$10^{-3}$	AdamW	4	OneCycle	8	8	128	32
AirfRANS	$L_{v}+L_{s}$	400	$10^{-3}$	Adam	1	OneCycle	8	8	128	32
ShapeNet-Car	$L_{v}+0.5L_{s}$	200	$10^{-3}$	Adam	1	OneCycle	8	8	128	32
AirCraft	$L_{v}+L_{s}$	400	$10^{-3}$	Adam	1	OneCycle	8	8	128	32
GCE-RT	Relative L2	500	$10^{-3}$	AdamW	32	OneCycle	8	8	128	32

Appendix

Table 10 : Training and model configurations for Advectra. Baseline configurations follow official or best-reported settings when available, as described in Appendix I.4 . Relative L2 denotes field reconstruction error.

L_{v}

and

L_{s}

denote volume and surface losses. Here

H_{r}

denotes the number of parallel latent routing heads, i.e., independent anchor groups used for source/target assignments and feature projections. It is not a multi-head self-attention parameter; the latent sequence core is a bidirectional SSM.

	OOD Reynolds		OOD Angles
Dataset	Range	Samples	Range	Samples
Training Set	$[3\times 10^{6},5\times 10^{6}]$	500	$[-2.5^{\circ},12.5^{\circ}]$	800
Test Set	$[2\times 10^{6},3\times 10^{6}]\cup[5\times 10^{6},6\times 10^{6}]$	500	$[-5^{\circ},-2.5^{\circ}]\cup[12.5^{\circ},15^{\circ}]$	200

Appendix

Table 11 : Settings for OOD generalization experiments on AirfRANS. The training and test sets contain disjoint ranges for Reynolds numbers and angles of attack.

	GCE-RT				NS-Tracer-PwC		FNS-KF
Model	$\rho$	$\mathbf{v}$	$p$	tracer	$\mathbf{v}$	tracer	$\mathbf{v}$
Best Baseline	$0.0171{\scriptstyle\pm 0.0008}$	$0.0671{\scriptstyle\pm 0.0012}$	$0.0015{\scriptstyle\pm 0.0001}$	$0.0551{\scriptstyle\pm 0.0014}$	$0.1577{\scriptstyle\pm 0.0021}$	$0.2047{\scriptstyle\pm 0.0068}$	$0.1317{\scriptstyle\pm 0.0010}$
	(Transolver++)	(Transolver++)	(LinearNO)	(Transolver++)	(FNO)	(U-Net)	(FNO)
Advectra (Ours)	$\mathbf{0.0149}{\scriptstyle\pm 0.0003}$	$\mathbf{0.0595}{\scriptstyle\pm 0.0007}$	$\mathbf{0.0014}{\scriptstyle\pm 0.0001}$	$\mathbf{0.0488}{\scriptstyle\pm 0.0012}$	$\mathbf{0.1209}{\scriptstyle\pm 0.0011}$	$\mathbf{0.1607}{\scriptstyle\pm 0.0025}$	$\mathbf{0.1202}{\scriptstyle\pm 0.0014}$

Appendix

Table 12 : Standard deviations for GCE-RT, NS-Tracer, and FNS-KF benchmarks. We report the mean

\pm

standard deviation over three independent runs. All values are relative errors. The best baseline model for each task is indicated in parentheses.

	Airfoil	ShapeNet-Car	AirfRANS
Model	(Rel. Error)	(Corr. %)	(Corr. %)
Best Baseline	$0.0047{\scriptstyle\pm 0.0001}$	$99.28{\scriptstyle\pm 0.02}$	$99.92{\scriptstyle\pm 0.01}$
	(LaMO)	(SpiderSolver)	(LinearNO)
Advectra (Ours)	$\mathbf{0.0041}{\scriptstyle\pm 0.0001}$	$\mathbf{99.41}{\scriptstyle\pm 0.02}$	$\mathbf{99.95}{\scriptstyle\pm 0.01}$

Appendix

Table 13 : Standard deviations for geometric and mesh-based benchmarks. We report the mean

\pm

standard deviation over three independent runs. Spearman’s rank correlations for ShapeNet-Car and AirfRANS are presented as percentages (scaled by 100). Airfoil values are relative errors. The best baseline model for each task is indicated in parentheses.

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Learnable composition for neural operators

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Neural operators are fast, differentiable surrogates for physical simulation, but their accuracy often degrades when domain geometry, size, or operating conditions differ from training. Supervised adaptation can recover accuracy, but even a small target set requires costly high-fidelity simulations. We therefore ask how pretraining and transfer can be designed together to reduce this deployment cost. LatentDDM first pretrains a neural operator to predict fields on small subdomains. For a new setting, it freezes this operator and trains only a lightweight module that composes the local predictions. We evaluate our method on two complementary problems: steady Darcy flow, where long-range pressure coupling must extend across increasingly large porous domains, and unsteady incompressible flow around a pitching airfoil, where rollout errors compound as target pitching frequencies exceed the training range. Compared with the capacity-matched models that process the full domain at once, LatentDDM's error is 36-56% lower on larger Darcy domains after adaptation with 16 target simulations. It also improves 20-step field rollouts in fast-pitching airfoil flow, both zero-shot and after few-shot calibration. These results identify the co-designed local pretraining and composition-level transfer as a promising design principle for physical foundation models.

Abstract

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