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Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogates

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Neural PDE surrogates become markedly more stable and generalize better when trained with both rotation/reflection symmetries and hard conservation laws, using new layers that make these constraints work on staggered fluid grids.

desk verdict Staggered-grid equivariant layers are a real construction and the empirical case for combining constraints is strong; the SWE circular-padding mismatch weakens but does not sink the closed-boundary claim. read the letter →

arxiv 2506.05513 v1 pith:ZHWJRXQX submitted 2025-06-05 cs.LG physics.flu-dyn

Yunfei Huang , David S. Greenberg This is my paper
classification cs.LGphysics.flu-dyn
keywords neuralPDEsurrogatessymmetryequivariancestaggeredC-gridconservationlawsshallowwaterequationsdecayingturbulenceautoregressiverolloutshardconstraints
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

This paper claims that imposing geometric symmetries and physical conservation laws together makes neural PDE surrogates markedly better at long-horizon forecasting. To do this on the staggered C-grids used by fluid solvers, the authors design new equivariant input and output layers, then test every combination of rotation/reflection equivariance and mass/momentum conservation on shallow water equations and decaying turbulence. The consistent result is that each constraint helps, symmetry helps more than conservation, and the two together beat unconstrained networks, data augmentation, and pushforward training while remaining compatible with pushforward. If right, these constraints offer a practical route to stable long-rollout surrogates without changing the underlying solver grid.

What carries the argument

The load-bearing object is a pair of custom convolutional layers that interface a regular representation with a staggered C-grid. A regular representation is a set of channels indexed by the symmetry group's elements, on which rotation or reflection acts by permuting channels and spatially transforming each channel. The input layer transforms rectangular filter banks according to the group action, so that rotating the staggered velocity field permutes the output channels exactly as the symmetry requires; the output layer reads out staggered velocities from surrounding cell-center regular-representation values using weight patterns that are proven equivariant. Hard conservation laws are then applied as output layers: global mean subtraction for mass and momentum, and a curl-of-a-vector-potential parameterization ($\nabla \times a$) for divergence-free mass conservation. This sandwich lets an off-the-shelf equivariant convolutional backbone produce surrogates that are both exactly equivariant and exactly conservative on staggered grids.

What would settle it

Seed a shallow water wave near one boundary and check whether a surrogate trained with circular padding produces a spurious wave entering from the opposite boundary before the physical reflection time; if it does, the reported closed-boundary results are partly an artifact of periodic wrap-around rather than the intended reflecting dynamics.

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Extended reading notes

Core claim

The central claim is that symmetry equivariance and hard conservation laws are complementary inductive biases for neural PDE surrogates, and that they can be imposed even on staggered Arakawa C-grids where existing equivariant layers fail. The paper introduces specialized input and output layers that map staggered velocity fields to and from a regular representation, so that rotations and reflections act correctly on the grid's cell-centered and interface-centered variables. With these layers, the authors show that a doubly constrained surrogate (full rotation/reflection equivariance plus mass and momentum conservation) consistently outperforms unconstrained baselines, symmetry-only and physics-only variants, and strong training techniques such as data augmentation and pushforward training, across two fluid tasks, two base architectures, multiple network sizes, multiple dataset sizes, and real ocean current forecasts. Symmetries alone help more than conservation laws alone, but the two together give the best accuracy, stability, and generalization to new initial conditions and rollout durations beyond the training range.

Load-bearing premise

The shallow-water results assume that the network's circular padding, which makes the domain wrap around, does not secretly replace the intended closed reflecting walls, and that the binary boundary mask cancels any wrap-around artifact.

Editorial extensions

If this is right

  • Doubly constrained surrogates maintain accuracy on closed-boundary shallow water rollouts for 50 simulated hours, while unconstrained, FNO, and Dilated ResNet baselines diverge to infinity well before the end of the rollout.
  • In decaying turbulence, the doubly constrained surrogate matches the reference velocity and energy spectra even after its field-level correlation with the reference has decayed to zero, indicating better long-term statistical fidelity.
  • Symmetry and physics constraints improve generalization to novel initial conditions, including L-shaped and overlapping-elevation shallow water waves and turbulence initial conditions at wavenumber 8, beyond the training distribution.
  • The doubly constrained surrogate also outperforms an existing equivariant U-Net baseline and the unconstrained network on real ocean current forecasting at 12-hour and 120-hour horizons.
  • Constraints help most for smaller networks and longer rollouts, and the benefits persist as network and dataset sizes grow, so the approach remains relevant in data-hungry regimes.

Reading between the lines

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

  • Beyond the paper, the same input/output layer construction should extend to other staggered-grid quantities such as fluxes and vertex-centered scalars, which could make equivariant conservative surrogates practical for operational ocean and atmosphere models without modifying their solvers.
  • Because global mean subtraction is a nonlocal operation, a finite-volume-style local flux parameterization, which the paper mentions as future work, could combine equivariance with domain-size transfer and deserves direct testing.
  • The SWE results could be partly affected by the backbone's circular padding, which makes the domain wrap around even though the intended boundary condition is closed and reflecting; a padding ablation would separate the constraint benefit from a possible wrap-around artifact.
  • If the constraint benefits transfer to higher Reynolds numbers and non-square domains, the approach could become a default recipe for learned fluid solvers; that transfer is not established by the paper.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes novel group-equivariant input and output layers for neural PDE surrogates on staggered Arakawa C-grids, extending a modern U-Net with p4/p4m symmetry constraints and hard mass/momentum conservation constraints. It reports a systematic comparison on two tasks: shallow water equations with closed Dirichlet boundaries and decaying incompressible turbulence. The authors find that symmetry constraints improve accuracy and stability relative to unconstrained baselines, physical constraints yield additional gains, and doubly constrained p4m/M+rho-u surrogates perform best, including on generalization to new initial conditions, longer rollouts, and real-world ocean current forecasting. Code is released and the numerical reference solutions come from external solvers.

Significance. If the empirical claims hold, this is a valuable contribution: it removes a practical obstacle to using equivariant convolutional layers in computational fluid dynamics with staggered grids, and it provides an unusually broad ablation across architectures, training modes, data sizes, and network sizes. The external-solver evaluation, the released code, and the explicit generalization tests are strengths that make the results checkable. However, the SWE experiment is confounded by circular padding in a closed-boundary task, and the p4 output-layer derivation in the appendix is internally inconsistent. These issues must be fixed before the paper's central claim about closed-boundary surrogates can be accepted.

major comments (3)
  1. [§4.1 and Table 8] The SWE task is defined with closed Dirichlet boundary conditions (u=0 on ∂Ω, Eq. 8), yet Table 8 lists Circular padding for all modern U-Net configurations, including the p4 and p4m equivariant variants. Circular padding makes the hidden representation periodic, so the network can use wrap-around information from opposite edges of the grid, and the binary boundary mask is itself circularly padded. This means the SWE experiment does not cleanly test the closed-boundary setting described in the text. The constrained-vs-unconstrained comparison is still fair because all SWE variants share the same padding, but the paper's claim that the doubly-constrained surrogate is the best closed-boundary surrogate requires a retest with non-periodic padding (e.g., zero or constant padding) or an explicit demonstration that the circular padding is benign for this task.
  2. [Appendix D.1.1, Eqs. (59)-(64) and main text Eq. (50)] The derivation of the p4 output layer in Appendix D.1.1 is inconsistent with the formula given in the main text. Eq. (50) defines u_{i+0.5,j}=p_{i+1,j,0}-p_{i,j,1} and v_{i,j+0.5}=p_{i,j+1,2}-p_{i,j,3}, while Eqs. (63)-(64) state u_{i+0.5,j}=p_{i+1,j,1}-p_{i,j,3} and v_{i,j+0.5}=p_{i,j+1,2}-p_{i,j,4}. The latter uses an index 4 that does not exist in a p4 regular representation (indices 0-3), and the constraint equations (59)-(62) refer to c4, d4, e4, f4, which are out of range. The derivation must be rewritten with consistent indexing and must conclude with the formula that is actually implemented in the released code.
  3. [Appendix C] Appendix C provides a full equivariance proof only for the p4 SWE input layer and states that the p4m and INS input layers are 'straightforward' extensions; these layers are used in the main experiments. Since the equivariance of these layers is load-bearing for the method, the authors should either provide complete proofs for the p4m and INS cases or explicitly state in the main text that those cases are verified only empirically (Figures 11-12 show agreement to numerical precision, which is reassuring but is not a derivation).
minor comments (5)
  1. [Table 3] The handling of NaN entries is unclear: do the reported means and standard errors over the 20 ICs exclude divergent rollouts, set them to infinity, or aggregate them in some other way? Please specify the aggregation rule so the 25h results can be interpreted correctly.
  2. [Section 5.4 and Fig. 6] Several panels in Figure 6 have partially truncated axis labels and legends; please ensure that all subplots are legible in the final version.
  3. [Eq. (5)] The rotation action on vector fields in Eq. (5) uses a particular sign convention; please define whether R is a clockwise or counterclockwise 90-degree rotation so the later proofs and code can be checked unambiguously.
  4. [Appendix B] The paragraph on padding options is vague; it should specify which layers use circular padding, which use zero or constant padding, and how the C-grid dimension mismatch at domain boundaries is handled for the input and output layers.
  5. [Table 5 and Appendix J.14] For the ocean-current experiment, the data source should cite the actual ocean forecast product (e.g., the CMEMS product) in addition to Marullo et al. 2014, so that the dataset is identifiable and reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: constraint layers are imposed by construction, not fitted, and all performance claims are benchmarked against an external numerical solver.

full rationale

The paper's central claims are measured against an external numerical reference solver rather than against quantities fitted by the authors, so the derivation chain is self-contained. The physical constraints are hard-imposed by construction: global-mean subtraction for mass and momentum (Appendix F, Eqs. 74-76) and a curl reparameterization for divergence-free velocity. The paper does not present the resulting conservation as a prediction; it reports these as constraint checks, while the headline accuracy metrics (NRMSE, correlation, spectra) are computed with respect to the solver's outputs. The output-layer coefficients (Appendix D, Eqs. 63-64 and 72-73) are fixed by hand after deriving the full equivariant family; the paper explicitly states that other equivariant choices or learned coefficients are possible, so no parameter is fitted to the evaluation data. Equivariance of the new layers is shown by proof and by numerical verification against the solver's symmetry transformations, which are sanity checks rather than circular reductions. The only self-citation (Huang & Greenberg, 2023) appears in a related-work list and is not used to justify any load-bearing premise, uniqueness theorem, or ansatz. A known architectural concern is the use of circular padding (Table 8) for the closed-boundary SWE task (Eq. 8), which may confound that specific experiment, but it is a correctness issue rather than circularity: the constrained and unconstrained variants share the same padding, and the conclusion would need a padding-aware retest rather than being equivalent to its inputs by construction. No step in the derivation reduces to its own inputs or to a self-citation chain.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central empirical claims rest on the assumed symmetries and conservation laws of the reference solvers and on the selected hard-constraint parameterizations. No new physical entities are introduced; the main burden is that the constraints are exactly valid for the data, including real ocean observations.

free parameters (1)
  • Equivariant output layer readout coefficients = c1=1, other ci, di, ei, fi = 0 for p4; analogous for p4m
    The general equivariant output layer family (Appendix D) allows learnable coefficients, but the paper fixes one member. This hand-chosen readout could affect the accuracy of constrained models, though it is not fitted to data.
assumptions (5)
  • domain assumption The numerical solvers for SWE and INS are exactly equivariant under p4m on their respective domains, as empirically verified in Figures 9-10.
    This is the core justification for enforcing hard symmetry in the surrogate. It is verified only for specific test cases, not proven.
  • domain assumption For incompressible flow, the divergence-free constraint exactly equals mass conservation, and any divergence-free velocity on the discrete C-grid can be written as the curl of a vertex-defined scalar potential.
    Used to impose mass conservation via curl parameterization (§3, Appendix F), following Wandel et al. 2021.
  • domain assumption Global mean subtraction of the update is a valid hard constraint for mass conservation in SWE and momentum conservation in INS.
    Applied in Appendix F equations (74)-(75). For SWE, this enforces global mass, not local conservation; for INS, it forces the domain-averaged velocity update to zero.
  • domain assumption Real ocean current observations, despite lacking exact p4m symmetry and momentum conservation due to Coriolis and boundaries, are suitable for the same hard constraints.
    Section 5.5 applies p4m/M+ρ⃗u directly to ocean data without relaxing constraints; no justification of approximate validity is given.
  • domain assumption The modern U-Net is a strong base architecture, and parameter-count-matched baselines are fair comparisons.
    Used throughout for comparisons; parameter counts are matched but computational cost and effective capacity differ. The authors do not report FLOPs.

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Pith. "Pith review of Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogates." pith.science (2026). https://pith.science/paper/ZHWJRXQX

@misc{pith2026250605513,
  author       = {Pith},
  title        = {Pith review of: Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZHWJRXQX}},
  note         = {Machine review of arXiv:2506.05513}
}
read the original abstract

Neural PDE surrogates can improve the cost-accuracy tradeoff of classical solvers, but often generalize poorly to new initial conditions and accumulate errors over time. Physical and symmetry constraints have shown promise in closing this performance gap, but existing techniques for imposing these inductive biases are incompatible with the staggered grids commonly used in computational fluid dynamics. Here we introduce novel input and output layers that respect physical laws and symmetries on the staggered grids, and for the first time systematically investigate how these constraints, individually and in combination, affect the accuracy of PDE surrogates. We focus on two challenging problems: shallow water equations with closed boundaries and decaying incompressible turbulence. Compared to strong baselines, symmetries and physical constraints consistently improve performance across tasks, architectures, autoregressive prediction steps, accuracy measures, and network sizes. Symmetries are more effective than physical constraints, but surrogates with both performed best, even compared to baselines with data augmentation or pushforward training, while themselves benefiting from the pushforward trick. Doubly-constrained surrogates also generalize better to initial conditions and durations beyond the range of the training data, and more accurately predict real-world ocean currents.

Figures

Figures reproduced from arXiv: 2506.05513 by the authors.

Figure 1
Figure 1. Symmetry- and physics-constrained neural surrogate for incompressible flow on a staggered grid. A rotation-equivariant input layer maps velocities onto a non-staggered regular representation, hidden layers employ steerable convolutions and the equivariant output layer enforces conservation laws on mass and momentum (M + ρu⃗ ) as it maps to staggered velocities. 2. Background and Related Work Neural PDE surrogates We… view at source ↗
Figure 2
Figure 2. Action of rotation-equivariant input layer on staggered velocity fields (top left). The filter bank is transformed by each g ∈ G to compute a G-indexed regular representation y. Rotation-transforming inputs (bottom left) yields permuted, rotated output channels. Staggered grids Fluid dynamical systems are often simu￾lated using staggered grids ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. p4m/M (symmetry+physics constraints) outperforms other networks with similar parameter counts on SWEs. (a) Refer￾ence surface disturbance ζ with predictions from p1/∅ and p4m/M. (b-c) Accuracy over 50h rollouts, with standard error of the mean over 20 ICs. (d) Training loss over iterations. (e) Histogram of EtNRMSE over 20 ICs. (f) Violation of mass conservation for all methods (black line shows reference simulation… view at source ↗
Figures from the paper (32 more)
Figure 4
Figure 4. Figure 4: p4m/M+ρu⃗ outperforms other networks with similar parameter counts on INS. (a) Reference horizontal velocity with predictions from p1/∅ and p4m/M+ρu⃗ . (b-c) Accuracy over 50h rollouts, with standard error of the mean over 30 ICs. (d-e) Log-log plots of the average vel…
Figure 5
Figure 5. Figure 5: Generalization beyond training data. (a) SWE rollouts from p1/∅ p4m/M on L-shaped ICs. (b-c) Accuracy of each network over six generalization tests (Appendix J.11). (d) INS rollouts from p1/∅ and p4m/M+ρu⃗ on ICs with peak wavenumber 8. (e-f) Velocity- and energy spect…
Figure 6
Figure 6. Figure 6: Accuracy of symmetry- and physics-constrained INS models across data and network sizes, at 4.2 and 12.6 s. (a￾b) NRMSE-u and ρ(ˆu, u) vs. network size for p1/∅ and p4m/M+ρu⃗ . (c-d) NRMSE-u and ρ(ˆu, u) for p1/∅ and p4m/M+ρu⃗ vs. training datasets size. (Figs. 4d-e, ad…
Figure 7
Figure 7. Figure 7: Prediction of ocean current observations by p1/∅, Wang’s Equrot (Unet) and p4m/M+ρu⃗ . (a) Zonal ocean currents from reference observations and predictions. (b-c) NRMSE-u and correlation for forecasts up to 300 h ahead (averaged over 30 ICs). We present additional resu…
Figure 8
Figure 8. Figure 8: Arakawa’s C-grid staggering approach has been employed for time integrating the of shallow water equations (SWEs) and incompressible Navier–Stokes equations (INS). (a) Red square points denote the vertical displacement of the free surface, denoted by ζ. The circular po…
Figure 9
Figure 9. Figure 9: Empirical validation of the symmetries of the numerical SWE solver is demonstrated through three transformations: flip, rotation, and flip-rotation. These plots correspond to Equations (11-13). A.3. Symmetries of the incompressible Navier–Stokes equations The incompres…
Figure 10
Figure 10. Figure 10: Empirical confirmation of the equivariance of the INS solver. B. Padding options In certain numerical solvers, although a C-grid staggering is employed, the software generates output of an identical size for each component of the vector field. This phenomenon necessit…
Figure 11
Figure 11. Figure 11: Example illustrating equivariance of the SWE input layer for groups p4 and p4m on staggered grid input data. The equivariance relation is observed to hold, to numerical precision. In this case, a G-averaged regular representation from the input layer’s output is emplo…
Figure 12
Figure 12. Figure 12: Group equivariance of INS input layers for p4 and p4m. The mathematical formulas of these input layers are given in equations 42-49. D. Group equivariant output layer on the C-grid staggering for INS As shown in [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Equivariance of neural PDE surrogates with (a) and without (b) specialized input layers for velocity fields on staggered C-grids. Both networks used the same modern u-net architecture, but for (b) a standard equivariant vector field input from escnn was used, which as…
Figure 14
Figure 14. Figure 14: shows a hybrid method used to predict the solution of a shallow water system. In our neural PDE surrogate, we have only one output ζ and we have three inputs u, v and ζ. An additional physics-based solution step is required to compute u t and v t from the ζ t produced…
Figure 15
Figure 15. Figure 15: Rollout predictions for the fluid surface elevation ζ for all surrogates on the closed boundary shallow water system at various forecast horizons. The predictions of ten trained models are compared to the reference. The results obtained from this analysis clearly demo…
Figure 16
Figure 16. Figure 16: Comparison of reference simulations and neural surrogates trained with input noise, denoted p1/∅ + ε and p4m/M+ε, as well as their their noise-free counterparts p1/∅ and p4m/M, based on the metrics NRMSE-ζ and ρ( ˆζ, ζ). The NRMSE-ζ metric reveals that p4m/M+ϵ exhibit…
Figure 17
Figure 17. Figure 17: shows the mass, momentum, and total energy for the closed boundary shallow water system for various surrogates and the reference solution over the course of 50 simulated hours 17. The mass and total energy remain constant, while momentum oscillates slightly (black cur…
Figure 18
Figure 18. Figure 18: Rollout predictions for u from thirteen models with approximately 0.1M parameters each for decaying turbulence at varying time steps. The top row shows the reference simulation. This plot extends [PITH_FULL_IMAGE:figures/full_fig_p031_18.png]
Figure 19
Figure 19. Figure 19: Rollout predictions for v from thirteen models with approximately 0.1M parameters each for decaying turbulence at varying time steps. The top row shows the reference simulation. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_19.png]
Figure 20
Figure 20. Figure 20: shows reference and predicted momentum for various surrogates. The non-physically-constrained models (p1/∅, p4/∅, and p4m/∅) exhibit a substantial increase in momentum compared to the reference. h e a b c d f g i reference p4m/ reference p4m/M+ reference p1/ referenc…
Figure 21
Figure 21. Figure 21: Velocity and energy spectra at additional time points. Results are consistent with the those presented in the main text, specifically Figures (4-d,e). p4m/M+ρu⃗ (blue curves) most closely matches the reference spectra (black). J.7. Training with input noise for decayi…
Figure 22
Figure 22. Figure 22: Comparison of surrogates trained with noise vs. clean data: p1/∅ + ε, p4m/M+ρu⃗ + ε, p1/∅ and p4m/M+ρu⃗ . We find that training with noise did not bring about clear improvements to accuracy, though unconstrained models were slightly more accurate for long rollouts. J.…
Figure 23
Figure 23. Figure 23: Rollouts of u by nine Dilated ResNet surrogates for decaying turbulence. The network size these surrogates was approximately 0.1M parameters. In this particular instance, the p4m/M+ρu⃗ model is the most similar to the reference solution. 35 [PITH_FULL_IMAGE:figures/f…
Figure 24
Figure 24. Figure 24: Rollouts of u by nine Dilated ResNet surrogates for decaying turbulence, same data as [PITH_FULL_IMAGE:figures/full_fig_p036_24.png]
Figure 25
Figure 25. Figure 25: NRMSE-u and ρ(ˆu, u) for the Dilated ResNet surrogates on the decaying turbulence task. Results are derived from 30 random initial conditions. We also computed velocity and energy spectra (u(k)k 5 and E(k)k 5 ) for the Dilated ResNet models of decaying turbulence. Dou…
Figure 26
Figure 26. Figure 26: Velocity and energy spectra (u(k)k 5 and E(k)k 5 ) for the Dilated ResNet models of decaying turbulence. J.9. Pushforward Training Here we analyze surrogates trained with the pushforward trick (Brandstetter et al., 2022c) in greater detail. We consider the modern U-Ne…
Figure 27
Figure 27. Figure 27: NRMSE-u and ρ(ˆu, u) for pushforward training of PDE surrogates. J.10. Data augmentation Data augmentation is a regularization technique that uses transformed copies of data points to train a neural network, to reduce overfitting and improve generalization. It has bee…
Figure 28
Figure 28. Figure 28: Performance of data-augmented models. J.11. Generalization tasks We tested surrogates on additional ICs that were not present in their training datasets, for the SWE and INS cases. J.11.1. GENERALIZATION FOR SWES We evaluated generalization capability for SWE surrogat…
Figure 29
Figure 29. Figure 29: Rollouts demonstrating generalization for the SWEs, generated by all surrogates from a single rectangular-shaped elevation IC, are shown at various time intervals. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_29.png]
Figure 30
Figure 30. Figure 30: Rollouts demonstrating generalization for the SWEs, generated by all surrogates from a single IC consisting of two rectangular elevations, are shown at various time intervals. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_30.png]
Figure 31
Figure 31. Figure 31: Rollouts demonstrating generalization for the SWEs, generated by all surrogates from a single IC consisting of two overlapping (summed) rectangular elevations, are shown at various time intervals. J.11.2. GENERALIZATION FOR DECAYING TURBULENCE We evaluated generalizat…
Figure 32
Figure 32. Figure 32: Rollout performance of networks with various physical and symmetry constraints for the generalization test of decaying turbulence. The figures depict the evolution of the field variable u. 42 [PITH_FULL_IMAGE:figures/full_fig_p042_32.png]
Figure 33
Figure 33. Figure 33: Rollout performance of networks with various physical and symmetry constraints for the generalization test of decaying turbulence. The figures depict the evolution of the field variable v [PITH_FULL_IMAGE:figures/full_fig_p043_33.png]
Figure 34
Figure 34. Figure 34: Velocity (u) and energy power spectra over an extended range of rollouts for the generalization of decaying turbulence case. This analysis extends the findings presented in [PITH_FULL_IMAGE:figures/full_fig_p044_34.png]
Figure 35
Figure 35. Figure 35: Top: effect of network size on NRMSE-u, ρ(ˆu, u) (a, b) and the energy spectrum (c-g) for PDE surrogates. Bottom: effect of training data size on NRMSE-u, ρ(ˆu, u) (h, i), and the energy spectrum (g,s). All results are reported for 99.7s rollouts. J.13. Inference time…

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    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.