Performance of Neural Operator Baselines on 'The Well' Dataset

Yash Pincha; Andy Dimnaku; Andrew Chiang; Adarsh Ganeshram; Valentin Duruisseaux; Anima Anandkumar

cs.LGOct 25, 2026

Performance of Neural Operator Baselines on 'The Well' Dataset

Authors: Yash Pincha, Andy Dimnaku, Andrew Chiang, Adarsh Ganeshram, Valentin Duruisseaux, Anima Anandkumar

Abstract

The Well, a 15-terabyte dataset spanning 16 physical simulation domains, provides an extensive benchmark for developing and evaluating machine learning surrogate models for complex physical systems. The baselines released with the benchmark were presented by their authors as illustrative examples obtained at small computational cost, trained with zero to very limited hyperparameter tuning and under a highly constraining 12-hour per-run wall-clock limit. These results have subsequently been treated as fully tuned benchmarks, and in some cases effectively interpreted as unconstrained performance baselines. We perform a short non-exhaustive hyperparameter search for the Tucker-factorized Fourier Neural Operator (TFNO) under the same 12-hour per-run wall-clock limit, while using only a single input timestep rather than the history of up to four used by the published baselines. Despite this more restrictive setup, tuning alone reduces VRMSE below both published FNO and TFNO baselines on 15 of 17 tasks, sometimes by several orders of magnitude. This sensitivity to a modest tuning effort underscores that the released neural operator baselines should not be treated as architectural benchmarks.

Figures & tables

Dataset	12h-tuned	vs. pub. TFNO	vs. pub. FNO	vs. Best Reported
acoustic_scattering_maze	0.022886	$22.1\times$	$22.1\times$	$0.7\times$
active_matter	0.077859	$4.6\times$	$4.7\times$	$1.3\times$
convective_envelope_rsg	0.030917	$0.9\times$	$0.9\times$	$0.9\times$
euler_multi_quadrants	0.060337	$6.9\times$	$6.8\times$	$2.5\times$
gray_scott_reaction_diffusion	0.022986	$15.8\times$	$5.9\times$	$5.9\times$
helmholtz_staircase	0.000280	$12.5\times$	$1.8\times$	$1.8\times$

Table 1: Sensitivity of the reported operator baselines to protocol and configuration. Column 2 gives the VRMSE of our 12h-tuned TFNO. Columns 3 and 4 give the ratio of the published TFNO and FNO VRMSEs from The Well [ 4 ] to ours, each computed as published value divided by tuned value ; column 5 gives the same ratio against the best (lowest) of all four published baselines on that dataset. Values above one indicate that our run scores lower than the reference; values below one indicate that it scores higher. The two baselines are reported separately rather than aggregated, since on gray_scott_reaction_diffusion and helmholtz_staircase the published FNO substantially outperforms the published TFNO, and any single aggregate would obscure this.

	Configuration and outcome							Ratios
Budget	Modes	Width	Batch	LR	Rank	Epochs	VRMSE	vs. 12h	vs. operator baseline
12 h	64	32	32	$2\times 10^{-3}$	0.20	17	0.022886	$1.0\times$	$22.1\times$
36 h	128	48	8	$8\times 10^{-4}$	1.00	38	0.004875	$4.7\times$	$103.8\times$

Table 2: Effect of increasing the per-run wall-clock budget from 12 to 36 hours on the dataset acoustic_scattering_maze . For each budget, the configuration and cyclic learning-rate schedule are re-selected, so the comparison captures the combined effect of additional training time and the different configurations that become competitive under the larger budget; it is not a continuation of the 12-hour run. Both runs are halted by their respective wall clocks while validation error is still decreasing and should therefore be treated as budget-limited. vs. 12h is

0.022886

divided by the row’s VRMSE. vs. operator baseline uses the larger of the published FNO and TFNO VRMSEs (

0.5062

) divided by the row’s VRMSE. Ratios against the published operator values quantify how far those protocol-specific figures can be displaced and are not evidence of architectural superiority.

Appendix figures & tables24 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameter	Candidate values
Layers	4, 6
Spectral modes	32, 64, 128, 256
Channel width	24, 30, 32, 40, 48, 50, 64, 96
Batch size	1–64, set by memory limits per dataset
Learning rate	$1\times 10^{-4}$ , $5\times 10^{-4}$ , $1\times 10^{-3}$ , $2\times 10^{-3}$
Tucker rank	0.05, 0.10, 0.20, 0.50, 1.00

Appendix

Table 3: Candidate values per varied hyperparameter. All other settings were fixed to the values in the caption of Table 4 . The number of spectral modes was chosen separately for each dataset based on resolution.

Dataset	Layers	Modes	Width	Batch Size	LR	Rank	Patience	Epochs	ES
acoustic_scattering_maze	4	64	32	32	$2\times 10^{-3}$	0.20	8	17	$\times$
active_matter	6	128	48	32	$1\times 10^{-3}$	0.20	4	171	✓
convective_envelope_rsg	6	64	24	1	$5\times 10^{-4}$	0.05	6	40	✓
euler_multi_quadrants	6	128	30	12	$5\times 10^{-4}$	1.00	4	4	$\times$
gray_scott_reaction_diffusion	6	64	64	32	$5\times 10^{-4}$	0.10	8	7	$\times$
helmholtz_staircase	6	128	64	8	$5\times 10^{-4}$	0.10	4	64	✓

Appendix

Table 4: Selected TFNO configurations and training outcomes for the 17 reported 12-hour runs. All runs use model=FNO , factorization=Tucker , optimizer=Adam , lr_method=cycle , normalize=1 , lifting_channels=projection_channels=64 , and max_train_hours=12 . Epochs is the number completed before termination. ES marks whether the early-stopping criterion fired within the budget (✓) or the run was halted by the 12-hour wall clock while validation error was still decreasing (

\times

). A ✓ records only the behavior of the budget-shaped schedule and is not evidence of budget-independent convergence (Section 3.2 ). A rank of 1.00 denotes the dense limit. Configurations were selected by the limited search described in Section 2.5 and are not claimed to be optimal.

Figure 1: acoustic_scattering_maze , pressure $p$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 2: active_matter , concentration $c$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 3: convective_envelope_rsg , radial velocity $v_{r}$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 4: euler_multi_quadrants , density $\rho$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 5: gray_scott_reaction_diffusion , species $A$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 6: helmholtz_staircase , real part of the pressure, $\mathrm{Re}\,p$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 7: MHD_64 , magnetic field $B_{x}$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 8: planetswe , height $h$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 9: post_neutron_star_merger , density $\rho$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 10: rayleigh_benard , buoyancy $b$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 11: rayleigh_taylor_instability , density $\rho$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 12: rayleigh_taylor_instability , velocity $v_{x}$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 13: rayleigh_taylor_instability , velocity $v_{y}$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 14: rayleigh_taylor_instability , velocity $v_{z}$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 15: shear_flow , passive tracer $c$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 16: supernova_explosion_64 , density $\rho$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 17: turbulence_gravity_cooling , density $\rho$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 18: turbulence_gravity_cooling , pressure $p$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 19: turbulence_gravity_cooling , temperature $T$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 20: turbulent_radiative_layer_2D , density $\rho$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 21: turbulent_radiative_layer_3D , density $\rho$ : ground truth (top) and tuned-TFNO prediction (bottom).

Figure 22: viscoelastic_instability , conformation tensor $c_{zz}$ : ground truth (top) and tuned-TFNO prediction (bottom).

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Sequential Physics-Constrained Neural Operator Forward Modeling for the $\textit{Norne}$ Reservoir System

Clement Etienam, Juntao Yang, Oleg Ovcharenko, Nick Luiken+3

NVIDIA Corporation

Fourier Neural Operator PDE Surrogate Modeling Physics-Informed Neural Operators Computational Fluid Dynamics

Open paper AskPDF arXiv

We develop a comprehensive mathematical and computational framework for sequential surrogate modeling of three-phase black-oil reservoir dynamics using neural operators, with particular emphasis on Fourier Neural Operators (FNO) and their physics-informed variant (PINO). The application focus is the Norne benchmark reservoir, defined on a heterogeneous

46\times112\times22

grid (

N=113,344

cells), with a production history spanning

T=30

timesteps covering 3298 days. Our theoretical contributions are organized around four interlocking problems: (1) functional-analytic formulation in a product-Sobolev-space setting, including well-posedness of the implicit timestep map and sharp local Lipschitz estimates; (2) covariate shift quantification, proving that the Wasserstein-2 distance grows as

W_2 \leq \varepsilon(L^n-1)/(L-1)

, with exponential population-risk discrepancy for

L>1

; (3) physics-constrained spectral stability, showing PINO training with

λ_R \geq λ^*_R

reduces the learned Jacobian spectral radius to

ρ_F + Cλ_R^{-1/2}

, yielding uniform-in-time rollout error

|δ_n| \leq \varepsilon/(1-ρ)

; and (4)

K

-step TBPTT gradient analysis, deriving geometric bias decay

O(ρ^K)

, optimal window

K^ = O(\log(T/σ^2))

, and Adam convergence

O(1/\sqrt{t}) + O(ρ^{K^*})

. Empirical validation confirms all theoretical predictions: autoregressive PINO surrogates sustain

R^2>0.99

(oil),

R^2>0.90

(gas),

R^2\approx 0.80

(pressure), and monotonically improving

R^2

(water) across the full 3298-day horizon, trained on eight NVIDIA B200 GPUs in under one hour. A 1000-member ensemble runs in under one minute on a single B200 GPU, giving a

{\sim}10^4\times

wall-clock speedup over the OPM finite-volume simulator.

Abstract

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