Fourier Neural Operator

Variants

FNO3D

20.1M

3D Fourier Neural Operator for volumetric reservoir datasets. Uses 3D FFT for spectral convolutions.

Modes: Truncated 3D Fourier modes

Library: neuraloperator

Exact params: 20,087,224

Benchmarks:

Flow Arena Two-Phase Oil-Water CO2 Nested Two-Phase 3D Channels

FNO2D

2.4M

2D Fourier Neural Operator for Cartesian grid datasets. Uses 2D FFT.

Modes: Truncated 2D Fourier modes

Library: neuraloperator

Exact params: 2,430,992

Benchmarks:

Two-Phase Oil-Water CO2 Radial

Training Provenance

These values are read from the generated benchmark result metadata, not from static method-page copy. When a model is retrained and the website data is regenerated, this section updates with the result records.

Arena training recipe

Epochs: 200

Batch size: 32

Learning rate: 0.0001

Loss: CombinedGradientLoss, MSEGradientLoss

Arena feature sets

Flow Arena input channels vary by fault complexity. Current result records for this method include 3 feature-set definitions.

PORO PERMX SWAT_0 PRESSURE_0 MULTX MULTY DEPTH

Strengths & Weaknesses

Strengths

Global receptive field from the first layer — captures long-range pressure communication naturally
Resolution-invariant in principle — can generalize across grid sizes
Very compact model (0.8M params for 2D, 8.9M for 3D) compared to convolutional alternatives
Strong theoretical foundation in operator learning and function space approximation
Spectral bias provides implicit regularization for smooth fields like pressure

Weaknesses

Spectral truncation can blur sharp discontinuities (saturation fronts, fault boundaries)
Assumes periodic boundary conditions implicitly — may introduce artifacts near domain boundaries
Requires uniform Cartesian grids for standard FFT — not directly applicable to unstructured meshes
Performance degrades on datasets with highly localized features that need many Fourier modes
Checkpoint saving/loading requires special handling (state_dict extraction) unlike standard PyTorch models

Results Across Benchmarks

All results for Fourier Neural Operator variants across all benchmark families. Sorted by the first field's rel-L² error by default.

Two-Phase Oil-Water

Loss

9 results

					Pres (bar)			Sat (frac)
#	Target	Model	Loss	Epochs	rel-L²▲	MRE	MAE	rel-L²	MRE	MAE	Params
1	pressure	FNO3D	BadawiSingleField	200	0.0344	0.0300	5.154885	-	-	-	5.6M
2	combined	FNO3D	BadawiCombined	200	0.0415	0.0357	6.102467	0.0532	0.0423	0.014022	5.6M
3	combined	FNO2D	RelLp	200	0.0434	0.0375	6.469421	0.0504	0.0379	0.012490	14.7M
4	combined	FNO2D	RelLp	200	0.0555	0.0482	8.634762	0.0539	0.0414	0.013596	2.4M
5	pressure	FNO2D	RelLp	200	0.0581	0.0521	8.838425	-	-	-	14.6M
6	pressure	FNO2D	RelLp	200	0.0596	0.0528	9.173145	-	-	-	2.4M
7	saturation	FNO2D	RelLp	200	-	-	-	0.0617	0.0465	0.015350	2.4M
8	saturation	FNO2D	RelLp	200	-	-	-	0.0565	0.0426	0.014027	14.6M
9	saturation	FNO3D	BadawiSingleField	200	-	-	-	0.0555	0.0431	0.014490	5.6M

CO2 Radial

Loss

5 results

					Pres (bar)			Sat (frac)
#	Target	Model	Loss	Epochs	rel-L²▲	MRE	MAE	rel-L²	MRE	MAE	Params
1	pressure	FNO2D	UFNODerivLoss	500	0.0057	0.0015	0.350251	-	-	-	20.2M
2	pressure	FNO2D	RelLp	500	0.0064	0.0015	0.376695	-	-	-	9.0M
3	combined	FNO2D	RelLp	500	0.0068	0.0019	0.433446	0.1152	9.9045	0.003185	9.0M
4	saturation	FNO2D	UFNODerivLoss	500	-	-	-	0.1197	9.9107	0.003067	20.2M
5	saturation	FNO2D	RelLp	500	-	-	-	0.1067	8.7535	0.002723	9.0M

Flow Arena

Property

Fault

Control

Loss

24 results

					Sat (frac)			Pres (bar)
#	Dataset	Model	Loss	Epochs	rel-L²▲	MRE	MAE	rel-L²	MRE	MAE	Params
1	geostat_nf_rate	FNO3D	AbsLp(p=2)	200	0.2176	0.4239	0.050780	0.0422	0.0361	7.765929	20.1M
2	geostat_nf_bhp	FNO3D	AbsLp(p=2)	200	0.2207	0.4422	0.051482	0.0371	0.0320	8.007223	20.1M
3	geostat_nf_bhp	FNO3D	MSE	200	0.2345	0.4861	0.054643	0.0392	0.0337	8.408319	20.1M
4	geostat_nf_rate	FNO3D	MSE	200	0.2413	0.4747	0.051482	0.0472	0.0398	8.519495	20.1M
5	channels_nf_bhp	FNO3D	AbsLp(p=2)	200	0.3062	0.6482	0.081473	0.0630	0.0522	13.104323	20.1M
6	geostat_zt_bhp	FNO3D	AbsLp(p=2)	200	0.3079	0.5782	0.069802	0.0604	0.0490	12.343442	20.1M
7	geostat_zt_bhp	FNO3D	MSE	200	0.3101	0.5740	0.067988	0.0616	0.0500	12.460086	20.1M
8	channels_nf_bhp	FNO3D	MSE	200	0.3125	0.7185	0.082872	0.0630	0.0519	13.005433	20.1M
9	geostat_zt_rate	FNO3D	MSE	200	0.3215	0.5966	0.072588	0.0999	0.0760	17.914213	20.1M
10	geostat_zt_rate	FNO3D	AbsLp(p=2)	200	0.3224	0.6041	0.075425	0.0993	0.0763	18.035299	20.1M
11	channels_nf_rate	FNO3D	AbsLp(p=2)	200	0.3508	0.6550	0.080835	0.0766	0.0612	13.280948	20.1M
12	channels_nf_rate	FNO3D	MSE	200	0.3562	0.7176	0.080030	0.0801	0.0639	13.712110	20.1M
13	geostat_vt_bhp	FNO3D	MSE	200	0.3630	0.6794	0.083388	0.0810	0.0665	16.656662	35.7M
14	geostat_vt_bhp	FNO3D	AbsLp(p=2)	200	0.3636	0.7398	0.093502	0.0778	0.0636	16.000710	35.7M
15	geostat_vt_rate	FNO3D	AbsLp(p=2)	200	0.3744	0.7287	0.098138	0.1098	0.0830	20.213245	35.7M
16	geostat_vt_rate	FNO3D	MSE	200	0.3791	0.7203	0.084728	0.1190	0.0903	21.365623	35.7M
17	channels_zt_bhp	FNO3D	AbsLp(p=2)	200	0.4080	0.7672	0.089509	0.0860	0.0697	17.261770	20.1M
18	channels_zt_bhp	FNO3D	MSE	200	0.4119	0.8244	0.090742	0.0856	0.0688	17.075972	20.1M
19	channels_zt_rate	FNO3D	MSE	200	0.4301	0.8042	0.091805	0.1241	0.0950	22.028023	20.1M
20	channels_zt_rate	FNO3D	AbsLp(p=2)	200	0.4328	0.7792	0.094815	0.1227	0.0940	21.869766	20.1M
21	channels_vt_bhp	FNO3D	AbsLp(p=2)	200	0.4708	0.9126	0.105809	0.1008	0.0799	20.026644	35.7M
22	channels_vt_bhp	FNO3D	MSE	200	0.4785	0.9393	0.102066	0.1039	0.0824	20.654514	35.7M
23	channels_vt_rate	FNO3D	AbsLp(p=2)	200	0.4823	0.9359	0.109471	0.1458	0.1106	26.306049	35.7M
24	channels_vt_rate	FNO3D	MSE	200	0.4886	0.9646	0.111228	0.1505	0.1138	26.803350	35.7M

CO2 Nested

1 results

				Pres (bar)
#	Model	Loss	Epochs	rel-L²▲	MRE	MAE	Params
1	FNO3D	RelLp	200	0.0003	8.79e-5	0.020232	13.2M

Two-Phase 3D Channels

6 results

					Pres (bar)			Sat (frac)
#	Target	Model	Loss	Epochs	rel-L²▲	MRE	MAE	rel-L²	MRE	MAE	Params
1	pressure	FNO3D	RelLp	500	0.0021	0.0015	0.491382	-	-	-	9.0M
2	combined	FNO3D	RelLp	500	0.0022	0.0017	0.539620	0.2223	0.1442	0.031166	9.0M
3	pressure	FNO3D	RelLp	200	0.0024	0.0018	0.582078	-	-	-	9.0M
4	combined	FNO3D	RelLp	200	0.0026	0.0020	0.640025	0.2308	0.1786	0.036466	9.0M
5	saturation	FNO3D	RelLp	200	-	-	-	0.2312	0.1648	0.034480	9.0M
6	saturation	FNO3D	RelLp	500	-	-	-	0.2310	0.1571	0.033396	9.0M

Training Curves

FNO2D - Two-Phase Oil-Water (combined)

FNO2D - Two-Phase Oil-Water (pressure)

FNO2D - Two-Phase Oil-Water (saturation)

FNO2D - CO2 Radial (pressure)

FNO2D - CO2 Radial (saturation)

Architecture

The Fourier Neural Operator (FNO), introduced by Li et al. (2021), represents a fundamentally different approach to learning PDE surrogates. Instead of operating directly in physical space like convolutional networks, FNO performs convolutions in Fourier space via the Fast Fourier Transform (FFT). This spectral approach enables global receptive fields at every layer — each Fourier mode interacts with the entire spatial domain — making FNO naturally suited for problems with long-range spatial correlations like pressure diffusion.

The architecture consists of a lifting layer (projecting inputs to a higher-dimensional channel space), a sequence of Fourier layers (each performing spectral convolution followed by a local linear transform and nonlinearity), and a projection layer mapping back to the output space. The spectral convolutions truncate high-frequency modes, acting as a learned low-pass filter that implicitly regularizes the solution.

FNO has a key theoretical advantage: it is resolution-invariant in principle, meaning a model trained on one grid resolution can be evaluated on another without retraining. In practice, this property is approximate and works best when the underlying physics is smooth.

We use the NVIDIA NeuralOperator library implementation, which provides optimized 2D and 3D variants. The 3D version (FNO3D) treats the spatial domain as a 3D volume, while the 2D version (FNO2D) is used for Cartesian 2D datasets.

Model Summary

FNO2D

FNO2DWrapper(
  (model): FNO(
    (positional_embedding): GridEmbeddingND()
    (fno_blocks): FNOBlocks(
      (convs): ModuleList(
        (0-3): 4 x SpectralConv(
          (weight): DenseTensor(shape=torch.Size([64, 64, 16, 9]), rank=None)
        )
      )
      (fno_skips): ModuleList(
        (0-3): 4 x Flattened1dConv(
          (conv): Conv1d(64, 64, kernel_size=(1,), stride=(1,), bias=False)
        )
      )
      (channel_mlp): ModuleList(
        (0-3): 4 x ChannelMLP(
          (fcs): ModuleList(
            (0): Conv1d(64, 32, kernel_size=(1,), stride=(1,))
            (1): Conv1d(32, 64, kernel_size=(1,), stride=(1,))
          )
        )
      )
      (channel_mlp_skips): ModuleList(
        (0-3): 4 x SoftGating()
      )
    )
    (lifting): ChannelMLP(
      (fcs): ModuleList(
        (0): Conv1d(86, 128, kernel_size=(1,), stride=(1,))
        (1): Conv1d(128, 64, kernel_size=(1,), stride=(1,))
      )
    )
    (projection): ChannelMLP(
      (fcs): ModuleList(
        (0): Conv1d(64, 128, kernel_size=(1,), stride=(1,))
        (1): Conv1d(128, 80, kernel_size=(1,), stride=(1,))
      )
    )
  )
)

FNO3D

FNO3DWrapper(
  (model): FNO(
    (positional_embedding): GridEmbeddingND()
    (fno_blocks): FNOBlocks(
      (convs): ModuleList(
        (0-3): 4 x SpectralConv(
          (weight): DenseTensor(shape=torch.Size([48, 48, 4, 32, 17]), rank=None)
        )
      )
      (fno_skips): ModuleList(
        (0-3): 4 x Flattened1dConv(
          (conv): Conv1d(48, 48, kernel_size=(1,), stride=(1,), bias=False)
        )
      )
      (channel_mlp): ModuleList(
        (0-3): 4 x ChannelMLP(
          (fcs): ModuleList(
            (0): Conv1d(48, 24, kernel_size=(1,), stride=(1,))
            (1): Conv1d(24, 48, kernel_size=(1,), stride=(1,))
          )
        )
      )
      (channel_mlp_skips): ModuleList(
        (0-3): 4 x SoftGating()
      )
    )
    (lifting): ChannelMLP(
      (fcs): ModuleList(
        (0): Conv1d(8, 96, kernel_size=(1,), stride=(1,))
        (1): Conv1d(96, 48, kernel_size=(1,), stride=(1,))
      )
    )
    (projection): ChannelMLP(
      (fcs): ModuleList(
        (0): Conv1d(48, 96, kernel_size=(1,), stride=(1,))
        (1): Conv1d(96, 40, kernel_size=(1,), stride=(1,))
      )
    )
  )
)