cs.LGOct 20, 2026

Neural Operator Splitting for Fluid Dynamics in Complex Geometry

Authors: Jianxiang Li, Xi Deng, Hong Chul Nam, Anima Anandkumar

Organizations: New York University

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Abstract

Solving dynamical systems in complex and diverse geometries remains challenging for both numerical solvers and learning-based surrogate models. We propose Neural Operator Splitting, a framework for learning fluid dynamics while preserving the physical structure of fractional-step numerical schemes. Instead of learning the full-step evolution monolithically, we decompose each time step into physically interpretable steps, and learn each operator separately using intermediate states from the underlying numerical solver. Guided by the linearity properties of the fractional steps, we introduce a conditional-linear Fourier neural operator (CL-FNO) whose linear state mappings are modulated by nonlinear conditioning on inputs such as boundary geometry. We further introduce two complementary strategies, symmetry sampling and corrective cycling, to mitigate error accumulation during autoregressive rollout. Experiments on 2D and 3D fluid flows with varying geometries and initial conditions show that our full model reduces mean relative L2L^2 rollout error by 68% in 2D and 40% in 3D compared with a single full-step neural operator. At comparable error levels, neural-operator inference is approximately 10310^3 times faster per step than the Monte Carlo solver.

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Figure 1: (a) Improved accuracy . We compare our neural operator splitting approach against a single operator with a matched network parameter count. Across four datasets, our method achieves a 2–3 ×\times reduction in relative L2L^{2} error. (b) 3D fluid rollout visualization . We visualize the predicted 3D velocity field using particle trajectories on a representative slice of the domain. Across unseen geometries, our model produces rollout dynamics that remain spatially close to the reference solution.
Figure 2: Overview of learned operator splitting . (a) Geometry-conditioned advection, diffusion, and projection operators advance the velocity field from frame u(n)u^{(n)} to frame u(n+1)u^{(n+1)} . (b) Monte Carlo solver targets supervise each stage during training; no solves are required at inference. (c) Each operator uses a geometry encoder to generate gates that modulate spectral convolution blocks processing the input velocity field.
Figure 3: Qualitative rollout comparison in 2D and 3D. Rows show the MC reference, a single full-step operator (Single), and our combined model (Ours). Ours uses splitting, CL, SS and CC. Columns show t=0.8t=0.8 , 1.61.6 , and 3.23.2 . (a) Flow around a circular obstacle, colored by signed vorticity with a shared scale. (b) Flow around a bunny-shaped obstacle. Blue volumes show vortex regions identified by the λ2\lambda_{2} criterion. All 3D panels use the same rendering parameters.
Figure 4: Vortex-ring interaction with a bunny-shaped obstacle. Rows show a single full-step operator (single), an operator-splitting model (split), and the MC reference. Columns show t=0.0t=0.0 , 0.40.4 , 0.80.8 , 1.21.2 , and 1.61.6 . Red volumes show the magnitude of azimuthal vorticity; brown surfaces show the solid obstacle.
DatasetSingleSingle 3 ×\timesSplit
Spectral-2D1.626×10−3\mathbf{1.626\times 10^{-3}}1.634×10−31.634\times 10^{-3} ( ++ 0.5%)2.630×10−32.630\times 10^{-3} ( ++ 61.8%)
Basilisk-2D5.286×10−25.286\times 10^{-2}5.251×10−25.251\times 10^{-2} ( −- 0.7%)4.919×10−2\mathbf{4.919\times 10^{-2}} ( −- 6.9%)
MC-2D7.78×10−27.78\times 10^{-2}7.96×10−27.96\times 10^{-2} ( ++ 2.2%)6.95×10−2\mathbf{6.95\times 10^{-2}} ( −- 10.7%)
MC-3D5.07×10−25.07\times 10^{-2}5.14×10−25.14\times 10^{-2} ( ++ 1.3%)3.86×10−2\mathbf{3.86\times 10^{-2}} ( −- 23.8%)
Table 1: Relative L2L^{2} rollout error of single-operator and split models. Single 3×3\times uses three times as many FNO layers as Single. Percentages are relative to Single.
Figure 5: Relative L2L^{2} rollout error of standard and conditional-linear (CL) single-operator and split models. Black dotted curves show MC noise; shaded regions mark extrapolation beyond the training horizon.
DatasetStandard SingleCL SingleStandard SplitCL Split
Spectral-2D1.626×10−31.626\times 10^{-3}2.416×10−32.416\times 10^{-3} ( ++ 48.6%)2.630×10−32.630\times 10^{-3}1.031×10−3\mathbf{1.031\times 10^{-3}} ( −- 60.8%)
Basilisk-2D5.286×10−25.286\times 10^{-2}4.999×10−24.999\times 10^{-2} ( −- 5.4%)4.919×10−24.919\times 10^{-2}1.901×10−2\mathbf{1.901\times 10^{-2}} ( −- 61.3%)
MC-2D7.78×10−27.78\times 10^{-2}7.59×10−27.59\times 10^{-2} ( −- 2.5%)6.95×10−26.95\times 10^{-2}3.97×10−2\mathbf{3.97\times 10^{-2}} ( −- 42.9%)
MC-3D5.07×10−25.07\times 10^{-2}4.09×10−24.09\times 10^{-2} ( −- 19.3%)3.86×10−23.86\times 10^{-2}3.17×10−2\mathbf{3.17\times 10^{-2}} ( −- 17.9%)
Table 2: Relative L2L^{2} rollout error of the standard and conditional-linear (CL) single and split models. Percentages indicate the relative change from the corresponding standard model.
Figure 6: Relative L2L^{2} rollout error of the single-operator model with plain inference, symmetry sampling (SS), and full group averaging. The symmetry group is D4D_{4} in 2D and OhO_{h} in 3D. Black dotted curves show MC noise; shaded regions mark extrapolation beyond the training horizon.
DatasetSingleSingle+SSSplitSplit+SS
Spectral-2D1.626×10−31.626\times 10^{-3}8.424×10−4\mathbf{8.424\times 10^{-4}} ( −- 48.2%)2.630×10−32.630\times 10^{-3}1.303×10−31.303\times 10^{-3} ( −- 50.5%)
Basilisk-2D5.286×10−25.286\times 10^{-2}4.176×10−24.176\times 10^{-2} ( −- 21.0%)4.919×10−24.919\times 10^{-2}3.838×10−2\mathbf{3.838\times 10^{-2}} ( −- 22.0%)
MC-2D7.78×10−27.78\times 10^{-2}5.08×10−25.08\times 10^{-2} ( −- 34.7%)6.95×10−26.95\times 10^{-2}4.73×10−2\mathbf{4.73\times 10^{-2}} ( −- 32.0%)
MC-3D5.07×10−25.07\times 10^{-2}4.18×10−24.18\times 10^{-2} ( −- 17.6%)3.86×10−23.86\times 10^{-2}3.33×10−2\mathbf{3.33\times 10^{-2}} ( −- 13.9%)
Table 3: Relative L2L^{2} rollout error with and without symmetry sampling (SS). Percentages indicate the relative change from the corresponding model without SS.
DatasetSingleSingle+CCSplitSplit+CC
Spectral-2D1.626×10−3\mathbf{1.626\times 10^{-3}}1.801×10−31.801\times 10^{-3} ( ++ 10.8%)2.630×10−32.630\times 10^{-3}2.232×10−32.232\times 10^{-3} ( −- 15.1%)
Basilisk-2D5.286×10−25.286\times 10^{-2}5.446×10−25.446\times 10^{-2} ( ++ 3.0%)4.919×10−2\mathbf{4.919\times 10^{-2}}5.065×10−25.065\times 10^{-2} ( ++ 3.0%)
MC-2D7.78×10−27.78\times 10^{-2}6.24×10−26.24\times 10^{-2} ( −- 19.9%)6.95×10−26.95\times 10^{-2}5.81×10−2\mathbf{5.81\times 10^{-2}} ( −- 16.5%)
MC-3D5.07×10−25.07\times 10^{-2}4.00×10−24.00\times 10^{-2} ( −- 21.0%)3.86×10−23.86\times 10^{-2}3.80×10−2\mathbf{3.80\times 10^{-2}} ( −- 1.7%)
Table 4: Relative L2L^{2} rollout error with and without corrective cycling (CC). Percentages indicate the relative change from the corresponding baseline without CC.
Figure 7: Combined rollout improvements. Relative L2L^{2} rollout error divided by the single-operator baseline error on each benchmark. Starting from Split, successive bars add the conditional-linear architecture (CL), symmetry sampling (SS), and corrective cycling (CC). Single 3×3\times uses three times as many FNO layers as Single. The dashed line marks the baseline ratio of 11 .
Figure 8: Runtime–accuracy tradeoff. Per-step runtime on an RTX 4090 versus relative L2L^{2} error, with logarithmic axes. Filled circles show variance-corrected neural-operator rollout errors. Open circles estimate the first-step sampling error of one MC realization from independent seeds (Appendix C ). MC path counts vary; other estimator settings follow Table 6 . Lines show Pareto fronts.
Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

 Spectral-2DBasilisk-2DMC-2DMC-3D
Grid64264^{2}64264^{2}64264^{2}64364^{3}
Train/validation/test500/50/50500/50/50500/50/50500/50/502000/200/1002000/200/100300/30/30300/30/30
Training trajectory length1001001001001001005050
Test trajectory length2002002002002002005050
Appendix
Table 5: Grid resolutions, trajectory counts, and trajectory lengths (in time steps).
 MC-2DMC-3D
Path length4444
MC paths (train/test)6.25×1046.25\times 10^{4}105/5×10610^{5}/5\times 10^{6}
Direct volume / pseudo-boundary samples1.25×1051.25\times 10^{5}10510^{5}
Indirect volume samples1010100100
Indirect pseudo-boundary samples10101010
Winding samples1100
Appendix
Table 6: MC estimator settings. Separate training and test values are shown where they differ.
 Standard FNOLinear state branchConditioning branch
Hidden width644848
Fourier modes242420
NonlinearityGELUNoneGELU
Channel MLPYes, expansion 0.5NoYes, expansion 0.5
StabilizerNoneNonetanh⁡\tanh
Appendix
Table 7: Architecture settings for the standard FNO and the two conditional-linear branches in the MC experiments. Fourier mode counts are per spatial dimension.
DatasetStandard FNOSingle 3×3\timesCL single/advectionCL diffusion/projection
Spectral-2D5.1635.163 M15.45415.454 M5.0485.048 M5.1455.145 M
Basilisk-2D5.1635.163 M15.45415.454 M5.0485.048 M5.0485.048 M
MC-2D5.1635.163 M15.45415.454 M5.0485.048 M5.0485.048 M
MC-3D122.735122.735 M368.169368.169 M109.705109.705 M109.705109.705 M
Appendix
Table 8: Trainable parameters per learned operator. Counts include both state and conditioning branches for conditional-linear models. Single 3×3\times is the 12-layer standard FNO from Table 1 .
DatasetMetricSingleSingle 3×3\timesSplit
Spectral-2DOne-step rel. L2L^{2}5.206×10−55.206\times 10^{-5}4.944×10−5\mathbf{4.944\times 10^{-5}}7.723×10−57.723\times 10^{-5}
 Rollout MSE1.005×10−7\mathbf{1.005\times 10^{-7}}1.015×10−71.015\times 10^{-7}2.631×10−72.631\times 10^{-7}
 Rollout abs div5.029×10−35.029\times 10^{-3}3.663×10−3\mathbf{3.663\times 10^{-3}}5.157×10−35.157\times 10^{-3}
Basilisk-2DOne-step rel. L2L^{2}1.533×10−31.533\times 10^{-3}1.499×10−3\mathbf{1.499\times 10^{-3}}2.138×10−32.138\times 10^{-3}
 Rollout MSE3.099×10−53.099\times 10^{-5}3.058×10−53.058\times 10^{-5}2.683×10−5\mathbf{2.683\times 10^{-5}}
 Rollout abs div5.826×10−25.826\times 10^{-2}5.605×10−2\mathbf{5.605\times 10^{-2}}6.602×10−26.602\times 10^{-2}
Appendix
Table 9: One-step relative L2L^{2} error, rollout MSE, and absolute divergence for the models in Table 1 .
DatasetMetricSingleCL SingleSplitCL Split
Spectral-2DOne-step rel. L2L^{2}5.206×10−55.206\times 10^{-5}4.984×10−5\mathbf{4.984\times 10^{-5}}7.723×10−57.723\times 10^{-5}6.561×10−56.561\times 10^{-5}
 Rollout MSE1.005×10−71.005\times 10^{-7}2.220×10−72.220\times 10^{-7}2.631×10−72.631\times 10^{-7}4.045×10−8\mathbf{4.045\times 10^{-8}}
 Rollout abs div5.029×10−35.029\times 10^{-3}5.519×10−35.519\times 10^{-3}5.157×10−35.157\times 10^{-3}4.276×10−3\mathbf{4.276\times 10^{-3}}
Basilisk-2DOne-step rel. L2L^{2}1.533×10−31.533\times 10^{-3}1.151×10−3\mathbf{1.151\times 10^{-3}}2.138×10−32.138\times 10^{-3}2.305×10−32.305\times 10^{-3}
 Rollout MSE3.099×10−53.099\times 10^{-5}2.772×10−52.772\times 10^{-5}2.683×10−52.683\times 10^{-5}4.010×10−6\mathbf{4.010\times 10^{-6}}
 Rollout abs div5.826×10−25.826\times 10^{-2}5.555×10−25.555\times 10^{-2}6.602×10−26.602\times 10^{-2}4.160×10−2\mathbf{4.160\times 10^{-2}}
Appendix
Table 10: One-step relative L2L^{2} error, rollout MSE, and absolute divergence for the models in Table 2 .
DatasetMetricSingleSingle+SSSplitSplit+SS
Spectral-2DRollout MSE1.005×10−71.005\times 10^{-7}2.699×10−8\mathbf{2.699\times 10^{-8}}2.631×10−72.631\times 10^{-7}6.456×10−86.456\times 10^{-8}
 Rollout abs div5.029×10−35.029\times 10^{-3}3.954×10−3\mathbf{3.954\times 10^{-3}}5.157×10−35.157\times 10^{-3}3.998×10−33.998\times 10^{-3}
Basilisk-2DRollout MSE3.099×10−53.099\times 10^{-5}1.934×10−51.934\times 10^{-5}2.683×10−52.683\times 10^{-5}1.634×10−5\mathbf{1.634\times 10^{-5}}
 Rollout abs div5.826×10−25.826\times 10^{-2}3.145×10−2\mathbf{3.145\times 10^{-2}}6.602×10−26.602\times 10^{-2}3.952×10−23.952\times 10^{-2}
MC-2DRollout MSE1.773×10−31.773\times 10^{-3}7.566×10−47.566\times 10^{-4}1.415×10−31.415\times 10^{-3}6.547×10−4\mathbf{6.547\times 10^{-4}}
 Rollout abs div1.146×10−11.146\times 10^{-1}1.082×10−1\mathbf{1.082\times 10^{-1}}1.153×10−11.153\times 10^{-1}1.119×10−11.119\times 10^{-1}
Appendix
Table 11: Rollout MSE and absolute divergence with and without symmetry sampling, corresponding to Table 3 .
DatasetMetricSingleSingle+CCSplitSplit+CC
Spectral-2DRollout MSE1.005×10−7\mathbf{1.005\times 10^{-7}}1.233×10−71.233\times 10^{-7}2.631×10−72.631\times 10^{-7}1.894×10−71.894\times 10^{-7}
 Rollout abs div5.029×10−35.029\times 10^{-3}5.286×10−35.286\times 10^{-3}5.157×10−35.157\times 10^{-3}4.886×10−3\mathbf{4.886\times 10^{-3}}
Basilisk-2DRollout MSE3.099×10−53.099\times 10^{-5}3.289×10−53.289\times 10^{-5}2.683×10−5\mathbf{2.683\times 10^{-5}}2.846×10−52.846\times 10^{-5}
 Rollout abs div5.826×10−25.826\times 10^{-2}5.335×10−2\mathbf{5.335\times 10^{-2}}6.602×10−26.602\times 10^{-2}6.417×10−26.417\times 10^{-2}
MC-2DRollout MSE1.773×10−31.773\times 10^{-3}1.139×10−31.139\times 10^{-3}1.415×10−31.415\times 10^{-3}9.874×10−4\mathbf{9.874\times 10^{-4}}
 Rollout abs div1.146×10−11.146\times 10^{-1}1.105×10−1\mathbf{1.105\times 10^{-1}}1.153×10−11.153\times 10^{-1}1.138×10−11.138\times 10^{-1}
Appendix
Table 12: Rollout MSE and absolute divergence with and without corrective cycling, corresponding to Table 4 .
DatasetMetricSingleSplitCL SplitSplit+CL +SSSplit+CL +SS+CC
Spectral-2DRollout rel. L2L^{2}1.626×10−31.626\times 10^{-3}2.630×10−32.630\times 10^{-3}1.031×10−31.031\times 10^{-3}5.908×10−45.908\times 10^{-4}5.511×10−4\mathbf{5.511\times 10^{-4}}
 Rollout MSE1.005×10−71.005\times 10^{-7}2.631×10−72.631\times 10^{-7}4.045×10−84.045\times 10^{-8}1.327×10−81.327\times 10^{-8}1.155×10−8\mathbf{1.155\times 10^{-8}}
 Rollout abs div5.029×10−35.029\times 10^{-3}5.157×10−35.157\times 10^{-3}4.276×10−34.276\times 10^{-3}4.108×10−34.108\times 10^{-3}3.629×10−3\mathbf{3.629\times 10^{-3}}
Basilisk-2DRollout rel. L2L^{2}5.286×10−25.286\times 10^{-2}4.919×10−24.919\times 10^{-2}1.901×10−21.901\times 10^{-2}1.660×10−21.660\times 10^{-2}1.616×10−2\mathbf{1.616\times 10^{-2}}
 Rollout MSE3.099×10−53.099\times 10^{-5}2.683×10−52.683\times 10^{-5}4.010×10−64.010\times 10^{-6}3.057×10−63.057\times 10^{-6}2.897×10−6\mathbf{2.897\times 10^{-6}}
 Rollout abs div5.826×10−25.826\times 10^{-2}6.602×10−26.602\times 10^{-2}4.160×10−24.160\times 10^{-2}3.955×10−23.955\times 10^{-2}3.719×10−2\mathbf{3.719\times 10^{-2}}
Appendix
Table 13: Relative L2L^{2} rollout error, MSE, and absolute divergence for the model configurations in Fig. 7 .
ModelMean local operator errorMean cosineRollout error
Single5.03×10−35.03\times 10^{-3}0.9971.17×10−31.17\times 10^{-3}
Single+SS2.92×10−32.92\times 10^{-3}0.9476.86×10−46.86\times 10^{-4}
Single+CC5.63×10−35.63\times 10^{-3}0.9901.27×10−31.27\times 10^{-3}
Appendix
Table 14: Local operator errors and rollout error on 50 Spectral-2D test trajectories over 100 steps. Mean local error averages 100 operator calls per trajectory; mean cosine similarity averages 99 consecutive error pairs. Rollout error is the pooled relative L2L^{2} error over these trajectories and steps.
Figure 9: The xx -component of two successive local operator errors, with a shared color scale. All variants share the first call, which uses no symmetry transform. The second call uses the baseline, SS, or CC. For this illustration, SS uses the 180∘180^{\circ} rotation that minimizes the absolute cosine similarity between consecutive error fields among non-identity D4D_{4} transforms; Table 14 uses independent random symmetry samples.
Figure 10: Relative L2L^{2} rollout error after training on 500, 50, or 10 trajectories. MC test errors use the paired-reference pooled variance correction of Appendix C ; BIE test errors use standard relative L2L^{2} error.
ComparisonMethodTimeRollout errorPeak GPU (GB)
ReferenceBase4:14:160.0780.0786.7276.727
Full trainingUnrolled9:13:050.055\mathbf{0.055}11.15711.157
 CC4:45:390.0620.0626.727\mathbf{6.727}
Fine-tuningUnrolled2:51:350.073\mathbf{0.073}11.07311.073
 CC1:38:190.0740.0746.727\mathbf{6.727}
 Pushforward2:18:180.0760.07612.80112.801
Appendix
Table 15: Training cost on one NVIDIA B200 and 200-step relative L2L^{2} rollout error on MC-2D NACA. Times are in hours:minutes:seconds. Fine-tuning and CC times exclude base training; CC times include data export. Full CC takes 8:59:55 including base training, still less than unrolled training. GPU memory is the peak training allocation. Bold marks the lowest values within each comparison.
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