We propose a conservative local neural operator (CLNO) for accurate and generalizable autoregressive prediction of compressible flows. Building upon the locality and translation invariance of the local neural operator (LNO), CLNO introduces a conservative evolution framework based on local subdomains of fixed physical size, separating local mean-state evolution from fine-scale reconstruction. The flow is encoded into subdomain-integrated conservative states and fine-scale features, while a shared local network predicts interface fluxes for finite-volume updates that enforce discrete conservation. A convolutional reconstruction network recovers fine-scale structures using zero-mean basis functions, preserving the updated local means and providing information for subsequent evolution. To accommodate different physical parameters, Reynolds--Mach number conditioning modulates the encoding, flux-prediction, and reconstruction networks, enabling a single model to learn parameter-dependent dynamics and generalize to unseen parameter combinations without retraining. CLNO supports prediction across spatial resolutions, computational domain sizes, and boundary conditions. Numerical experiments demonstrate improved generalization and competitive or higher accuracy compared with LNO, including order-of-magnitude error reductions in representative cases, at approximately three to five times the inference cost on periodic benchmarks. These results demonstrate accurate, generalizable, and physically consistent flow prediction through conservative local evolution and fine-scale reconstruction.
Figures & tables
Figure 1: Overall framework of CLNO. The current flow field is encoded on local subdomains of fixed physical size and advanced through shared interface fluxes. Reconstruction adds zero-mean fine-scale details to the updated local means, and the predicted field is re-encoded for the next time step. The Reynolds and Mach numbers condition all three modules. Panels (a)–(c) detail local encoding, conservative interface-flux updates, and mean-preserving Legendre reconstruction, respectively.
Flow configuration
(N,L)
Relative L2 error (%)
Inference time (s)
Conservation error (%)
FNO
DeepONet
LNO
CLNO
FNO
DeepONet
LNO
CLNO
FNO
DeepONet
LNO
CLNO
Periodic multiscale wave
(1024,1)
6.392
4.594
7.563
1.840
0.1893
0.1190
0.1499
0.6432
1.308
0.054
4.690
1.59e-6
(1536,1)
6.663
4.839
27.989
1.346
0.2101
0.1114
0.1302
0.6436
1.310
0.053
4.974
2.33e-6
(1536,1.5)
29.267
28.885
6.939
1.220
0.2047
0.1320
0.1344
0.6342
1.310
0.053
4.974
4.55e-6
Transient reflecting obstacle
(1024,1)
NaN
16.960
11.050
3.045
—
0.3750
0.1759
2.2425
NaN
4.637
5.269
0.365
(1536,1)
NaN
17.154
26.103
2.993
—
0.2186
0.1548
2.2206
NaN
4.606
6.154
0.353
Table 1: One-dimensional Navier–Stokes equation forecasts over 100 frames. (N,L) denotes grid cells and domain length. Relative L2 errors are percentages and inference times are seconds. NaN indicates that a state became nonfinite before the final frame; partial runtimes are omitted. Finite forecasts are scored regardless of state positivity. Boldface marks the lowest finite value in each row and metric.
Figure 2: Space–time density, velocity, and pressure for nonperiodic contact transport. Columns show the reference, LNO, CLNO, and their absolute errors. The velocity and pressure field colorbars use a centered inverse-hyperbolic-sine scale to resolve variations near their nearly constant reference values; tick labels give the original physical values.
Figure 3: Contact-discontinuity profiles at t=0 , 0.25 , 0.5 , 0.75 , and 1 .
Figure 4: Space–time prediction with a temporary reflecting obstacle on the refined 1536-cell grid. The solid interval is shaded gray during its active period.
Figure 5: Density, velocity, and pressure profiles of the refined-grid obstacle test at t=0 , 0.25 , 0.5 , 0.75 , and 1 .
Flow configuration
Relative L2 error (%)
Inference time (s)
Conservation error (%)
FNO
DeepONet
LNO
CLNO
FNO
DeepONet
LNO
CLNO
FNO
DeepONet
LNO
CLNO
Held-out vortical flows (64,1) , T=8
4.946
14.124
15.901
3.306
0.3066
0.1747
0.2858
0.9649
0.476
0.608
4.172
2.39e-6
High-wavenumber perturbation (64,1) , T=1
8.529
18.855
12.005
7.101
0.0363
0.0244
0.0352
0.1211
0.754
0.399
1.773
1.20e-6
Vortical flow, standard grid (64,1) , T=1
1.428
10.354
5.789
1.193
0.0419
0.0154
0.0462
0.1580
0.293
0.207
1.704
9.29e-7
Vortical flow, refined grid (96,1) , T=1
1.551
10.660
13.895
1.560
0.0420
0.0153
0.0452
0.1362
0.287
0.214
2.717
5.17e-7
Vortical flow, expanded domain (96,1.5) , T=1
14.842
14.439
9.035
3.328
0.0365
0.0135
0.0393
0.1364
0.287
0.214
2.718
2.59e-7
Table 2: Representative two-dimensional Navier–Stokes equation forecasts. Relative L2 errors are computed over each complete space–time trajectory. The held-out row reports the mean over ten initial fields; the final three rows share one initial field. (N,L) gives the grid size per direction and domain side length; T is the final physical time. Boldface marks the lowest finite value in each row and metric.
Figure 6: Held-out two-dimensional vortical flow at the final forecast time t=8 on the 64×64 grid. Columns show the reference, LNO, CLNO, and their absolute errors; rows show density, both velocity components, and pressure.
Figure 7: The matched vortical realization on a refined 96×96 grid over [0,1]2 at t=0.75 .
Figure 8: The matched vortical realization over the expanded domain [0,1.5]2 on a 96×96 grid at t=0.75 .
Re
Ma
Relative L2 error (%)
Inference time (s)
Conservation error (%)
FNO
DeepONet
LNO
CLNO
FNO
DeepONet
LNO
CLNO
FNO
DeepONet
LNO
CLNO
5500
0.35
0.309
1.089
94.809
0.066
0.1219
0.0556
0.0882
0.7106
0.169
0.055
224.594
8.24e-7
5500
0.45
0.237
0.909
12.994
0.077
0.1222
0.0559
0.0882
0.7108
0.115
0.033
30.610
8.24e-7
6500
0.35
0.320
1.097
159.986
0.064
0.1220
0.0558
0.0882
0.7104
0.134
0.065
476.535
6.66e-7
6500
0.45
0.234
0.912
41.951
0.078
0.1225
0.0556
0.0879
0.7096
0.061
0.032
97.441
4.24e-6
7500
0.35
0.330
1.100
129.474
0.063
0.1226
0.0556
0.0883
0.7106
0.116
0.067
335.866
1.80e-6
Table 3: One-dimensional Reynolds–Mach parameter sweep on a periodic 1024-cell grid. The first eleven rows use the same paired initial field: a validation initial field at seen parameter pairs, a test initial field at the held-out pair (7500,0.45) , and the corresponding independent initial field at the outside-range pair (5000,0.50) . The final two rows report means over ten test initial fields at the two unseen pairs. All models predict 64 future frames. Errors are in percent; time is per complete trajectory in seconds. Boldface marks the lowest finite value in each row and metric.
Model
Relative L2 error (%)
Inference time (s)
Conservation error (%)
CLNO (w/ both components)
1.384
1.050
2.13e-6
CLNO (w/o conservative flux)
1.953
0.962
0.337
CLNO (w/o fine reconstruction)
3.351
0.657
3.14e-6
Table 4: Two-dimensional 64×64 Navier–Stokes equation ablation on two representative 160-frame held-out trajectories. The trajectories maximize the smaller of the two ablation-to-full error ratios among ten complete forecasts. Values are means over these two trajectories. Boldface marks the lowest value in each metric.
Appendix figures & tables10 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 9: Space–time evolution of density, velocity, and pressure for the held-out periodic multiscale wave. Reference, LNO, CLNO, and absolute-error fields are compared.
Figure 10: Density, velocity, and pressure profiles of the periodic multiscale wave at t=0 , 0.25 , 0.5 , 0.75 , and 1 .
Figure 11: Space–time density, velocity, and pressure for the standard-grid temporary-obstacle test. The solid interval is shaded gray during its active period.
Figure 12: Flow profiles with obstacle appearance and removal at t=0 , 0.25 , 0.5 , 0.75 , and 1 .
Figure 13: Pressure-discontinuity profiles at t=0 , 0.25 , 0.5 , 0.75 , and 1 .
Figure 14: Space–time density, velocity, and pressure for the pressure-discontinuity Riemann problem. Columns show the reference, LNO, CLNO, and their absolute errors.
Figure 15: Matched vortical flow on the standard 64×64 grid over [0,1]2 at t=0.75 . Columns show the reference, LNO, CLNO, and their absolute errors, respectively.
Table 5: Architecture settings of the reported fixed-parameter models. n,m,k are the local-resolution parameters of the LNO implementation. FNO modes count retained Fourier modes per spatial direction; DeepONet rank is the branch–trunk contraction dimension.
Benchmark
Model
Min. epochs
Batch
Initial LR
Final horizon
1D NS
CLNO
50
16
3×10−4
10
1D NS
FNO / DeepONet
50
16
3×10−4
10
1D NS
LNO
200
4
10−3
10
1D Re–Ma
CLNO (w/ FiLM)
50
16
3×10−4
10
1D Re–Ma
FNO / DeepONet
50
16
3×10−4
10
1D Re–Ma
LNO
200
4
10−3
10
Appendix
Table 6: Training settings for the reported checkpoints. The 2D NS rows report a common minimum of 100 epochs; the corresponding runs continued to 101 or 120 epochs. Final horizon is the last curriculum horizon used for checkpoint selection, rather than the number of frames in every evaluation.
Benchmark
Reference interval
Forecast frames
CLNO substeps
Boundary
1D fixed NS
0.01
100
2
periodic / test-specific
2D fixed NS
0.05
20 or 160
2
periodic
1D Re–Ma
0.01
64
2
periodic
Appendix
Table 7: Inference settings. The frame count is the number of autoregressively predicted frames; all trajectories start at the reference initial frame.
We present CoFINN (Conservation Flux Informed Neural Networks), a physics-informed deep learning framework for predicting compressible flow fields governed by conservation laws. Unlike conventional data-driven convolutional neural networks (CNNs), which optimize only pixel-wise similarity metrics, CoFINN embeds finite-volume conservation physics directly into the training process. Unlike classical physics-informed methods which enforce differential-equation residuals at collocation points through automatic differentiation, CoFINN adopts a finite-volume perspective consistent with modern CFD methodology. CoFINN interprets CNN output fields as structured computational grids, where each pixel represents a finite-volume cell, and enforces conservation consistency through sophisticated numerical flux calculations. The framework is evaluated on transonic flow prediction around airfoils at (M=0.7, Re=6 * 10^6), including challenging conditions involving shock waves and high angles of attack. Results show that CoFINN improves aerodynamic force prediction accuracy, reducing drag prediction error by up to 34% at extreme angles of attack and by approximately 15% on average across the test set. Improvements are particularly significant in limited-data regimes, demonstrating that the conservation-based loss acts as an effective physical regularizer. The proposed approach maintains the computational efficiency advantages of CNN surrogates while significantly improving physical consistency and conservation behavior. The framework is architecture-agnostic and extensible to broader classes of conservation-law-governed physical systems.
Adnan Harun Doğan, Mert Deniz, Hande Alemdar +1
Dept. of Computer Engineering Middle East Technical University Ankara, Turkey · Dept. of Mechanical Engineering Middle East Technical University Ankara, Turkey
Deep learning surrogates for CFD flow-field prediction often rely on large, complex models, which can be slow and fragile when data are noisy or incomplete. We introduce FlowForge, a staged local rollout engine that predicts future flow fields by compiling a locality-preserving update schedule and executing it with a shared lightweight local predictor. Rather than producing the next frame in a single global pass, FlowForge rewrites spatial sites stage by stage so that each update conditions only on bounded local context exposed by earlier stages. This compile-execute design aligns inference with short-range physical dependence, keeps latency predictable, and limits error amplification from global mixing. Across PDEBench, CFDBench, and BubbleML, FlowForge matches or improves upon strong baselines in pointwise accuracy, delivers consistently better robustness to noise and missing observations, and maintains stable multi-step rollout behavior while reducing per-step latency.
Xiaowen Zhang, Ziming Zhou, Fengnian Zhao +1
Shanghai Jiao Tong University · University of Michigan
Neural operators enable fast PDE forecasting, but repeated predictions accumulate errors and fine-scale structures remain difficult to resolve. We introduce a frequency-decomposed finite-time flow-map neural operator (F3NO) that leverages updated low-frequency features to guide nonlinear refinement of high-frequency information. Within each layer, this cross-scale conditioning connects global spectral processing with local detail refinement. The model directly predicts states at specified future times and adjusts the contributions of the two branches according to the prediction interval. For longer trajectories, it combines parallel predictions within short temporal segments with recursive propagation between segments. Experiments on five PDE benchmarks demonstrate improved forecasting accuracy over autoregressive and direct-prediction baselines. Ablations show that frequency-decomposed refinement can improve accuracy with fewer parameters, while the benefits of segmentation depend on spatial resolution and dynamical regime.