CLNO: Conservative Local Neural Operator for Compressible Flow Prediction

Buxuan Wang; Rongzu Li; Qingsong Zou

math.NAOct 12, 2026

CLNO: Conservative Local Neural Operator for Compressible Flow Prediction

Authors: Buxuan Wang, Rongzu Li, Qingsong Zou

Organizations: School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou, China

Abstract

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 $L^{2}$ 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

L^{2}

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 $L^{2}$ 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

L^{2}

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 $64\times 64$ 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 $96\times 96$ 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 $96\times 96$ grid at $t=0.75$ .

$\mathrm{Re}$	$\mathrm{Ma}$	Relative $L^{2}$ 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 $L^{2}$ 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\times 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 $64\times 64$ grid over $[0,1]^{2}$ at $t=0.75$ . Columns show the reference, LNO, CLNO, and their absolute errors, respectively.

Benchmark	Model	Feature width	Spatial representation / depth
1D NS	CLNO	256	128 latent cells; 3 message blocks; 4 Legendre modes
1D NS	LNO	20	4 blocks; $(n,m,k)=(12,6,2)$
1D NS	FNO	40	4 spectral blocks; 16 Fourier modes
1D NS	DeepONet	64	branch pool 32; trunk rank 64
2D NS	CLNO	256	$16\times 16$ latent cells; 3 blocks; 24 modes
2D NS	LNO	40	4 blocks; $(n,m,k)=(16,8,2)$

Appendix

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	$3\times 10^{-4}$	10
1D NS	FNO / DeepONet	50	16	$3\times 10^{-4}$	10
1D NS	LNO	200	4	$10^{-3}$	10
1D Re–Ma	CLNO (w/ FiLM)	50	16	$3\times 10^{-4}$	10
1D Re–Ma	FNO / DeepONet	50	16	$3\times 10^{-4}$	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.

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