Mesh-Free Convolution: Learned Spectral Attenuation and Transport

Niloufar Zakariaei; Manav Manav; Eldad Haber

cs.LGOct 11, 2026

Mesh-Free Convolution: Learned Spectral Attenuation and Transport

Authors: Niloufar Zakariaei, Manav Manav, Eldad Haber

Organizations: University of British Columbia · Computational Geoscience Inc.

Abstract

Learning solution operators for partial differential equations (PDEs) requires models that can represent spatial structure across grids and scales. This becomes especially important in time-dependent problems, where the learned operator is often used autoregressively: each predicted state is fed back as input to predict future states. In this setting, small spatial errors can be repeatedly propagated through the learned dynamics, leading to instability and drift, vitiating long-term solution fidelity. We propose Mesh-Free Convolutions (MFCs), filters defined in continuous space and discretized only at evaluation time. This allows the same learned filters to be applied across resolutions. Rather than imposing a grid-dependent stencil or a fixed spectral cutoff, MFCs learn smooth scale-dependent filtering and include directional terms for transport-like behavior. Across fluid-dynamics and weather-forecasting benchmarks, MFC-based models produce stable long-horizon rollouts, transfer reliably across resolutions, and better preserve long-time statistics. These results suggest that continuous, mesh-free filtering is a useful inductive bias for PDE operator learning, especially in autoregressive prediction.

Figures & tables

Figure 1: Autoregressive rollout on the PDEBench 1D advection benchmark [ 32 ] . Both FNOs are trained as one-step predictors and differ only in the number of retained Fourier modes. A small mode budget leads to phase drift, amplitude distortion, and oscillatory artifacts, while increasing the number of modes improves the final-state prediction.

Figure 2: Kernel family produced by the MFC parameterization. Varying $\tau$ controls scale (localized to diffuse); the diffusion-tensor entries set anisotropy and orientation; $\gamma_{x},\gamma_{y}$ introduce directional, derivative-like lobes; and the shift $\delta$ translates the kernel center. A compact parameter vector ${\bm{\xi}}$ thus spans smooth, anisotropic, and transport-like filters without hand-crafted stencils.

Dataset	PDE / Data	Resolution	Baselines	Evaluation focus
Navier–Stokes [ 12 ]	2D incomp	$128^{2}$	FNO , F-FNO , UNet	Rollout stability
WeatherBench2 [ 29 ]	ERA5	$64{\times}32\!\to\!240{\times}121$	AFNO, FNO	Long-time statistics
Burgers 2D [ 36 ]	2D Burgers	$64^{2}\!\to\!128^{2}$	FNO, F-FNO	Localized structures

Table 1: Summary of experimental benchmarks.

Panel of Figure 3 — Figure 3: Autoregressive rollout on the 2D Navier–Stokes benchmark. MFC-UNet shows the slowest error growth and highest spatial correlation, while FNO becomes unstable and F-FNO loses correlation faster. At $\Delta t=40$ , the energy spectrum shows that MFC-UNet better preserves the ground-truth scale structure, whereas FNO develops high-wavenumber artifacts.

Variable	Horizon $T$	$\sigma_{\mathrm{GT}}$	RMSE	NRMSE (%)	RMSE	NRMSE (%)	RMSE	NRMSE (%)
			MFC-resnet		AFNO		FNO
$U_{10}$ ( $\mathrm{m}\mathrm{/}\mathrm{s}$ )	$100$	$3.509$	0.569	$16.2$	$0.572$	$16.3$	$8.227$	$234.5$
	$300$	$3.514$	0.836	$23.8$	$1.134$	$32.3$	$10.159$	$289.1$
	$500$	$3.525$	1.029	$29.2$	$1.422$	$40.3$	$10.075$	$285.8$
$V_{10}$ ( $\mathrm{m}\mathrm{/}\mathrm{s}$ )	$100$	$1.703$	0.388	$22.8$	$0.412$	$24.2$	$3.679$	$216.0$
	$300$	$1.708$	0.512	$30.0$	$0.856$	$50.1$	$5.292$	$309.9$

Table 2: Latitude-weighted climatology error on WeatherBench2. NRMSE is RMSE normalized by the ground-truth spatial standard deviation,

\sigma_{\mathrm{GT}}

, and reported as a percentage.

	Ground truth	MFC-resnet	AFNO	MFC $-$ GT	AFNO $-$ GT
$U10$

$V10$

$T850$

Figure 4: WeatherBench2 500-step time-mean fields. Rows show

U10

V10

, and

T850

. The first three columns show the ground truth, MFC-resnet, and AFNO time means with a shared color scale. The last two columns show the corresponding climatological biases,

\text{MFC}-\text{GT}

and

\text{AFNO}-\text{GT}

, with a separate shared bias scale.

	MFC-resnet			AFNO			FNO
Variable	$64{\times}32$	$240{\times}121$	Ratio	$64{\times}32$	$240{\times}121$	Ratio	$64{\times}32$	$240{\times}121$	Ratio
$U_{10}$ (m/s)	4.47	4.80	1.07	4.50	9.40	2.09	5.54	6.06	1.09
$V_{10}$ (m/s)	4.66	4.93	1.06	4.71	6.06	1.29	5.53	5.60	1.01
$T_{850}$ (K)	4.25	4.33	1.02	4.43	13.14	2.97	7.01	10.53	1.50

Table 3: Cross-resolution zero-shot transfer at step 50, reported as median RMSE over initial conditions. Models are trained at

64{\times}32

and evaluated at both native and

240{\times}121

resolution. The degradation ratio

\mathrm{RMSE}_{240{\times}121}/\mathrm{RMSE}_{64{\times}32}

measures sensitivity to resolution change, while the absolute

240{\times}121

RMSE is the main high-resolution accuracy metric.

Test	Res.	FNO	FFNO	MFC	FNO	FFNO	MFC	FNO	FFNO	MFC
		GMean@20			GMean@50			GMean@100
A	$64^{2}$	0.4153	0.0135	0.0841	1381.5232	0.1969	0.6660	$5.75{\times}10^{9}$	8.7762	1.7009
B	$64^{2}$	0.5353	0.0694	0.2183	1414.6782	0.3758	0.8718	$4.97{\times}10^{9}$	9.7672	1.5673
C	$128^{2}$	0.7712	0.2592	0.2681	2384.3485	1.2758	1.1350	$9.20{\times}10^{9}$	44.6557	2.3423
D	$128^{2}$	0.9035	0.3454	0.4090	2261.6899	1.2914	1.2314	$7.15{\times}10^{9}$	39.7726	2.1275

Table 4: Aggregated rollout error on 2D Burgers across the four test sets. We report the geometric mean of normalized RMSE at 20, 50, and 100 autoregressive rollout steps [ 9 ] ; lower is better.

			GMean@20
Variant	Modification	Val. rel. $\ell_{2}$	A	B	C	D
MFC-full (released)	—	0.0113	0.084	0.218	0.268	0.409
MFC-full (retrain)	—	0.0131	0.128	0.279	0.332	0.486
$-$ shift	$\delta\equiv 0$	0.0144	0.214	0.304	0.359	0.461
$-$ first-order	$\gamma_{x}=\gamma_{y}\equiv 0$	0.0127	0.070	0.173	1.012	0.998
$-$ anisotropy	$\alpha_{xy}\equiv 0$	0.0166	0.115	0.253	1.017	0.970

Table 5: Component ablations on 2D Burgers. Validation relative

\ell_{2}

is the 5-step unrolled average used for checkpoint selection. GMean@50/@100 and checkpoint provenance: Appendix D .

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

	GMean@20				GMean@50				GMean@100
Variant	A	B	C	D	A	B	C	D	A	B	C	D
MFC-full (released)	0.084	0.218	0.268	0.409	0.666	0.872	1.135	1.231	1.701	1.567	2.342	2.128
MFC-full (retrain)	0.128	0.279	0.332	0.486	0.621	0.806	1.133	1.195	0.972	0.998	1.359	1.287
$-$ shift	0.214	0.304	0.359	0.461	0.834	0.852	1.075	1.086	1.512	1.266	1.833	1.623
$-$ first-order	0.070	0.173	1.012	0.998	0.459	0.601	1.149	1.099	1.139	1.074	1.311	1.200
$-$ anisotropy	0.115	0.253	1.017	0.970	0.764	0.934	1.618	1.433	1.933	1.705	2.228	1.847

Appendix

Table 6: Component ablations on 2D Burgers at all horizons (test sets A/B/C/D).

Figure 5: Learned MFC parameters: Burgers (left) and Navier–Stokes (right).

Figure 6: Spectral amplification on Burgers test A. Vertical line: FNO mode cutoff

k=18

; shaded: unresolved

k>28

	ms / iteration					Peak memory
Model	$64^{2}$	$128^{2}$	$256^{2}$	$64{\times}32$	$240{\times}121$	$64^{2}$	$128^{2}$	$256^{2}$
FNO	0.9	2.3	18.8	0.8	4.8	0.12	0.44	1.12
F-FNO	2.8	19.5	83.0	1.5	38.2	1.23	4.81	2.67
MFC-UNet	4.5	12.9	75.9	4.2	29.7	0.42	1.14	4.51
AFNO	28.7	133.3	547.9	10.7	207.8	0.53	2.09	8.29

Appendix

Table 7: Runtime (ms/iter) and peak memory (GiB).

Figure 7: Per-latitude unweighted RMSE at step 50. Top: zero-shot

240{\times}121

(poles exist). Bottom: native

64{\times}32

. Shaded:

|{\rm lat}|\geq 60^{\circ}

Panel of Figure 8 — Figure 8: Qualitative autoregressive rollout comparison on the 2D Navier–Stokes benchmark. Columns show prediction horizons $\Delta t\in\{3,6,9,12\}$ . For each model, the upper image in each cell is the predicted vorticity field and the lower image is the corresponding error map. MFC-UNet remains closer to the ground truth, while FNO and UNet accumulate larger errors over longer rollouts.

Panel of Figure 9 — Figure 9: Representative $U10$ rollout at step 50. The initial condition is selected by the median-improvement criterion described above. MFC-parabolic remains closer to the ground truth at both resolutions, while FNO shows stronger distortion after rollout.

Panel of Figure 10 — Figure 10: Representative $V10$ rollout at step 50. The initial condition is selected by the median-improvement criterion described above. MFC-parabolic better preserves the spatial structure across both the training and zero-shot resolutions.

Panel of Figure 11 — Figure 11: Representative $T850$ rollout at step 50. The initial condition is selected by the median-improvement criterion described above. The gap between MFC-parabolic and FNO becomes more visible under zero-shot transfer.

Setting	Value
Resolution	$128\times 128$
Context frames	10
Prediction	Single-step teacher-forced
Optimizer	Adam
Learning rate	$5\times 10^{-4}$
Scheduler	StepLR, step size $50$ , $\gamma=0.5$

Appendix

Table 8: Training configuration for the Navier–Stokes Kolmogorov-flow experiment.

Setting	Value
Training resolution	$64\times 32$
Zero-shot test resolution	$240\times 121$
History frames	4
Prediction	Single-step
Optimizer	Adam
Learning rate	$10^{-3}$

Appendix

Table 9: Training configuration for the WeatherBench2 experiment (ERA5; U10, V10, T850).

Setting	Value
Training resolution	$64\times 64$
Cross-resolution test	$128\times 128$
Context frames	5
Unroll steps (training)	5
Optimizer	Adam
Learning rate	$5\times 10^{-4}$

Appendix

Table 10: Training configuration for the 2D Burgers experiment.

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