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Mesh-Free Convolution: Learned Spectral Attenuation and Transport

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural operators for general geometries

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Proposes Mesh-Free Convolutions (MFCs), continuous-space filters defined via a parabolic PDE solution operator with directional readout and learned shift, discretized via FFT at evaluation time, improving autoregressive PDE rollouts and zero-shot cross-resolution transfer.

Problem

Autoregressive learned PDE solution operators accumulate spatial errors, suffer instability and drift, and hard Fourier-mode truncation in FNO limits preservation of fine-scale and transport-driven structure and cross-resolution transfer.

Approach

Parameterize convolution kernels as the time-tau solution operator of an anisotropic parabolic PDE (learned diffusion tensor, evolution time), followed by a zeroth/first-order directional readout and learned continuous shift; the Fourier symbol is a continuous function evaluated on whatever FFT grid is used, replacing hard mode truncation.

Key results

• On WeatherBench2 with 35 initial conditions, FNO diverges on 11 of 35 rollouts while MFC-resnet and AFNO complete all 35; MFC-resnet has the lowest climatology NRMSE across variables and horizons (e.g., U10 at T=500 NRMSE 29.2% vs AFNO 40.3%).
• Zero-shot transfer from 64x32 to 240x121 gives MFC-resnet U10 RMSE 4.80 vs AFNO 9.40 (degradation ratio 1.07 vs 2.09).
• On 2D Burgers at 100-step GMean, MFC has lowest error on sets B, C, and D with values like 2.34 on C versus 44.7 for F-FNO, while F-FNO is best at short horizons on sets A and B.

Surprising findings

• FNO retrained with 32 modes (effectively untruncated in the resolved band) still diverges on all trajectories, suggesting unconstrained spectral weights rather than truncation drive instability.
• The paper's own stability theory does not apply to released checkpoints: certified Lipschitz bound is about 10^59, and observed stability is empirical and on-distribution.

Limitations

• Implementation targets periodic rectangular grids with FFT; irregular domains and 3D are future work.
• Stability theory is layer-level; the conditional rollout bound's hypotheses (L<=1) are not satisfied by released checkpoints.
• F-FNO is better at short horizons in-distribution on Burgers; MFC advantage is mainly long-horizon and OOD.

Experimental setup

Datasets: Navier-Stokes Re=5000 Kolmogorov flow WeatherBench2 ERA5 2D Burgers (synthetic) PDEBench 1D advection

Benchmarks: Navier-Stokes Re=5000 WeatherBench2 2D Burgers

Metrics: relative l2 error spatial correlation energy spectrum latitude-weighted NRMSE geometric mean normalized RMSE median RMSE runtime peak memory

Baselines: FNO F-FNO UNet AFNO

Builds on: Fourier Neural Operator UNet AFNO F-FNO

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