Mesh-Free Convolution: Learned Spectral Attenuation and Transport

Key points

• We propose Mesh-Free Convolutions (MFCs), filters defined in continuous space and discretized only at evaluation time.1
• 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.1
• 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.1
• These results suggest that continuous, mesh-free filtering is a useful inductive bias for PDE operator learning, especially in autoregressive prediction.1

Sources (1)

• [1]Mesh-Free Convolution: Learned Spectral Attenuation and Transport
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 11, 09:28 PM

  Learning solution operators for partial differential equations (PDEs) requires models that can represent spatial structure across grids and scales.

  We propose Mesh-Free Convolutions (MFCs), filters defined in continuous space and discretized only at evaluation time.

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