Latent Dynamics · 2026-10-14
Mesh-Free Convolution: Learned Spectral Attenuation and Transport
Niloufar Zakariaei, Manav Manav, Eldad Haber
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TL;DR
The authors propose Mesh-Free Convolutions (MFCs), continuous-space filters discretized only at evaluation time. MFCs replace hard Fourier-mode truncation with learned smooth spectral attenuation, add first-order directional readout for transport-like behavior, and use a continuous symbol so the same learned filters transfer across resolutions. They report more stable long-horizon rollouts and better long-time statistics than Fourier-based baselines on fluid-dynamics and weather-forecasting benchmarks.
Why it matters
Autoregressive PDE learning can drift or become unstable because small spatial/spectral errors are repeatedly fed back. By learning a continuous Fourier symbol with smooth attenuation and directional/shift terms, MFCs aim to preserve dynamically relevant structure under repeated time-stepping and enable zero-shot cross-resolution evaluation.
Method
- MFCs define convolution through a continuous-space Fourier symbol and discretize it only on the evaluation mesh.
- They replace hard Fourier truncation with learned smooth spectral attenuation via an anisotropic parabolic parameterization.
- They add a directional readout with first-order terms plus a learned continuous shift to model transport-like, asymmetric behavior.
Abstract (from arXiv)
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.
Related papers
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Publication record
Bibliographic evidence describes where and when a record appeared. It does not validate scientific claims.
First public release: 2026-10-11 · Indexed by Pub-AI: 2026-10-14 UTC
Mesh-Free Convolution: Learned Spectral Attenuation and Transport
Preprint posted: 2026-10-11 · Original record ↗
- arxiv
- 2610.15931
Explicit source versions (1)
- v1 · 2026-10-11
- Preprint available
Source date: 2026-10-11 · First detected: 2026-10-25 UTC · Historical baseline
Source evidence (1)
- Official evidence ↗ · atom:published (official metadata)
Abstract · arxiv
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.