Mesh-free convolution improves long-term predictions for spatial equations
Mesh-Free Convolution: Learned Spectral Attenuation and Transport
Machine Learning
Summary
Predicting how things like fluids or weather change over time can be tricky because small mistakes can build up quickly. This paper introduces a new method called Mesh-Free Convolutions that works smoothly across different scales and resolutions without relying on fixed grids. The authors show that their method makes long-term predictions more stable and accurate for problems like fluid dynamics and weather forecasting. This approach helps models remember complex spatial patterns better over time.
What this means in practice
- •For weather prediction teams: Create weather forecasting models that maintain accuracy over long periods and across different grid resolutions by using mesh-free convolution filters.
- •For computational fluid dynamics engineers: Improve simulations of fluid flows with stable long-term predictions that adapt to varying spatial resolutions using learned transport-aware filters.
Authors
Niloufar Zakariaei, Manav Manav, Eldad Haber
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.