DeepVekua: Geometric-Spectral Representation Learning for Physics-Informed Fields

TL;DR

DeepVekua introduces a hybrid geometric-spectral deep learning approach that effectively solves PDEs in complex geometries with sparse data, outperforming existing methods by leveraging spectral analysis and learned coordinate transformations.

Contribution

It unifies geometric deep learning with spectral analysis, learning a coordinate transformation to improve PDE solving in complex geometries, surpassing state-of-the-art implicit methods.

Findings

Achieves 100x improvement over spectral baselines.

Outperforms state-of-the-art implicit representations.

Effectively handles complex geometries in PDEs.

Abstract

We present DeepVekua, a hybrid architecture that unifies geometric deep learning with spectral analysis to solve partial differential equations (PDEs) in sparse data regimes. By learning a diffeomorphic coordinate transformation that maps complex geometries to a latent harmonic space, our method outperforms state-of-the-art implicit representations on advection-diffusion systems. Unlike standard coordinate-based networks which struggle with spectral bias, DeepVekua separates the learning of geometry from the learning of physics, solving for optimal spectral weights in closed form. We demonstrate a 100x improvement over spectral baselines. The code is available at https://github.com/VladimerKhasia/vekuanet.