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arXiv · 2512.12402

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

Vladimer Khasia
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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.

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Vladimer Khasia. 2025-12-13. DeepVekua: Geometric-Spectral Representation Learning for Physics-Informed Fields. https://doi.org/10.5281/zenodo.17918695

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