{"ID":2829590,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.12402","arxiv_id":"2512.12402","title":"DeepVekua: Geometric-Spectral Representation Learning for Physics-Informed Fields","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.","short_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-th...","url_abs":"https://arxiv.org/abs/2512.12402","url_pdf":"https://arxiv.org/pdf/2512.12402v1","authors":"[\"Vladimer Khasia\"]","published":"2025-12-13T17:29:55Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[\"Diffusion Model\"]","has_code":false,"code_links":[{"ID":605959,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_id":2829590,"paper_url":"https://arxiv.org/abs/2512.12402","paper_title":"DeepVekua: Geometric-Spectral Representation Learning for Physics-Informed Fields","repo_url":"https://github.com/VladimerKhasia/vekuanet","is_official":false,"mentioned_in_paper":false,"mentioned_in_github":true,"github_stars":0}]}