{"ID":2829947,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.11776","arxiv_id":"2512.11776","title":"The Adaptive Vekua Cascade: A Differentiable Spectral-Analytic Solver for Physics-Informed Representation","abstract":"Coordinate-based neural networks have emerged as a powerful tool for representing continuous physical fields, yet they face two fundamental pathologies: spectral bias, which hinders the learning of high-frequency dynamics, and the curse of dimensionality, which causes parameter explosion in discrete feature grids. We propose the Adaptive Vekua Cascade (AVC), a hybrid architecture that bridges deep learning and classical approximation theory. AVC decouples manifold learning from function approximation by using a deep network to learn a diffeomorphic warping of the physical domain, projecting complex spatiotemporal dynamics onto a latent manifold where the solution is represented by a basis of generalized analytic functions. Crucially, we replace the standard gradient-descent output layer with a differentiable linear solver, allowing the network to optimally resolve spectral coefficients in a closed form during the forward pass. We evaluate AVC on a suite of five rigorous physics benchmarks, including high-frequency Helmholtz wave propagation, sparse medical reconstruction, and unsteady 3D Navier-Stokes turbulence. Our results demonstrate that AVC achieves state-of-the-art accuracy while reducing parameter counts by orders of magnitude (e.g., 840 parameters vs. 4.2 million for 3D grids) and converging 2-3x faster than implicit neural representations. This work establishes a new paradigm for memory-efficient, spectrally accurate scientific machine learning. The code is available at https://github.com/VladimerKhasia/vecua.","short_abstract":"Coordinate-based neural networks have emerged as a powerful tool for representing continuous physical fields, yet they face two fundamental pathologies: spectral bias, which hinders the learning of high-frequency dynamics, and the curse of dimensionality, which causes parameter explosion in discrete feature grids. We p...","url_abs":"https://arxiv.org/abs/2512.11776","url_pdf":"https://arxiv.org/pdf/2512.11776v1","authors":"[\"Vladimer Khasia\"]","published":"2025-12-12T18:41:35Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[]","has_code":false,"code_links":[{"ID":605985,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_id":2829947,"paper_url":"https://arxiv.org/abs/2512.11776","paper_title":"The Adaptive Vekua Cascade: A Differentiable Spectral-Analytic Solver for Physics-Informed Representation","repo_url":"https://github.com/VladimerKhasia/vecua","is_official":false,"mentioned_in_paper":false,"mentioned_in_github":true,"github_stars":0}]}