The Adaptive Vekua Cascade: A Differentiable Spectral-Analytic Solver for Physics-Informed Representation

• URL: http://arxiv.org/abs/2512.11776v1
• Date: Fri, 12 Dec 2025 18:41:35 GMT
• Title: The Adaptive Vekua Cascade: A Differentiable Spectral-Analytic Solver for Physics-Informed Representation
• Authors: Vladimer Khasia,
• Abstract summary: Adaptive-based neural networks have emerged as a powerful tool for representing continuous physical fields.<n>They face two fundamental pathologies: spectral bias and the curse of dimensionality.<n>We propose a hybrid architecture that bridges analytic analytic decouples manifold learning from function approximation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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