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.

翻译：基于坐标的神经网络已成为表示连续物理场的强大工具，但其面临两个基本病理问题：谱偏差（阻碍高频动力学学习）和维度灾难（导致离散特征网格参数爆炸）。我们提出了自适应Vekua级联（AVC），这是一种连接深度学习与经典逼近理论的混合架构。AVC通过使用深度网络学习物理域的微分同胚扭曲，将流形学习与函数逼近解耦，将复杂时空动力学投影到潜在流形上，其中解由广义解析函数基表示。关键的是，我们用可微分线性求解器替代标准梯度下降输出层，使网络能在前向传播中以闭式形式最优解析谱系数。我们在五个严谨的物理基准测试中评估AVC，包括高频亥姆霍兹波传播、稀疏医学重建和非定常三维纳维-斯托克斯湍流。结果表明，AVC在实现最先进精度的同时，将参数数量减少数个数量级（例如三维网格仅需840个参数，而非420万个），且收敛速度比隐式神经表示快2-3倍。这项工作为内存高效、谱精确的科学机器学习建立了新范式。代码发布于https://github.com/VladimerKhasia/vecua。