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

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.

💡 Research Summary

DeepVekua is a novel hybrid architecture that tackles the long‑standing challenge of reconstructing continuous physical fields from sparse observations, especially in domains with complex geometry and advection‑diffusion dynamics. The core idea is to decouple geometry from physics by learning a diffeomorphic coordinate warping and, in the warped latent space, representing the field with a radially‑modulated Fourier basis whose coefficients are solved analytically during the forward pass.

The geometric branch consists of a sinusoidal MLP that predicts a displacement field uℓ(x). Adding this displacement to the original coordinates yields a smooth, invertible map Φℓ(x)=x+uℓ(x). In two dimensions the warped coordinates are embedded as a complex number ζℓ = zℓ₁ + i zℓ₂, which enables compact representation of rotations and scalings. This warping can be interpreted as an approximation of the Lagrangian flow that removes the advective component of the underlying PDE.

In the physics branch, a set of learnable complex frequencies Fℓ={fk = uk+ivk} defines a Radially‑Modulated Fourier Basis (RMFB). For each frequency the inner product ϕk = Re(ζ·¯fk) is computed, and the basis functions are ψk(ζ) =