The Vekua Layer: Exact Physical Priors for Implicit Neural Representations via Generalized Analytic Functions

TL;DR

The paper introduces the Vekua Layer, a spectral method based on classical analytic functions, that improves implicit neural representations by providing exact physical priors, reducing optimization complexity, and enabling stable, physics-informed extrapolation.

Contribution

The Vekua Layer transforms the learning of physical fields into a convex least-squares problem using classical analytic function theory, offering exact priors and enhanced stability over traditional neural methods.

Findings

Achieves near-perfect reconstruction with MSE ≈ 10^{-33}

Provides stable performance under sensor noise with MSE ≈ 0.03

Enables holographic extrapolation from partial boundary data

Abstract

Implicit Neural Representations (INRs) have emerged as a powerful paradigm for parameterizing physical fields, yet they often suffer from spectral bias and the computational expense of non-convex optimization. We introduce the Vekua Layer (VL), a differentiable spectral method grounded in the classical theory of Generalized Analytic Functions. By restricting the hypothesis space to the kernel of the governing differential operator -- specifically utilizing Harmonic and Fourier-Bessel bases -- the VL transforms the learning task from iterative gradient descent to a strictly convex least-squares problem solved via linear projection. We evaluate the VL against Sinusoidal Representation Networks (SIRENs) on homogeneous elliptic Partial Differential Equations (PDEs). Our results demonstrate that the VL achieves machine precision ($\text{MSE} \approx 10^{-33}$) on exact reconstruction tasks and exhibits superior stability in the presence of incoherent sensor noise ($\text{MSE} \approx 0.03$), effectively acting as a physics-informed spectral filter. Furthermore, we show that the VL enables "holographic" extrapolation of global fields from partial boundary data via analytic continuation, a capability absent in standard coordinate-based approximations.