The Vekua Layer: Exact Physical Priors for Implicit Neural Representations via Generalized Analytic Functions
This addresses computational inefficiency and noise sensitivity in physics-based machine learning, offering a novel spectral method for PDEs, though it is incremental as it builds on existing INR frameworks.
The paper tackles the spectral bias and computational expense in Implicit Neural Representations (INRs) for physical fields by introducing the Vekua Layer, which transforms learning into a convex least-squares problem using Harmonic and Fourier-Bessel bases, achieving machine precision (MSE ≈ 10^-33) on exact reconstruction and improved stability (MSE ≈ 0.03) with noise.
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.