Recovering Sharp Conductivity Features in the Finite-Data Calderón Problem with Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) have recently emerged as a promising framework for addressing the Calderón inverse problem from limited boundary data. In this work, we revisit neural Calderón inversion by introducing multiscale boundary excitations based on randomized wavelet functions and investigating the role of Fourier-feature encoding (FFE) for representing sharp conductivity variations. We propose a physics-informed reconstruction framework that represents the unknown conductivity and the associated family of electric potentials with separate neural networks conditioned on the applied boundary excitations. The governing elliptic PDE is enforced through physics-informed residuals, while finite Dirichlet-to-Neumann (DtN) data are incorporated through boundary losses. Using synthetic data from a finite-difference forward solver, we evaluate the method on conductivity fields with inclusions, sharp interfaces, smooth profiles, and heterogeneous media. Results show that the framework recovers dominant conductivity structures from finite boundary measurements with relative errors between $3\%-12\%$ approximately. We show that FFE improves the reconstruction of localized sharp features, particularly for inclusions and interfaces, but are not universally optimal, with raw-coordinate networks performing competitively for smoother fields. These results highlight coordinate representations and boundary excitation design as key factors in neural Calderón inversion.

Learning embeddings of non-linear PDEs: the Burgers' equation

Embeddings provide low-dimensional representations that organize complex function spaces and support generalization. They provide a geometric representation that supports efficient retrieval, comparison, and generalization. In this work we generalize the concept to Physics Informed Neural Networks. We present a method to construct solution embedding spaces of nonlinear partial differential equations using a multi-head setup, and extract non-degenerate information from them using principal component analysis (PCA). We test this method by applying it to viscous Burgers' equation, which is solved simultaneously for a family of initial conditions and values of the viscosity. A shared network body learns a latent embedding of the solution space, while linear heads map this embedding to individual realizations. By enforcing orthogonality constraints on the heads, we obtain a principal-component decomposition of the latent space that is robust to training degeneracies and admits a direct physical interpretation. The obtained components for Burgers' equation exhibit rapid saturation, indicating that a small number of latent modes captures the dominant features of the dynamics.