# Recovering Sharp Conductivity Features in the Finite-Data Calder'on Problem with Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2606.28158 · HTML: https://mlsanity.com/paper/2606.28158/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 37 references that are in the map.
- First posted on arXiv 26 Jun 2026.

## Builds on (most cited first, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [Physical Symmetries Embedded in Neural Networks](https://mlsanity.com/paper/1904.08991/) (2019)
- [Solving Differential Equations Using Neural Network Solution Bundles](https://mlsanity.com/paper/2006.14372/) (2020)
- [Forecasting Hamiltonian dynamics without canonical coordinates](https://mlsanity.com/paper/2010.15201/) (2020)
- [Gravitational Duals from Equations of State](https://mlsanity.com/paper/2403.14763/) (2024)
- [Efficient PINNs via Multi-Head Unimodular Regularization of the Solutions Space](https://mlsanity.com/paper/2501.12116/) (2025)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) have recently emerged as a promising framework for addressing the Calder'on inverse problem from limited boundary data. In this work, we revisit neural Calder'on 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 Calder'on inversion.