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The Calderón's problem via DeepONets

Javier Castro, Claudio Muñoz, Nicolás Valenzuela

TL;DR

The paper investigates Calderón's problem for isotropic conductivities on a smooth bounded domain and introduces DeepONets as rigorous, infinite-dimensional approximators of operator-valued maps. It establishes universal approximation results for both the Dirichlet-to-Neumann map $\Lambda_a$ with a fixed conductivity and the full direct map $a\mapsto\Lambda_a$, as well as for the inverse Calderón mapping, within a Hilbert-space framework using projections, extensions, and probabilistic error measures. The contributions combine a detailed functional-analytic foundation with a general theory for learning nonlinear operators between infinite-dimensional spaces, showing that DeepONets can approximate the Calderón mappings to arbitrary accuracy in an $L^2$ sense under suitable measures. This provides a rigorous bridge between inverse problems in PDEs and operator-learning, with potential implications for numerical reconstruction and broader Calderón-type problems. The work also outlines future directions for numerical realization and extensions to related operators and geometries, as well as connections to PINNs and other learning paradigms.

Abstract

We consider the Dirichlet-to-Neumann operator and the direct and inverse Calderón's mappings appearing in the Inverse Problem of recovering a smooth bounded and positive isotropic conductivity of a material filling a smooth bounded domain in space. Using deep learning techniques, we prove that these mappings are rigorously approximated by DeepONets, infinite-dimensional counterparts of standard artificial neural networks.

The Calderón's problem via DeepONets

TL;DR

The paper investigates Calderón's problem for isotropic conductivities on a smooth bounded domain and introduces DeepONets as rigorous, infinite-dimensional approximators of operator-valued maps. It establishes universal approximation results for both the Dirichlet-to-Neumann map with a fixed conductivity and the full direct map , as well as for the inverse Calderón mapping, within a Hilbert-space framework using projections, extensions, and probabilistic error measures. The contributions combine a detailed functional-analytic foundation with a general theory for learning nonlinear operators between infinite-dimensional spaces, showing that DeepONets can approximate the Calderón mappings to arbitrary accuracy in an sense under suitable measures. This provides a rigorous bridge between inverse problems in PDEs and operator-learning, with potential implications for numerical reconstruction and broader Calderón-type problems. The work also outlines future directions for numerical realization and extensions to related operators and geometries, as well as connections to PINNs and other learning paradigms.

Abstract

We consider the Dirichlet-to-Neumann operator and the direct and inverse Calderón's mappings appearing in the Inverse Problem of recovering a smooth bounded and positive isotropic conductivity of a material filling a smooth bounded domain in space. Using deep learning techniques, we prove that these mappings are rigorously approximated by DeepONets, infinite-dimensional counterparts of standard artificial neural networks.
Paper Structure (38 sections, 19 theorems, 102 equations, 1 figure)

This paper contains 38 sections, 19 theorems, 102 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting: the Calderón's problem
  3. Brief review of results on the Calderón's problem
  4. Main goal
  5. Deep Learning techniques in PDEs
  6. Brief overview
  7. Deep Learning techniques in Inverse Problems
  8. First introduction to DeepONets
  9. General continuous Banach case
  10. Finite dimensional deep neural networks
  11. Infinite dimensional deep neural networks
  12. Main Results
  13. Approximation of $\Lambda_a$, $a$ fixed
  14. Approximation of the full Calderón's mapping $a\mapsto \Lambda_a$
  15. Approximation of the inverse Calderón's mapping
  16. ...and 23 more sections

Key Result

Lemma 3.1

Let $(\varepsilon_k)_{k\in\mathbb{N}}$ be any collection of strictly positive numbers such that $\varepsilon_k\to 0$. Let $V$ be a compact set of functions in $C(K)$. Then, for every $k$ there exists an operator $T_k\colon C(K)\to C(K)$ such that every function $f\in V$ is $\varepsilon_k$-close to $ is compact in $C(K)$.

Figures (1)

  • Figure 1: Schematic description of how the DeepONet \ref{['eq:DO-def']} is approximating a general mapping $G:H\to W$.

Theorems & Definitions (53)

  • Lemma 3.1: CC95
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Remark 3.1: On quantitative bounds for the projection operator $P_{H,d}$
  • Lemma 3.5: Cas22
  • Remark 3.2
  • Definition 4.1: Admissible conductivities
  • Theorem 4.2
  • Theorem 4.3
  • ...and 43 more