# A Mathematical Framework for Deep Learning in Elastic Source Imaging

## TL;DR

This paper introduces a mathematical framework for elastic source imaging that integrates low-dimensional manifold regularization, improving reconstruction performance with sparse data, and links it to deep learning techniques.

## Contribution

It presents a novel framework combining manifold regularization with inverse problem algorithms, connecting it to deep convolutional framelet expansion in machine learning.

## Key findings

- Enhanced source reconstruction with sparse measurements.
- The framework is mathematically equivalent to deep convolutional framelet expansion.
- Numerical examples demonstrate improved performance.

## Abstract

An inverse elastic source problem with sparse measurements is of concern. A generic mathematical framework is proposed which incorporates a low- dimensional manifold regularization in the conventional source reconstruction algorithms thereby enhancing their performance with sparse datasets. It is rigorously established that the proposed framework is equivalent to the so-called \emph{deep convolutional framelet expansion} in machine learning literature for inverse problems. Apposite numerical examples are furnished to substantiate the efficacy of the proposed framework.