Special Session 78: Learning Approaches for PDE Forward and Inverse Problems

Point Source Identification Using Singularity Enriched Neural Networks

Tianhao Hu The Chinese University of Hong Kong Hong Kong Co-Author(s):    Tianhao Hu, Bangti Jin, Zhi Zhou

Abstract: Neural network-based methods have shown great promise in stably solving ill-posed inverse problems. In this work, we focus on the inverse problem of recovering point sources, an important class of applied inverse problems. Despite their potential, neural network-based methods for identifying point sources remain underdeveloped, primarily due to the inherent singularity of the solution. To address this challenge, we develop a novel neural algorithm for identifying point sources, utilizing the singularity enrichment technique. We employ the fundamental solution and neural networks to represent the singular and regular parts, respectively, and then minimize an empirical loss involving the intensities and locations of unknown point sources and the parameters of the neural network. Moreover, by combining the conditional stability argument of the inverse problem with the generalization error of the empirical loss, we conduct a rigorous error analysis of the algorithm. We demonstrate the effectiveness of the method with several challenging experiments.

Leveraging Multiple Scattering for Inverse Problems in Reflection Microscopy

Thomas Wasik CNRS/ Ecole polytechnique France Co-Author(s):    Thomas WASIK, Victor BAROLLE, Alexandre AUBRY, Josselin GARNIER

Abstract: In microscopy, reflection-mode configurations offer greater flexibility than transmission ones, enabling imaging of thicker samples and in vivo applications. Reflection-mode Optical Diffraction Tomography (ODT), however, faces a well-known limitation: the corresponding inverse problem is ill-posed. In single-scattering regime, reflected waves from the medium capture only the spatial high-frequency components, while, in the multiple-scattering regime, the forward model is non-linear, complicating optimization-based reconstruction. Nevertheless, it can be shown that this multiple scattering effect reduces the filtering effect associated with reflection measurements. For instance, light scattered by background structures also encodes the spatial low-frequencies of foreground objects, making the reconstruction possible. We present a novel optimization framework for 3D refractive index reconstruction in reflection-mode microscopy. Our method leverages backscattered light from unknown background structures to reconstruct foreground refractive index distributions. Regularization, an appropriate solver and a temporal loss ensure convergence despite the non-convexity of the problem. Simulations demonstrate accurate recovery of both low and high spatial frequencies, overcoming the missing-cone problem occurring in transmission-mode imaging. We also show that multi-wavelength illumination is necessary for convergence, making reflection ODT computationally intensive but rewarding.