Yimin Zhong

Hongkai Zhao, Yimin Zhong

We present a hybrid imaging method for a challenging travel time tomography problem which includes both unknown medium and unknown scatterers in a bounded domain. The goal is to recover both the medium and the boundary of the scatterers from the scattering relation data on the domain boundary. Our method is composed of three steps: 1) preprocess the data to classify them into three different categories of measurements corresponding to non-broken rays, broken-once rays, and others, respectively, 2) use the the non-broken ray data and an effective data-driven layer stripping strategy--an optimization based iterative imaging method--to recover the medium velocity outside the convex hull of the scatterers, and 3) use selected broken-once ray data to recover the boundary of the scatterers--a direct imaging method. By numerical tests, we show that our hybrid method can recover both the unknown medium and the not-too-concave scatterers efficiently and robustly.

Shijun Zhang, Hongkai Zhao, Yimin Zhong et al.

In this work, we present a comprehensive study combining mathematical and computational analysis to explain why a two-layer neural network struggles to handle high frequencies in both approximation and learning, especially when machine precision, numerical noise, and computational cost are significant factors in practice. Specifically, we investigate the following fundamental computational issues: (1) the minimal numerical error achievable under finite precision, (2) the computational cost required to attain a given accuracy, and (3) the stability of the method with respect to perturbations. The core of our analysis lies in the conditioning of the representation and its learning dynamics. Explicit answers to these questions are provided, along with supporting numerical evidence.

Wei Li, Yang Yang, Yimin Zhong

We investigate a hybrid inverse problem in fluorescence ultrasound modulated optical tomography (fUMOT) in the diffusive regime. We prove that the absorption coefficient of the fluorophores at the excitation frequency and the quantum efficiency coefficient can be uniquely and stably reconstructed from boundary measurement of the photon currents, provided that some background medium parameters are known. Reconstruction algorithms are proposed and numerically implemented as well.

Yi Liu, Yanzhao Cao, Junshan Lin et al.

In this work, we propose a multi-level machine learning framework for solving inverse scattering problems with multi-frequency data. The multi-level neural network is built along the frequency axis of the scattering problem, wherein at each fixed frequency, a new level of network is added to the existing architecture to update the reconstruction. By marching through the frequency levels, the proposed multi-level computational framework is able to obtain higher-order Fourier modes of the imaging target as the depth of the neural network grows and higher-frequency data are used. Furthermore, the overall learning problem is decomposed into a sequence of simpler local tasks, each associated with a single frequency. This decomposition significantly reduces the complexity of the optimization problem and mitigates the risk of convergence to undesirable local minima, resulting in a robust and reliable training procedure for solving inverse scattering problems. We conduct various numerical experiments for the inverse source scattering problem and the inverse medium scattering problem to illustrate the effectiveness and robustness of the proposed machine learning framework. In addition, theoretical analysis in the neural tangent kernel regime shows that the proposed multi-level architecture progressively recovers the higher-order Fourier components of the imaging target.

Shijun Zhang, Hongkai Zhao, Yimin Zhong et al.

The architecture of a neural network and the selection of its activation function are both fundamental to its performance. Equally vital is ensuring these two elements are well-matched, as their alignment is key to achieving effective representation and learning. In this paper, we introduce the Fourier Multi-Component and Multi-Layer Neural Network (FMMNN), a novel model that creates a strong synergy between them. We demonstrate that FMMNNs are highly effective and flexible in modeling high-frequency components. Our theoretical results demonstrate that FMMNNs have exponential expressive power for function approximation. We also analyze the optimization landscape of FMMNNs and find it to be much more favorable than that of standard fully connected neural networks, especially when dealing with high-frequency features. In addition, we propose a scaled random initialization method for the first layer's weights in FMMNNs, which significantly speeds up training and enhances overall performance. Extensive numerical experiments support our theoretical insights, showing that FMMNNs consistently outperform traditional approaches in accuracy and efficiency across various tasks.

Shijun Zhang, Hongkai Zhao, Yimin Zhong et al.

In this work, we propose a balanced multi-component and multi-layer neural network (MMNN) structure to accurately and efficiently approximate functions with complex features, in terms of both degrees of freedom and computational cost. The main idea is inspired by a multi-component approach, in which each component can be effectively approximated by a single-layer network, combined with a multi-layer decomposition strategy to capture the complexity of the target function. Although MMNNs can be viewed as a simple modification of fully connected neural networks (FCNNs) or multi-layer perceptrons (MLPs) by introducing balanced multi-component structures, they achieve a significant reduction in training parameters, a much more efficient training process, and improved accuracy compared to FCNNs or MLPs. Extensive numerical experiments demonstrate the effectiveness of MMNNs in approximating highly oscillatory functions and their ability to automatically adapt to localized features.

Yiran Wang, Yimin Zhong

The limited angle Radon transform is notoriously difficult to invert due to its ill-posedness. In this work, we give a mathematical explanation that data-driven approaches can stably reconstruct more information compared to traditional methods like filtered backprojection. In addition, we use experiments based on the U-Net neural network to validate our theory.