arXiv subjects

Qiao Zhu

Publications and source records attributed to Qiao Zhu.

7 recordsLinked to original sources

arXiv2026

Shallow neural network yields regularization for ill-posed inverse problems

In this paper, we develop a regularization theory for neural network approximations of general ill-posed operator equations with noisy data. Within the framework of iterative regularization, we introduce two expanding neural network methods (ENNs) under different a priori assumptions on the exact solution. Instead of prescribing a fixed architecture, ENNs adaptively select the number of neurons through an a posteriori stopping rule, so that the selected network size serves as a regularization parameter balancing approximation accuracy and stability with respect to data noise. We prove the regularization properties of the proposed ENNs and establish quantitative relationships between the selected network size and the noise level. Within the framework of variational regularization, we propose a neural network-based Tikhonov scheme and derive both convergence and convergence-rate results under mild assumptions. The resulting estimates account for the noise level, the network size, and the underlying smoothness expressed through general variational source conditions, thereby allowing greater flexibility than existing results. Numerical experiments demonstrate the effectiveness and robustness of the proposed algorithms. In particular, they show that, for highly noisy data, relatively small network architectures can already produce stable reconstructions, whereas excessively large architectures may degrade stability due to overfitting.

math.NA↗

arXiv2026

Deep neural network yields regularization for ill-posed inverse problems

This paper studies the regularization of ill-posed inverse problems by deep neural networks (DNNs). We extend architecture-based regularization from shallow networks to deep models by developing a deterministic framework in which the admissible network class is enlarged adaptively and the resulting architecture complexity acts as the regularization mechanism. We propose two discrepancy-principle-driven expanding DNN algorithms to treat the cases where an explicit parameter-radius bound is available and unavailable, respectively. For both algorithms, we prove the finite termination of the adaptive expansion procedure and the convergence of the regularized solutions as the noise level vanishes. In addition, we derive explicit asymptotic bounds on the terminal network architecture, thereby quantifying how the required network complexity scales with the noise level. Numerical experiments on several representative linear and non-linear inverse problems support the theoretical findings and illustrate the practical usefulness of the proposed framework.

math.NA↗

arXiv2025

Deep asymptotic expansion method for solving singularly perturbed time-dependent reaction-advection-diffusion equations

Physics-informed neural network (PINN) has shown great potential in solving partial differential equations. However, it faces challenges when dealing with problems involving steep gradients. The solutions to singularly perturbed time-dependent reaction-advection-diffusion equations exhibit internal moving transition layers with sharp gradients, and thus the standard PINN becomes ineffective. In this work, we propose a deep asymptotic expansion (DAE) method, which is inspired by asymptotic analysis and leverages deep learning to approximate the smooth part of the expansion. We first derive the governing equations for transition layers, which are then solved using PINN. Numerical experiments show that the DAE outperforms the standard PINN, gPINN, and PINN with adaptive sampling. We also show its robustness with respect to training point distributions, network architectures, and random seeds.

math.NA↗

arXiv2025

Multi-layer 5D Optical Data Storage: Mathematical Modeling and Deep Learning-Based Reconstruction of Birefringent Parameters

Five-dimensional (5D) optical data storage has emerged as a promising technology for ultra-high-density, long-term data archiving. However, its practical realization is hindered by noise and interference during data readout. In this work, we develop a high-precision mathematical model for multi-layer 5D optical storage, grounded in the Jones matrix framework, to accurately capture polarization transformations induced by stacked birefringent nanostructures. Building on this model, we propose a 20-frame FiLM-conditioned U-Net algorithm to reconstruct birefringence parameters--specifically, slow-axis orientation and retardance magnitude-directly from measured intensity patterns. Trained on both ideal and noisy datasets, the network demonstrates robust reconstruction performance under challenging measurement conditions. Compared with conventional frame-based retrieval approaches, our method achieves over an order-of-magnitude improvement in reconstruction accuracy. The proposed model and algorithm can be readily integrated into existing 5D optical readout systems, offering both a solid theoretical foundation and practical tools for precise data recovery.

physics.optics↗

arXiv2024

Double-activation neural network for solving parabolic equations with time delay

This paper presents the double-activation neural network (DANN), a novel network architecture designed for solving parabolic equations with time delay. In DANN, each neuron is equipped with two activation functions to augment the network's nonlinear expressive capacity. Additionally, a new parameter is introduced for the construction of the quadratic terms in one of two activation functions, which further enhances the network's ability to capture complex nonlinear relationships. To address the issue of low fitting accuracy caused by the discontinuity of solution's derivative, a piecewise fitting approach is proposed by dividing the global solving domain into several subdomains. The convergence of the loss function is proven. Numerical results are presented to demonstrate the superior accuracy and faster convergence of DANN compared to the traditional physics-informed neural network (PINN).

math.NA↗

arXiv2022

Rate-Splitting Multiple Access in Multi-cell Dense Networks: A Stochastic Geometry Approach

In this paper, the potential benefits of applying the Rate-Splitting Multiple Access (RSMA) in multi-cell dense networks are explored. Using tools of stochastic geometry, the sum-rate of RSMA-enhanced multi-cell dense networks is evaluated mathematically based on a Moment Generating Function (MGF) based framework to prove that RSMA is a general and powerful strategy for multi-antenna downlink systems. Further elaboration of the systematic performance metrics is undertaken by developing analytical expressions for area spectral efficiency and sum-rate in the RSMA-enhanced multi-cell dense networks. Based on the tractable expressions, we then offer an optimization framework for energy efficiency in terms of the number of antennas. Additionally, simulation results are shown to verify the accuracy of our analytical results and provide some insightful insights into system design. Analytically, it has been shown that: 1) the sum-rate of RSMA-enhanced multi-cell dense networks is significantly influenced by the power splitting ratio, and there is a unique value that maximizes the sum-rate; 2) the RSMA-enhanced multi-cell dense networks transmission scheme has superior sum-rate performance compared with Non-Orthogonal Multiple Access (NOMA) and Space-Division Multiple Access (SDMA) in a wide range of power splitting ratio; 3) By increasing the number of antennas and BS density in an RSMA-enhanced multi-cell dense network, the area spectral efficiency can be substantially enhanced; 4) As for energy efficiency, there exists an optimal antenna number for maximizing this performance metric.

cs.IT↗

arXiv2022

A Modification Piecewise Convexification Method for Box-Constrained Non-Convex Optimization Programs

This paper presents a piecewise convexification method to approximate the whole approximate optimal solution set of non-convex optimization problems with box constraints. In the process of box division, we first classify the sub-boxes and only continue to divide only some sub-boxes in the subsequent division. At the same time, applying the $α$-based Branch-and-Bound ({\rm$α$BB}) method, we construct a series of piecewise convex relax sub-problems, which are collectively called the piecewise convexification problem of the original problem. Then, we define the (approximate) solution set of the piecewise convexification problem based on the classification result of sub-boxes. Subsequently, we derive that these sets can be used to approximate the global solution set with a predefined quality. Finally, a piecewise convexification algorithm with a new selection rule of sub-box for the division and two new termination tests is proposed. Several instances verify that these techniques are beneficial to improve the performance of the algorithm.

math.OC↗