arXiv subjects

Quanhui Zhu

Publications and source records attributed to Quanhui Zhu.

6 recordsLinked to original sources

arXiv2026

A General First- and Second-Order Numerical Solver for Non-Markovian Quantum State Diffusion

The numerical simulation of non-Markovian open quantum systems based on the non-Markovian quantum state diffusion (NMQSD) equation is complicated by functional derivatives with respect to the stochastic process. A general numerical framework that directly treats these functional derivatives without relying on prescribed decompositions of the bath correlation function is still lacking. In this work, we derive an analytical solution of the linear NMQSD equation that reveals three elementary structures of the non-Markovian stochastic dynamics: stochastic propagation, functional-derivative insertion, and memory pairing. Based on this structure, we construct a general auxiliary-state framework for arbitrary bath correlation functions. The framework separates the numerical construction into time discretization, memory quadrature, and hierarchy truncation. We then construct first- and second-order schemes and provide diagrammatic transition rules for their explicit implementation. Numerical results verify the expected temporal accuracy and demonstrate the applicability of the proposed methods to different bath correlation functions and multi-level quantum systems.

quant-ph↗

arXiv2026

Energy Dissipation Preserving Feature-based DNN Galerkin Methods for Gradient Flows

In recent years, deep learning methods, exemplified by Physics-Informed Neural Networks (PINNs), have been widely applied to the numerical solution of differential equations. However, these methods may suffer from limited accuracy, high training costs, and lack of robustness, particularly their inability to preserve the intrinsic physical structures of continuous PDE models, such as the energy dissipation property in gradient flow systems. To address these challenges, we propose a feature-based Deep Neural Network Galerkin (DNN-G) framework designed for structure-preserving simulations of gradient flows. Instead of treating neural networks merely as optimization-driven solvers, we employ them as adaptive feature generators that define nonlinear trial spaces within a Galerkin projection formulation.This formulation guarantees semi-discrete energy dissipation and can be naturally combined with energy stable time integration schemes. Several strategies for constructing neural basis functions are investigated, including random features, structured initialization, and problem-informed pre-training. Numerical experiments demonstrate that the proposed method preserves robust energy stability in high-dimensional settings and accurately captures complex topological transitions. With equivalent degrees of freedom, the DNN-G framework achieves higher accuracy than classical spectral methods, highlighting the effectiveness of neural feature representations for the numerical solution of partial differential equations.

math.NA↗

arXiv2025

$ε$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics

Understanding the training dynamics of deep neural networks (DNNs), particularly how they evolve low-dimensional features from high-dimensional data, remains a central challenge in deep learning theory. In this work, we introduce the concept of $ε$-rank, a novel metric quantifying the effective feature of neuron functions in the terminal hidden layer. Through extensive experiments across diverse tasks, we observe a universal staircase phenomenon: during training process implemented by the standard stochastic gradient descent methods, the decline of the loss function is accompanied by an increase in the $ε$-rank and exhibits a staircase pattern. Theoretically, we rigorously prove a negative correlation between the loss lower bound and $ε$-rank, demonstrating that a high $ε$-rank is essential for significant loss reduction. Moreover, numerical evidences show that within the same deep neural network, the $ε$-rank of the subsequent hidden layer is higher than that of the previous hidden layer. Based on these observations, to eliminate the staircase phenomenon, we propose a novel pre-training strategy on the initial hidden layer that elevates the $ε$-rank of the terminal hidden layer. Numerical experiments validate its effectiveness in reducing training time and improving accuracy across various tasks. Therefore, the newly introduced concept of $ε$-rank is a computable quantity that serves as an intrinsic effective metric characteristic for deep neural networks, providing a novel perspective for understanding the training dynamics of neural networks and offering a theoretical foundation for designing efficient training strategies in practical applications.

cs.LG↗

arXiv2025

Structured First-Layer Initialization Pre-Training Techniques to Accelerate Training Process Based on $\varepsilon$-Rank

Training deep neural networks for scientific computing remains computationally expensive due to the slow formation of diverse feature representations in early training stages. Recent studies identify a staircase phenomenon in training dynamics, where loss decreases are closely correlated with increases in $\varepsilon$-rank, reflecting the effective number of linearly independent neuron functions. Motivated by this observation, this work proposes a structured first-layer initialization (SFLI) pre-training method to enhance the diversity of neural features at initialization by constructing $\varepsilon$-linearly independent neurons in the input layer. We present systematic initialization schemes compatible with various activation functions and integrate the strategy into multiple neural architectures, including modified multi-layer perceptrons and physics-informed residual adaptive networks. Extensive numerical experiments on function approximation and PDE benchmarks, demonstrate that SFLI significantly improves the initial $\varepsilon$-rank, accelerates convergence, mitigates spectral bias, and enhances prediction accuracy. With the help of SILP, we only need to add one line of code to conventional existing algorithms.

math.NA↗

arXiv2022

Priori Error Estimate of Deep Mixed Residual Method for Elliptic PDEs

In this work, we derive a priori error estimate of the mixed residual method when solving some elliptic PDEs. Our work is the first theoretical study of this method. We prove that the neural network solutions will converge if we increase the training samples and network size without any constraint on the ratio of training samples to the network size. Besides, our results suggest that the mixed residual method can recover high order derivatives better than the deep Ritz method, which has also been verified by our numerical experiments.

math.NA↗

arXiv2021

A Local Deep Learning Method for Solving High Order Partial Differential Equations

At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly, and high order derivatives lack robustness for training purposes. We propose a novel approach to solving PDEs with high order derivatives by simultaneously approximating the function value and derivatives. We introduce intermediate variables to rewrite the PDEs into a system of low order differential equations as what is done in the local discontinuous Galerkin method. The intermediate variables and the solutions to the PDEs are simultaneously approximated by a multi-output deep neural network. By taking the residual of the system as a loss function, we can optimize the network parameters to approximate the solution. The whole process relies on low order derivatives. Numerous numerical examples are carried out to demonstrate that our local deep learning is efficient, robust, flexible, and is particularly well-suited for high-dimensional PDEs with high order derivatives.

math.NA↗