arXiv subjects

Jerry Zhijian Yang

Publications and source records attributed to Jerry Zhijian Yang.

At least 37 records · Page 2Linked to original sources

arXiv2025

Deep conditional distribution learning via conditional Föllmer flow

We introduce an ordinary differential equation (ODE) based deep generative method for learning conditional distributions, named Conditional Föllmer Flow. Starting from a standard Gaussian distribution, the proposed flow could approximate the target conditional distribution very well when the time is close to 1. For effective implementation, we discretize the flow with Euler's method where we estimate the velocity field nonparametrically using a deep neural network. Furthermore, we also establish the convergence result for the Wasserstein-2 distance between the distribution of the learned samples and the target conditional distribution, providing the first comprehensive end-to-end error analysis for conditional distribution learning via ODE flow. Our numerical experiments showcase its effectiveness across a range of scenarios, from standard nonparametric conditional density estimation problems to more intricate challenges involving image data, illustrating its superiority over various existing conditional density estimation methods.

stat.ML↗

arXiv2025

Online Quantum State Tomography via Stochastic Gradient Descent

We initiate the study of online quantum state tomography (QST), where the matrix representation of an unknown quantum state is reconstructed by sequentially performing a batch of measurements and updating the state estimate using only the measurement statistics from the current round. Motivated by recent advances in non-convex optimization algorithms for solving low-rank QST, we propose non-convex mini-batch stochastic gradient descent (SGD) algorithms to tackle online QST, which leverage the low-rank structure of the unknown quantum state and are well-suited for practical applications. Our main technical contribution is a rigorous convergence analysis of these algorithms. With proper initialization, we demonstrate that the SGD algorithms for online low-rank QST achieve linear convergence both in expectation and with high probability. Our algorithms achieve nearly optimal sample complexity while remaining highly memory-efficient. In particular, their time complexities are better than the state-of-the-art non-convex QST algorithms, in terms of the rank and the logarithm of the dimension of the unknown quantum state.

quant-ph↗

arXiv2025

CRPE: Expanding The Reasoning Capability of Large Language Model for Code Generation

We introduce CRPE (Code Reasoning Process Enhancer), an innovative three-stage framework for data synthesis and model training that advances the development of sophisticated code reasoning capabilities in large language models (LLMs). Building upon existing system-1 models, CRPE addresses the fundamental challenge of enhancing LLMs' analytical and logical processing in code generation tasks. Our framework presents a methodologically rigorous yet implementable approach to cultivating advanced code reasoning abilities in language models. Through the implementation of CRPE, we successfully develop an enhanced COT-Coder that demonstrates marked improvements in code generation tasks. Evaluation results on LiveCodeBench (20240701-20240901) demonstrate that our COT-Coder-7B-StepDPO, derived from Qwen2.5-Coder-7B-Base, with a pass@1 accuracy of 21.88, exceeds all models with similar or even larger sizes. Furthermore, our COT-Coder-32B-StepDPO, based on Qwen2.5-Coder-32B-Base, exhibits superior performance with a pass@1 accuracy of 35.08, outperforming GPT4O on the benchmark. Overall, CRPE represents a comprehensive, open-source method that encompasses the complete pipeline from instruction data acquisition through expert code reasoning data synthesis, culminating in an autonomous reasoning enhancement mechanism.

cs.SE↗

arXiv2025

Semi-Supervised Deep Sobolev Regression: Estimation and Variable Selection by ReQU Neural Network

We propose SDORE, a Semi-supervised Deep Sobolev Regressor, for the nonparametric estimation of the underlying regression function and its gradient. SDORE employs deep ReQU neural networks to minimize the empirical risk with gradient norm regularization, allowing the approximation of the regularization term by unlabeled data. Our study includes a thorough analysis of the convergence rates of SDORE in $L^{2}$-norm, achieving the minimax optimality. Further, we establish a convergence rate for the associated plug-in gradient estimator, even in the presence of significant domain shift. These theoretical findings offer valuable insights for selecting regularization parameters and determining the size of the neural network, while showcasing the provable advantage of leveraging unlabeled data in semi-supervised learning. To the best of our knowledge, SDORE is the first provable neural network-based approach that simultaneously estimates the regression function and its gradient, with diverse applications such as nonparametric variable selection. The effectiveness of SDORE is validated through an extensive range of numerical simulations.

stat.ML↗

arXiv2025

Deep Transfer Learning: Model Framework and Error Analysis

This paper presents a framework for deep transfer learning, which aims to leverage information from multi-domain upstream data with a large number of samples $n$ to a single-domain downstream task with a considerably smaller number of samples $m$, where $m \ll n$, in order to enhance performance on downstream task. Our framework offers several intriguing features. First, it allows the existence of both shared and domain-specific features across multi-domain data and provides a framework for automatic identification, achieving precise transfer and utilization of information. Second, the framework explicitly identifies upstream features that contribute to downstream tasks, establishing clear relationships between upstream domains and downstream tasks, thereby enhancing interpretability. Error analysis shows that our framework can significantly improve the convergence rate for learning Lipschitz functions in downstream supervised tasks, reducing it from $\tilde{O}(m^{-\frac{1}{2(d+2)}}+n^{-\frac{1}{2(d+2)}})$ ("no transfer") to $\tilde{O}(m^{-\frac{1}{2(d^*+3)}} + n^{-\frac{1}{2(d+2)}})$ ("partial transfer"), and even to $\tilde{O}(m^{-1/2}+n^{-\frac{1}{2(d+2)}})$ ("complete transfer"), where $d^* \ll d$ and $d$ is the dimension of the observed data. Our theoretical findings are supported by empirical experiments on image classification and regression datasets.

cs.LG↗

arXiv2024

A weighted scalar auxiliary variable method for solving gradient flows: bridging the nonlinear energy-based and Lagrange multiplier approaches

Two primary scalar auxiliary variable (SAV) approaches are widely applied for simulating gradient flow systems, i.e., the nonlinear energy-based approach and the Lagrange multiplier approach. The former guarantees unconditional energy stability through a modified energy formulation, whereas the latter preserves original energy stability but requires small time steps for numerical solutions. In this paper, we introduce a novel weighted SAV method which integrates these two approaches for the first time. Our method leverages the advantages of both approaches: (i) it ensures the existence of numerical solutions for any time step size with a sufficiently large weight coefficient; (ii) by using a weight coefficient smaller than one, it achieves a discrete energy closer to the original, potentially ensuring stability under mild conditions; and (iii) it maintains consistency in computational cost by utilizing the same time/spatial discretization formulas. We present several theorems and numerical experiments to validate the accuracy, energy stability and superiority of our proposed method.

math.NA↗

arXiv2024

Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuits

Parameterized quantum circuits (PQCs) have emerged as a promising approach for quantum neural networks. However, understanding their expressive power in accomplishing machine learning tasks remains a crucial question. This paper investigates the expressivity of PQCs for approximating general multivariate function classes. Unlike previous Universal Approximation Theorems for PQCs, which are either nonconstructive or rely on parameterized classical data processing, we explicitly construct data re-uploading PQCs for approximating multivariate polynomials and smooth functions. We establish the first non-asymptotic approximation error bounds for these functions in terms of the number of qubits, quantum circuit depth, and number of trainable parameters. Notably, we demonstrate that for approximating functions that satisfy specific smoothness criteria, the quantum circuit size and number of trainable parameters of our proposed PQCs can be smaller than those of deep ReLU neural networks. We further validate the approximation capability of PQCs through numerical experiments. Our results provide a theoretical foundation for designing practical PQCs and quantum neural networks for machine learning tasks that can be implemented on near-term quantum devices, paving the way for the advancement of quantum machine learning.

quant-ph↗

arXiv2024

DRM Revisited: A Complete Error Analysis

In this work, we address a foundational question in the theoretical analysis of the Deep Ritz Method (DRM) under the over-parameteriztion regime: Given a target precision level, how can one determine the appropriate number of training samples, the key architectural parameters of the neural networks, the step size for the projected gradient descent optimization procedure, and the requisite number of iterations, such that the output of the gradient descent process closely approximates the true solution of the underlying partial differential equation to the specified precision?

math.NA↗

arXiv2024

Solving the inverse source problem of the fractional Poisson equation by MC-fPINNs

In this paper, we effectively solve the inverse source problem of the fractional Poisson equation using MC-fPINNs. We construct two neural networks $ u_{NN}(x;θ)$ and $f_{NN}(x;ψ)$ to approximate the solution $u^{*}(x)$ and the forcing term $f^{*}(x)$ of the fractional Poisson equation. To optimize these two neural networks, we use the Monte Carlo sampling method mentioned in MC-fPINNs and define a new loss function combining measurement data and the underlying physical model. Meanwhile, we present a comprehensive error analysis for this method, along with a prior rule to select the appropriate parameters of neural networks. Several numerical examples are given to demonstrate the great precision and robustness of this method in solving high-dimensional problems up to 10D, with various fractional order $α$ and different noise levels of the measurement data ranging from 1$\%$ to 10$\%$.

math.NA↗

arXiv2024

A Stabilized Physics Informed Neural Networks Method for Wave Equations

In this article, we propose a novel Stabilized Physics Informed Neural Networks method (SPINNs) for solving wave equations. In general, this method not only demonstrates theoretical convergence but also exhibits higher efficiency compared to the original PINNs. By replacing the $L^2$ norm with $H^1$ norm in the learning of initial condition and boundary condition, we theoretically proved that the error of solution can be upper bounded by the risk in SPINNs. Based on this, we decompose the error of SPINNs into approximation error, statistical error and optimization error. Furthermore, by applying the approximating theory of $ReLU^3$ networks and the learning theory on Rademacher complexity, covering number and pseudo-dimension of neural networks, we present a systematical non-asymptotic convergence analysis on our method, which shows that the error of SPINNs can be well controlled if the number of training samples, depth and width of the deep neural networks have been appropriately chosen. Two illustrative numerical examples on 1-dimensional and 2-dimensional wave equations demonstrate that SPINNs can achieve a faster and better convergence than classical PINNs method.

math.NA↗

arXiv2023

Current density impedance imaging with PINNs

In this paper, we introduce CDII-PINNs, a computationally efficient method for solving CDII using PINNs in the framework of Tikhonov regularization. This method constructs a physics-informed loss function by merging the regularized least-squares output functional with an underlying differential equation, which describes the relationship between the conductivity and voltage. A pair of neural networks representing the conductivity and voltage, respectively, are coupled by this loss function. Then, minimizing the loss function provides a reconstruction. A rigorous theoretical guarantee is provided. We give an error analysis for CDII-PINNs and establish a convergence rate, based on prior selected neural network parameters in terms of the number of samples. The numerical simulations demonstrate that CDII-PINNs are efficient, accurate and robust to noise levels ranging from $1\%$ to $20\%$.

math.NA↗

arXiv2023

Deep Neural Network Approximation of Composition Functions: with application to PINNs

In this paper, we focus on approximating a natural class of functions that are compositions of smooth functions. Unlike the low-dimensional support assumption on the covariate, we demonstrate that composition functions have an intrinsic sparse structure if we assume each layer in the composition has a small degree of freedom. This fact can alleviate the curse of dimensionality in approximation errors by neural networks. Specifically, by using mathematical induction and the multivariate Faa di Bruno formula, we extend the approximation theory of deep neural networks to the composition functions case. Furthermore, combining recent results on the statistical error of deep learning, we provide a general convergence rate analysis for the PINNs method in solving elliptic equations with compositional solutions. We also present two simple illustrative numerical examples to demonstrate the effect of the intrinsic sparse structure in regression and solving PDEs.

math.NA↗

arXiv2023

GAS: A Gaussian Mixture Distribution-Based Adaptive Sampling Method for PINNs

With the recent study of deep learning in scientific computation, the Physics-Informed Neural Networks (PINNs) method has drawn widespread attention for solving Partial Differential Equations (PDEs). Compared to traditional methods, PINNs can efficiently handle high-dimensional problems, but the accuracy is relatively low, especially for highly irregular problems. Inspired by the idea of adaptive finite element methods and incremental learning, we propose GAS, a Gaussian mixture distribution-based adaptive sampling method for PINNs. During the training procedure, GAS uses the current residual information to generate a Gaussian mixture distribution for the sampling of additional points, which are then trained together with historical data to speed up the convergence of the loss and achieve higher accuracy. Several numerical simulations on 2D and 10D problems show that GAS is a promising method that achieves state-of-the-art accuracy among deep solvers, while being comparable with traditional numerical solvers.

cs.LG↗

arXiv2023

A sturcture-preserving, upwind-SAV scheme for the degenerate Cahn--Hilliard equation with applications to simulating surface diffusion

This paper establishes a structure-preserving numerical scheme for the Cahn--Hilliard equation with degenerate mobility. First, by applying a finite volume method with upwind numerical fluxes to the degenerate Cahn--Hilliard equation rewritten by the scalar auxiliary variable (SAV) approach, we creatively obtain an unconditionally bound-preserving, energy-stable and fully-discrete scheme, which, for the first time, addresses the boundedness of the classical SAV approach under $H^{-1}$-gradient flow. Then, a dimensional-splitting technique is introduced in high-dimensional cases, which greatly reduces the computational complexity while preserves original structural properties. Numerical experiments are presented to verify the bound-preserving and energy-stable properties of the proposed scheme. Finally, by applying the proposed structure-preserving scheme, we numerically demonstrate that surface diffusion can be approximated by the Cahn--Hilliard equation with degenerate mobility and Flory--Huggins potential when the absolute temperature is sufficiently low, which agrees well with the theoretical result by using formal asymptotic analysis.wn theoretically by formal matched asymptotics.

math.NA↗

arXiv2022

Stability and convergence analysis of high-order numerical schemes with DtN-type absorbing boundary conditions for nonlocal wave equations

The stability and convergence analysis of high-order numerical approximations for the one- and two-dimensional nonlocal wave equations on unbounded spatial domains are considered. We first use the quadrature-based finite difference schemes to discretize the spatially nonlocal operator, and apply the explicit difference scheme to approximate the temporal derivative to achieve a fully discrete infinity system. After that, we construct the Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs) to reduce the infinite discrete system into a finite discrete system. To do so, we first adopt the idea in [Du, Zhang and Zheng, \emph{Commun. Comput. Phys.}, 24(4):1049--1072, 2018 and Du, Han, Zhang and Zheng, \emph{SIAM J. Sci. Comp.}, 40(3):A1430--A1445, 2018] to derive the Dirichlet-to-Dirichlet (DtD)-type mappings for one- and two-dimensional cases, respectively. We then use the discrete nonlocal Green's first identity to achieve the discrete DtN-type mappings from the DtD-type mappings. The resulting DtN-type mappings make it possible to perform the stability and convergence analysis of the reduced problem. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed approach.

math.NA↗

arXiv2022

Global Optimization via Schr{ö}dinger-F{ö}llmer Diffusion

We study the problem of finding global minimizers of $V(x):\mathbb{R}^d\rightarrow\mathbb{R}$ approximately via sampling from a probability distribution $μ_σ$ with density $p_σ(x)=\dfrac{\exp(-V(x)/σ)}{\int_{\mathbb R^d} \exp(-V(y)/σ) dy }$ with respect to the Lebesgue measure for $σ\in (0,1]$ small enough. We analyze a sampler based on the Euler-Maruyama discretization of the Schr{ö}dinger-F{ö}llmer diffusion processes with stochastic approximation under appropriate assumptions on the step size $s$ and the potential $V$. We prove that the output of the proposed sampler is an approximate global minimizer of $V(x)$ with high probability at cost of sampling $\mathcal{O}(d^{3})$ standard normal random variables. Numerical studies illustrate the effectiveness of the proposed method and its superiority to the Langevin method.

math.OC↗

arXiv2022

Deep Neural Networks with ReLU-Sine-Exponential Activations Break Curse of Dimensionality in Approximation on Hölder Class

In this paper, we construct neural networks with ReLU, sine and $2^x$ as activation functions. For general continuous $f$ defined on $[0,1]^d$ with continuity modulus $ω_f(\cdot)$, we construct ReLU-sine-$2^x$ networks that enjoy an approximation rate $\mathcal{O}(ω_f(\sqrt{d})\cdot2^{-M}+ω_{f}\left(\frac{\sqrt{d}}{N}\right))$, where $M,N\in \mathbb{N}^{+}$ denote the hyperparameters related to widths of the networks. As a consequence, we can construct ReLU-sine-$2^x$ network with the depth $5$ and width $\max\left\{\left\lceil2d^{3/2}\left(\frac{3μ}ε\right)^{1/α}\right\rceil,2\left\lceil\log_2\frac{3μd^{α/2}}{2ε}\right\rceil+2\right\}$ that approximates $f\in \mathcal{H}_μ^α([0,1]^d)$ within a given tolerance $ε>0$ measured in $L^p$ norm $p\in[1,\infty)$, where $\mathcal{H}_μ^α([0,1]^d)$ denotes the Hölder continuous function class defined on $[0,1]^d$ with order $α\in (0,1]$ and constant $μ> 0$. Therefore, the ReLU-sine-$2^x$ networks overcome the curse of dimensionality on $\mathcal{H}_μ^α([0,1]^d)$. In addition to its supper expressive power, functions implemented by ReLU-sine-$2^x$ networks are (generalized) differentiable, enabling us to apply SGD to train.

cs.LG↗

arXiv2022

A rate of convergence of Physics Informed Neural Networks for the linear second order elliptic PDEs

In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition, by establishing the upper bounds on the number of training samples, depth and width of the deep neural networks to achieve desired accuracy. The error of PINNs is decomposed into approximation error and statistical error, where the approximation error is given in $C^2$ norm with $\mathrm{ReLU}^{3}$ networks (deep network with activations function $\max\{0,x^3\}$) and the statistical error is estimated by Rademacher complexity. We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with $\mathrm{ReLU}^{3}$ network, which is of immense independent interest.

math.NA↗