Chenguang Duan

Chenguang Duan, Yuling Jiao, Xiliang Lu et al.

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 $20\%$.

Zhao Ding, Chenguang Duan, Yuling Jiao et al.

This paper introduces score-based sequential Langevin sampling (SSLS), a novel approach to nonlinear data assimilation within a recursive Bayesian filtering framework. The proposed method decomposes the assimilation process into alternating prediction and update steps, using dynamic models for state prediction and incorporating observational data via score-based Langevin Monte Carlo during the updates. To overcome inherent challenges in highly non-log-concave posterior sampling, we integrate an annealing strategy into the update mechanism. Theoretically, we establish convergence guarantees for SSLS in total variation (TV) distance, yielding concrete insights into the algorithm's error behavior with respect to key hyperparameters. Crucially, our derived error bounds demonstrate the asymptotic stability of SSLS, guaranteeing that local posterior sampling errors do not accumulate indefinitely over time. Extensive numerical experiments across challenging scenarios, including high-dimensional systems, strong nonlinearity, and sparse observations, highlight the robust performance of the proposed method. Furthermore, SSLS effectively quantifies the uncertainty associated with state estimates, rendering it particularly valuable for reliable error calibration.

Chenguang Duan, Yuling Jiao, Gabriele Steidl et al.

We propose a novel method for sampling from unnormalized Boltzmann densities based on a probability-flow ordinary differential equation (ODE) derived from linear stochastic interpolants. The key innovation of our approach is the use of a sequence of Langevin samplers to enable efficient simulation of the flow. Specifically, these Langevin samplers are employed (i) to generate samples from the interpolant distribution at intermediate times and (ii) to construct, starting from these intermediate times, a robust estimator of the velocity field governing the flow ODE. For both applications of the Langevin diffusions, we establish convergence guarantees. Extensive numerical experiments demonstrate the efficiency of the proposed method on challenging multimodal distributions across a range of dimensions, as well as its effectiveness in Bayesian inference tasks.

Jinyuan Chang, Chenguang Duan, Yuling Jiao et al.

This paper proposes a novel diffusion-based posterior sampling method within a plug-and-play (PnP) framework. Our approach constructs a probability transport from an easy-to-sample terminal distribution to the target posterior, using a warm-start strategy to initialize the particles. To approximate the posterior score, we develop a Monte Carlo estimator in which particles are generated using Langevin dynamics, avoiding the heuristic approximations commonly used in prior work. The score governing the Langevin dynamics is learned from data, enabling the model to capture rich structural features of the underlying prior distribution. On the theoretical side, we provide non-asymptotic error bounds, showing that the method converges even for complex, multi-modal target posterior distributions. These bounds explicitly quantify the errors arising from posterior score estimation, the warm-start initialization, and the posterior sampling procedure. Our analysis further clarifies how the prior score-matching error and the condition number of the Bayesian inverse problem influence overall performance. Finally, we present numerical experiments demonstrating the effectiveness of the proposed method across a range of inverse problems.

Chenguang Duan, Yuling Jiao, Huazhen Lin et al.

Learning transferable data representations from abundant unlabeled data remains a central challenge in machine learning. Although numerous self-supervised learning methods have been proposed to address this challenge, a significant class of these approaches aligns the covariance or correlation matrix with the identity matrix. Despite impressive performance across various downstream tasks, these methods often suffer from biased sample risk, leading to substantial optimization shifts in mini-batch settings and complicating theoretical analysis. In this paper, we introduce a novel \underline{\bf Adv}ersarial \underline{\bf S}elf-\underline{\bf S}upervised Representation \underline{\bf L}earning (Adv-SSL) for unbiased transfer learning with no additional cost compared to its biased counterparts. Our approach not only outperforms the existing methods across multiple benchmark datasets but is also supported by comprehensive end-to-end theoretical guarantees. Our analysis reveals that the minimax optimization in Adv-SSL encourages representations to form well-separated clusters in the embedding space, provided there is sufficient upstream unlabeled data. As a result, our method achieves strong classification performance even with limited downstream labels, shedding new light on few-shot learning.

Zhao Ding, Chenguang Duan, Yuling Jiao et al.

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.

Zhao Ding, Chenguang Duan, Yuling Jiao et al.

We propose the characteristic generator, a novel one-step generative model that combines the efficiency of sampling in Generative Adversarial Networks (GANs) with the stable performance of flow-based models. Our model is driven by characteristics, along which the probability density transport can be described by ordinary differential equations (ODEs). Specifically, we first estimate the underlying velocity field and use the Euler method to solve the probability flow ODE, generating discrete approximations of the characteristics. A deep neural network is then trained to fit these characteristics, creating a one-step map that pushes a simple Gaussian distribution to the target distribution. In the theoretical aspect, we provide a comprehensive analysis of the errors arising from velocity matching, Euler discretization, and characteristic fitting to establish a non-asymptotic convergence rate in the 2-Wasserstein distance under mild data assumptions. Crucially, we demonstrate that under a standard manifold assumption, this convergence rate depends only on the intrinsic dimension of data rather than the much larger ambient dimension, proving our model's ability to mitigate the curse of dimensionality. To our knowledge, this is the first rigorous convergence analysis for a flow-based one-step generative model. Experiments on both synthetic and real-world datasets demonstrate that the characteristic generator achieves high-quality and high-resolution sample generation with the efficiency of just a single neural network evaluation.