Pingwen Zhang

Yixiang Luo, Jie Xu, Pingwen Zhang

We propose a fast algorithm for evaluating the moments of Bingham distribution. The calculation is done by piecewise rational approximation, where interpolation and Gaussian integrals are utilized. Numerical test shows that the algorithm reaches the maximal absolute error less than 5e-8 remarkably faster than adaptive numerical quadrature. We apply the algorithm to a model for liquid crystals with the Bingham distribution to examine the defect patterns of rod-like molecules confined in a sphere, and find a different pattern from the Landau-de Gennes theory.

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.

Yuling Jiao, Ruoxuan Li, Peiying Wu et al.

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?

Zhuoyuan Li, Bin Dong, Pingwen Zhang

Data assimilation is crucial in a wide range of applications, but it often faces challenges such as high computational costs due to data dimensionality and incomplete understanding of underlying mechanisms. To address these challenges, this study presents a novel assimilation framework, termed Latent Assimilation with Implicit Neural Representations (LAINR). By introducing Spherical Implicit Neural Representations (SINR) along with a data-driven uncertainty estimator of the trained neural networks, LAINR enhances efficiency in assimilation process. Experimental results indicate that LAINR holds certain advantage over existing methods based on AutoEncoders, both in terms of accuracy and efficiency.

Zhuoyuan Li, Bin Dong, Pingwen Zhang

Data assimilation has become a key technique for combining physical models with observational data to estimate state variables. However, classical assimilation algorithms often struggle with the high nonlinearity present in both physical and observational models. To address this challenge, a novel generative model, termed the State-Observation Augmented Diffusion (SOAD) model is proposed for data-driven assimilation. The marginal posterior associated with SOAD has been derived and then proved to match the true posterior distribution under mild assumptions, suggesting its theoretical advantages over previous score-based approaches. Experimental results also indicate that SOAD may offer improved performance compared to existing data-driven methods.

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.