arXiv subjects

Xiaodong Feng

Publications and source records attributed to Xiaodong Feng.

12 recordsLinked to original sources

arXiv2026

FLUID: Flow-based Unified Inference for Dynamics

Bayesian filtering and smoothing for high-dimensional nonlinear dynamical systems are fundamental yet challenging problems in many areas of science and engineering. In this work, we propose FLUID, a flow-based unified amortized inference framework for filtering and smoothing dynamics. The core idea is to encode each observation history into a fixed-dimensional summary statistic and use this shared representation to learn both a forward flow for the filtering distribution and a backward flow for the backward transition kernel. Specifically, a recurrent encoder maps each observation history to a fixed-dimensional summary statistic whose dimension does not depend on the length of the time series. Conditioned on this shared summary statistic, the forward flow approximates the filtering distribution, while the backward flow approximates the backward transition kernel. The smoothing distribution over an entire trajectory is then recovered by combining the terminal filtering distribution with the learned backward flow through the standard backward recursion. By learning the underlying temporal evolution structure, FLUID also supports extrapolation beyond the training horizon. Moreover, by coupling the two flows through shared summary statistics, FLUID induces an implicit regularization across latent state trajectories and improves trajectory-level smoothing. In addition, we develop a flow-based particle filtering variant that provides an alternative filtering procedure and enables ESS-based diagnostics when explicit model factors are available. Numerical experiments demonstrate that FLUID provides accurate approximations of both filtering distributions and smoothing paths.

stat.ML↗

arXiv2026

Latent representation learning based model correction and uncertainty quantification for PDEs

Model correction is essential for reliable PDE learning when the governing physics is misspecified due to simplified assumptions or limited observations. In the machine learning literature, existing correction methods typically operate in parameter space, where uncertainty is often quantified via sampling or ensemble-based methods, which can be prohibitive and motivates more efficient representation-level alternatives. To this end, we develop a latent-space model-correction framework by extending our previously proposed LVM-GP solver, which couples latent-variable model with Gaussian processes (GPs) for uncertainty-aware PDE learning. Our architecture employs a shared confidence-aware encoder and two probabilistic decoders, with the solution decoder predicting the solution distribution and the correction decoder inferring a discrepancy term to compensate for model-form errors. The encoder constructs a stochastic latent representation by balancing deterministic features with a GP prior through a learnable confidence function. Conditioned on this shared latent representation, the two decoders jointly quantify uncertainty in both the solution and the correction under soft physics constraints with noisy data. An auxiliary latent-space regularization is introduced to control the learned representation and enhance robustness. This design enables joint uncertainty quantification of both the solution and the correction within a single training procedure, without parameter sampling or repeated retraining. Numerical experiments show accuracy comparable to Ensemble PINNs and B-PINNs, with improved computational efficiency and robustness to misspecified physics.

math.NA↗

arXiv2025

Overcoming Spectral Bias via Cross-Attention

Spectral bias implies an imbalance in training dynamics, whereby high-frequency components may converge substantially more slowly than low-frequency ones. To alleviate this issue, we propose a cross-attention-based architecture that adaptively reweights a scaled multiscale random Fourier feature bank with learnable scaling factors. The learnable scaling adjusts the amplitudes of the multiscale random Fourier features, while the cross-attention residual structure provides an input-dependent mechanism to emphasize the most informative scales. As a result, the proposed design accelerates high-frequency convergence relative to comparable baselines built on the same multiscale bank. Moreover, the attention module supports incremental spectral enrichment: dominant Fourier modes extracted from intermediate approximations via discrete Fourier analysis can be appended to the feature bank and used in subsequent training, without modifying the backbone architecture. We further extend this framework to PDE learning by introducing a linear combination of two sub-networks: one specialized in capturing high-frequency components of the PDE solution and the other in capturing low-frequency components, with a learnable (or optimally chosen) mixing factor to balance the two contributions and improve training efficiency in oscillatory regimes. Numerical experiments on high-frequency and discontinuous regression problems, image reconstruction tasks, as well as representative PDE examples, demonstrate the effectiveness and robustness of the proposed method.

math.NA↗

arXiv2025

LVM-GP: Uncertainty-Aware PDE Solver via coupling latent variable model and Gaussian process

We propose a novel probabilistic framework, termed LVM-GP, for uncertainty quantification in solving forward and inverse partial differential equations (PDEs) with noisy data. The core idea is to construct a stochastic mapping from the input to a high-dimensional latent representation, enabling uncertainty-aware prediction of the solution. Specifically, the architecture consists of a confidence-aware encoder and a probabilistic decoder. The encoder implements a high-dimensional latent variable model based on a Gaussian process (LVM-GP), where the latent representation is constructed by interpolating between a learnable deterministic feature and a Gaussian process prior, with the interpolation strength adaptively controlled by a confidence function learned from data. The decoder defines a conditional Gaussian distribution over the solution field, where the mean is predicted by a neural operator applied to the latent representation, allowing the model to learn flexible function-to-function mapping. Moreover, physical laws are enforced as soft constraints in the loss function to ensure consistency with the underlying PDE structure. Compared to existing approaches such as Bayesian physics-informed neural networks (B-PINNs) and deep ensembles, the proposed framework can efficiently capture functional dependencies via merging a latent Gaussian process and neural operator, resulting in competitive predictive accuracy and robust uncertainty quantification. Numerical experiments demonstrate the effectiveness and reliability of the method.

stat.ML↗

arXiv2025

Integral regularization PINNs for evolution equations

Evolution equations, including both ordinary differential equations (ODEs) and partial differential equations (PDEs), play a pivotal role in modeling dynamic systems. However, achieving accurate long-time integration for these equations remains a significant challenge. While physics-informed neural networks (PINNs) provide a mesh-free framework for solving PDEs, they often suffer from temporal error accumulation, which limits their effectiveness in capturing long-time behaviors. To alleviate this issue, we propose integral regularization PINNs (IR-PINNs), a novel approach that enhances temporal accuracy by incorporating an integral-based residual term into the loss function. This method divides the entire time interval into smaller sub-intervals and enforces constraints over these sub-intervals, thereby improving the resolution and correlation of temporal dynamics. Furthermore, IR-PINNs leverage adaptive sampling to dynamically refine the distribution of collocation points based on the evolving solution, ensuring higher accuracy in regions with sharp gradients or rapid variations. Numerical experiments on benchmark problems demonstrate that IR-PINNs outperform original PINNs and other state-of-the-art methods in capturing long-time behaviors, offering a robust and accurate solution for evolution equations.

math.NA↗

arXiv2025

Wearable intelligent throat enables natural speech in stroke patients with dysarthria

Wearable silent speech systems hold significant potential for restoring communication in patients with speech impairments. However, seamless, coherent speech remains elusive, and clinical efficacy is still unproven. Here, we present an AI-driven intelligent throat (IT) system that integrates throat muscle vibrations and carotid pulse signal sensors with large language model (LLM) processing to enable fluent, emotionally expressive communication. The system utilizes ultrasensitive textile strain sensors to capture high-quality signals from the neck area and supports token-level processing for real-time, continuous speech decoding, enabling seamless, delay-free communication. In tests with five stroke patients with dysarthria, IT's LLM agents intelligently corrected token errors and enriched sentence-level emotional and logical coherence, achieving low error rates (4.2% word error rate, 2.9% sentence error rate) and a 55% increase in user satisfaction. This work establishes a portable, intuitive communication platform for patients with dysarthria with the potential to be applied broadly across different neurological conditions and in multi-language support systems.

eess.AS↗

arXiv2024

A hybrid FEM-PINN method for time-dependent partial differential equations

In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based formulation where the neural network is defined on a spatiotemporal domain, our methodology utilizes finite element basis functions in the time direction where the space-dependent coefficients are defined as the output of a neural network. We then apply the Galerkin or collocation projection in the time direction to obtain a system of PDEs for the space-dependent coefficients which is approximated in the framework of PINN. The advantages of such a hybrid formulation are twofold: statistical errors are avoided for the integral in the time direction, and the neural network's output can be regarded as a set of reduced spatial basis functions. To further alleviate the difficulties from high dimensionality and low regularity, we have developed an adaptive sampling strategy that refines the training set. More specifically, we use an explicit density model to approximate the distribution induced by the PDE residual and then augment the training set with new time-dependent random samples given by the learned density model. The effectiveness and efficiency of our proposed method have been demonstrated through a series of numerical experiments.

math.NA↗

arXiv2022

MC-Nonlocal-PINNs: handling nonlocal operators in PINNs via Monte Carlo sampling

We propose, Monte Carlo Nonlocal physics-informed neural networks (MC-Nonlocal-PINNs), which is a generalization of MC-fPINNs in \cite{guo2022monte}, for solving general nonlocal models such as integral equations and nonlocal PDEs. Similar as in MC-fPINNs, our MC-Nonlocal-PINNs handle the nonlocal operators in a Monte Carlo way, resulting in a very stable approach for high dimensional problems. We present a variety of test problems, including high dimensional Volterra type integral equations, hypersingular integral equations and nonlocal PDEs, to demonstrate the effectiveness of our approach.

math.NA↗

arXiv2022

Gradient-enhanced deep neural network approximations

We propose in this work the gradient-enhanced deep neural networks (DNNs) approach for function approximations and uncertainty quantification. More precisely, the proposed approach adopts both the function evaluations and the associated gradient information to yield enhanced approximation accuracy. In particular, the gradient information is included as a regularization term in the gradient-enhanced DNNs approach, for which we present similar posterior estimates (by the two-layer neural networks) as those in the path-norm regularized DNNs approximations. We also discuss the application of this approach to gradient-enhanced uncertainty quantification, and present several numerical experiments to show that the proposed approach can outperform the traditional DNNs approach in many cases of interests.

cs.LG↗

arXiv2022

Solving time dependent Fokker-Planck equations via temporal normalizing flow

In this work, we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck (TFP) equations. It is well known that solutions of such equations are probability density functions, and thus our approach relies on modelling the target solutions with the temporal normalizing flows. The temporal normalizing flow is then trained based on the TFP loss function, without requiring any labeled data. Being a machine learning scheme, the proposed approach is mesh-free and can be easily applied to high dimensional problems. We present a variety of test problems to show the effectiveness of the learning approach.

cs.LG↗

arXiv2017

Robust Sparse Coding via Self-Paced Learning

Sparse coding (SC) is attracting more and more attention due to its comprehensive theoretical studies and its excellent performance in many signal processing applications. However, most existing sparse coding algorithms are nonconvex and are thus prone to becoming stuck into bad local minima, especially when there are outliers and noisy data. To enhance the learning robustness, in this paper, we propose a unified framework named Self-Paced Sparse Coding (SPSC), which gradually include matrix elements into SC learning from easy to complex. We also generalize the self-paced learning schema into different levels of dynamic selection on samples, features and elements respectively. Experimental results on real-world data demonstrate the efficacy of the proposed algorithms.

cs.LG↗

arXiv2015

Predicting Small Group Accretion in Social Networks: A topology based incremental approach

Small Group evolution has been of central importance in social sciences and also in the industry for understanding dynamics of team formation. While most of research works studying groups deal at a macro level with evolution of arbitrary size communities, in this paper we restrict ourselves to studying evolution of small group (size $\leq20$) which is governed by contrasting sociological phenomenon. Given a previous history of group collaboration between a set of actors, we address the problem of predicting likely future group collaborations. Unfortunately, predicting groups requires choosing from $n \choose r$ possibilities (where $r$ is group size and $n$ is total number of actors), which becomes computationally intractable as group size increases. However, our statistical analysis of a real world dataset has shown that two processes: an external actor joining an existing group (incremental accretion (IA)) or collaborating with a subset of actors of an exiting group (subgroup accretion (SA)), are largely responsible for future group formation. This helps to drastically reduce the $n\choose r$ possibilities. We therefore, model the attachment of a group for different actors outside this group. In this paper, we have built three topology based prediction models to study these phenomena. The performance of these models is evaluated using extensive experiments over DBLP dataset. Our prediction results shows that the proposed models are significantly useful for future group predictions both for IA and SA.

cs.SI↗