Xiaofei Guan

Xiaofei Guan, Xintong Wang, Hao Wu et al.

In this paper, we introduce an innovative approach for addressing Bayesian inverse problems through the utilization of physics-informed invertible neural networks (PI-INN). The PI-INN framework encompasses two sub-networks: an invertible neural network (INN) and a neural basis network (NB-Net). The primary role of the NB-Net lies in modeling the spatial basis functions characterizing the solution to the forward problem dictated by the underlying partial differential equation. Simultaneously, the INN is designed to partition the parameter vector linked to the input physical field into two distinct components: the expansion coefficients representing the forward problem solution and the Gaussian latent noise. If the forward mapping is precisely estimated, and the statistical independence between expansion coefficients and latent noise is well-maintained, the PI-INN offers a precise and efficient generative model for Bayesian inverse problems, yielding tractable posterior density estimates. As a particular physics-informed deep learning model, the primary training challenge for PI-INN centers on enforcing the independence constraint, which we tackle by introducing a novel independence loss based on estimated density. We support the efficacy and precision of the proposed PI-INN through a series of numerical experiments, including inverse kinematics, 1-dimensional and 2-dimensional diffusion equations, and seismic traveltime tomography. Specifically, our experimental results showcase the superior performance of the proposed independence loss in comparison to the commonly used but computationally demanding kernel-based maximum mean discrepancy loss.

Yuzhou Nan, Yajun Wang, Changqing Ye et al.

Multiscale thermomechanical problems in highly heterogeneous media are challenging because the elastic, thermal, and coupling coefficients may vary on unresolved spatial scales. We propose a multiphysics-embedding localized orthogonal decomposition (ME-LOD) method in which displacement and temperature correctors are generated by a coupled static operator. The corrector problems are localized to coarse-grid patches and solved in the kernel of a projective quasi-interpolation operator. We prove uniform inf-sup stability on the global fine-scale kernel and on all zero-extension patch kernels, establish exponential decay of the coupled correctors and the resulting multiscale basis functions, and derive spatial approximation and fully discrete reduction estimates. Numerical experiments demonstrate that, for the tested periodic, random, and high-contrast coefficient fields, ME-LOD attains smaller errors than the comparison method at the same coarse resolution and patch size and can reach a prescribed accuracy with fewer oversampling layers. Although each coupled local corrector is more expensive than a decoupled corrector, the improved localization yields a favorable overall accuracy-to-cost balance in the reported tests.

Xiaofei Guan, Lijian Jiang, Yajun Wang et al.

This paper proposes localized subspace iteration (LSI) methods to construct generalized finite element basis functions for elliptic problems with multiscale coefficients. The key components of the proposed method consist of the localization of the original differential operator and the subspace iteration of the corresponding local spectral problems, where the localization is conducted by enforcing the local homogeneous Dirichlet condition and the partition of the unity functions. From a novel perspective, some multiscale methods can be regarded as one iteration step under approximating the eigenspace of the corresponding local spectral problems. Vice versa, new multiscale methods can be designed through subspaces of spectral problem algorithms. Then, we propose the efficient localized standard subspace iteration (LSSI) method and the localized Krylov subspace iteration (LKSI) method based on the standard subspace and Krylov subspace, respectively. Convergence analysis is carried out for the proposed method. Various numerical examples demonstrate the effectiveness of our methods. In addition, the proposed methods show significant superiority in treating long-channel cases over other well-known multiscale methods.

Shan Zhang, Yajun Wang, Xiaofei Guan

In this paper, we propose and analyze a new semi-implicit stochastic multiscale method for the radiative heat transfer problem with additive noise fluctuation in composite materials. In the proposed method, the strong nonlinearity term induced by heat radiation is first approximated, by a semi-implicit predictor-corrected numerical scheme, for each fixed time step, resulting in a spatially random multiscale heat transfer equation. Then, the infinite-dimensional stochastic processes are modeled and truncated using a complete orthogonal system, facilitating the reduction of the model's dimensionality in the random space. The resulting low-rank random multiscale heat transfer equation is approximated and computed by using efficient spatial basis functions based multiscale method. The main advantage of the proposed method is that it separates the computational difficulty caused by the spatial multiscale properties, the high-dimensional randomness and the strong nonlinearity of the solution, so they can be overcome separately using different strategies. The convergence analysis is carried out, and the optimal rate of convergence is also obtained for the proposed semi-implicit stochastic multiscale method. Numerical experiments on several test problems for composite materials with various microstructures are also presented to gauge the efficiency and accuracy of the proposed semi-implicit stochastic multiscale method.

Hang Qi, Changqing Ye, Xiaofei Guan

The Landau--Lifshitz--Gilbert equation poses significant challenges for numerical simulation due to its nonlinearity, nonconvex unit-length constraint, and nonlocal field contributions. Existing implicit or semi-implicit schemes exhibit unconditional stability but require repeated solution of nonlinear or linearized systems, resulting in high computational costs. In this work, we revisit the tangent plane formulation and show that the resulting discrete problem at each time step can be written as a generalized saddle-point system. Exploiting this structure, we develop a matrix-free preconditioner for the implicit Euler scheme by combining FFT-based techniques with a splitting iteration method. We show that each component in the splitting method can be efficiently implemented in a direct fashion via fast Poisson solvers, enabling the design of an efficient preconditioner for iterative solvers. To enhance geometric consistency of the tangent vector and reduce projection steps, a Crank--Nicolson scheme is further investigated, retaining the same algorithmic structure as the Euler scheme, thus allowing direct reuse of preconditioners and solvers. Extensive 2D and 3D numerical experiments are conducted to validate the framework, demonstrating accurate constraint preservation, improved computational efficiency, and robust solver performance. The proposed approach is shown to scale effectively for large-scale micromagnetic simulations, making it suitable for practical applications in complex magnetic systems.

Xintong Wang, Xiaofei Guan, Ling Guo et al.

Bayesian filtering for high-dimensional nonlinear stochastic dynamical systems is a fundamental yet challenging problem in many fields of science and engineering. Existing methods face significant obstacles: Gaussian-based filters struggle with non-Gaussian distributions, while sequential Monte Carlo methods are computationally intensive and prone to particle degeneracy in high dimensions. Although generative models in machine learning have made significant progress in modeling high-dimensional non-Gaussian distributions, their inefficiency in online updating limits their applicability to filtering problems. To address these challenges, we propose a flow-based Bayesian filter (FBF) that integrates normalizing flows to construct a novel latent linear state-space model with Gaussian filtering distributions. This framework facilitates efficient density estimation and sampling using invertible transformations provided by normalizing flows, and it enables the construction of filters in a data-driven manner, without requiring prior knowledge of system dynamics or observation models. Numerical experiments demonstrate the superior accuracy and efficiency of FBF.

Guoxin Fan, Huaqing Liu, Zhenhua Wu et al.

Background and objective: Combined evaluation of lumbosacral structures (e.g. nerves, bone) on multimodal radiographic images is routinely conducted prior to spinal surgery and interventional procedures. Generally, magnetic resonance imaging is conducted to differentiate nerves, while computed tomography (CT) is used to observe bony structures. The aim of this study is to investigate the feasibility of automatically segmenting lumbosacral structures (e.g. nerves & bone) on non-contrast CT with deep learning. Methods: a total of 50 cases with spinal CT were manually labeled for lumbosacral nerves and bone with Slicer 4.8. The ratio of training: validation: testing is 32:8:10. A 3D-Unet is adopted to build the model SPINECT for automatically segmenting lumbosacral structures. Pixel accuracy, IoU, and Dice score are used to assess the segmentation performance of lumbosacral structures. Results: the testing results reveals successful segmentation of lumbosacral bone and nerve on CT. The average pixel accuracy is 0.940 for bone and 0.918 for nerve. The average IoU is 0.897 for bone and 0.827 for nerve. The dice score is 0.945 for bone and 0.905 for nerve. Conclusions: this pilot study indicated that automatic segmenting lumbosacral structures (nerves and bone) on non-contrast CT is feasible and may have utility for planning and navigating spinal interventions and surgery.