Xiaozhe Hu

Machine learning techniques have recently been of great interest for solving differential equations. Training these models is classically a data-fitting task, but knowledge of the expression of the differential equation can be used to supplement the training objective, leading to the development of physics-informed scientific machine learning. In this article, we focus on one class of models called nonlinear vector autoregression (NVAR) to solve ordinary differential equations (ODEs). Motivated by connections to numerical integration and physics-informed neural networks, we explicitly derive the physics-informed NVAR (piNVAR) which enforces the right-hand side of the underlying differential equation regardless of NVAR construction. Because NVAR and piNVAR completely share their learned parameters, we propose an augmented procedure to jointly train the two models. This yields a statistical framework to capture and predict dynamic relationships between multiple time series variables with the extra knowledge that the observed data are from ODEs. Then, using both data-driven and ODE-driven metrics, we evaluate the ability of the piNVAR model to predict solutions to various ODE systems, such as the undamped spring, a Lotka–Volterra predator-prey nonlinear model, and the chaotic Lorenz system. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/samuelhocking/pinvar-sisc-2024 and in the supplementary materials ( pinvar-sisc-2024-main.zip [6.77MB]). [Formula: see text]

We consider the Stokes–Darcy coupled problem, which models the interaction between free flow and porous medium flow. By enforcing the normal flux continuity interface condition directly within the finite-element space, we establish unified well-posedness results for the coupled system under various boundary condition scenarios. Using the operator preconditioning framework, we develop a parameter-robust preconditioner that avoids the use of fractional operators. Numerical experiments employing both [Formula: see text]-conforming and nonconforming finite-element methods are presented to confirm the theoretical findings and demonstrate the robustness of the proposed block preconditioner with respect to the physical parameters and mesh size. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/xuesdu/SD.git and in the supplementary materials ( SD-master.zip [341KB]). [Formula: see text]

In this paper, we propose a parameter-robust preconditioner for the coupled Stokes–Darcy problem equipped with various boundary conditions, enforcing the mass conservation at the interface via a Lagrange multiplier. We rigorously establish that the coupled system is well-posed with respect to physical parameters and mesh size and provides a framework for constructing parameter-robust preconditioners. Furthermore, we analyze the convergence behavior of the minimal residual method in the presence of small outlier eigenvalues linked to specific boundary conditions, which can lead to slow convergence or stagnation. To address this issue, we employ deflation techniques to accelerate the convergence. Finally, numerical experiments confirm the effectiveness and robustness of the proposed approach.

An alternative to the fully implicit or monolithic methods used for the solution of the coupling of fluid flow and deformation in porous media is a sequential approach in which the fully coupled system is broken into subproblems (flow and mechanics problems) that are solved one after the other. This fully explicit coupling approach is a very simple scheme which allows much flexibility in the implementation and has a lower computational cost, making it quite attractive in practice since smaller linear systems need to be solved in order to obtain the solution for the whole coupled poroelastic system. Due to the appealing advantages of these methods, intensive research is currently being carried out in this direction, as in the present work. Although the application of this type of method is very common in practice, there exist only a few works devoted to their theoretical analysis. In this work, we consider the so-called explicit fixed-stress split scheme, which consists of solving the flow problem first with time-lagging the displacement term, followed by the solution of the mechanics problem. To the best of our knowledge, we provide the first convergence analysis of the explicit fixed-stress split scheme for Biot's equations. In particular, we prove that this algorithm is optimally convergent if the considered finite element discretization satisfies an inf-sup condition. In addition, with the aim of designing the simplest scheme for solving Biot's model, we also propose a similar decoupled algorithm for piecewise linear finite elements for both variables which arises from the novel stabilization recently proposed in \cite{Pe2025}, and is demonstrated to be optimally convergent.

Protein function prediction from multiplex protein-protein association networks is a crucial approach to extending functional annotation. Current methods use embeddings of the heterogeneous network data that aim to place related proteins near each other in embedding space. However, such embeddings suffer from spurious protein proximity as well, reducing function prediction accuracy. Because heterogeneous input networks often have very different structures, it is hard to confidently declare proteins to be dissimilar using the network structure or the resulting embeddings. Here we address this problem with REPEL, a function prediction tool using a random graph augmentation method that applies a uniform weak force to push nodes apart. We assess this method on simulated networks with planted overlapping communities, as well as on real multiplex yeast and E.coli protein association networks. Surprisingly, we find that this method consistently improves protein function prediction over competing methods Mashup, deepNF, and BIONIC. The random repelling nature of the augmented graphs has a denoising effect on the learning process, distancing node pairs with spurious proximity while preserving true functional connections, thus increasing robustness. This graph augmentation principle may generalize to denoising and improving robustness in other graph-based learning algorithms.

In this work, we present a new stabilization method aimed at removing spurious oscillations in the pressure approximation of Biot’s model for poroelasticity with low permeabilities and/or small time steps. We consider different finite-element discretizations and illustrate how not only does such a stabilized scheme provide numerical solutions that are free of nonphysical oscillations, but it also allows one to iterate the fluid and mechanics problems in a fashion similar to the well-known fixed-stress split method. The resulting iterative solution method is convergent without adding extra stabilization terms. Finally, we present numerical results illustrating the robust behavior of both the stabilization and iterative solver with respect to the physical and discretization parameters of the model.

When developing robust preconditioners for multiphysics problems, fractional functions of the Laplace operator often arise and need to be inverted. Rational approximation in the uniform norm can be used to convert inverting those fractional operators into inverting a series of shifted Laplace operators. Care must be taken in the approximation so that the shifted Laplace operators remain symmetric positive definite, making them better conditioned. In this work, we study two greedy algorithms for finding rational approximations to such fractional operators. The first algorithm improves the orthogonal greedy algorithm discussed in [Y. Li, L. Zikatanov, and C. Zuo, SIAM J. Sci. Comput., 46 (2024), pp. S68–S87] by adding one minimization step in the uniform norm to the procedure. The second approach employs the weak Chebyshev greedy algorithm in the uniform norm. Both methods yield nonincreasing error. Numerical results confirm the effectiveness of our proposed algorithms, which are also flexible and applicable to other approximation problems. Moreover, with effective rational approximations to the fractional operator, the resulting algorithms show good performance in preconditioning a Darcy–Stokes coupled problem.

In this work, we present a new stabilization method aimed at removing spurious oscillations in the pressure approximation of Biot's model for poroelasticity with low permeabilities and/or small time steps. We consider different finite-element discretizations and illustrate how not only does such a stabilized scheme provide numerical solutions that are free of non-physical oscillations, but it also allows one to iterate the fluid and mechanics problems in a fashion similar to the well-known fixed-stress split method. The resulting solution method is convergent without the necessity for additional terms to stabilize the iteration. Finally, we present numerical results illustrating the robust behavior of both the stabilization and iterative solver with respect to the physical and discretization parameters of the model.