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Chen-Song Zhang

Publications and source records attributed to Chen-Song Zhang.

At least 19 recordsLinked to original sources
arXiv2026

An Adaptive Subdomain Coupling Approach in Domain Decomposition for Multiphase Porous Media Flow

The numerical simulation of large-scale multiphase flow in porous media is of considerable importance across various application fields, particularly in the petroleum industry. The fully implicit method is preferred in reservoir simulations owing to its superior numerical stability and more relaxed time step constraints. However, this method requires solving a large nonlinear system, which becomes highly nonlinear in complex heterogeneous media with small grid scales, emphasizing the need for efficient and convergent numerical methods to accelerate nonlinear solvers on parallel computing systems. In this paper, we present an adaptively coupled subdomain framework based on domain decomposition methods. This framework effectively handles strong local nonlinearities in global problems by solving subproblems within the coupled regions. Furthermore, we propose several adaptive coupling strategies and present a novel method for calculating initial guesses, aimed at improving the convergence and scalability of nonlinear solvers. A series of numerical experiments validate the effectiveness and robustness of the proposed framework. Additionally, large-scale reservoir simulations demonstrate that the proposed method achieves competitive parallel performance.

math.NA↗
arXiv2026

Physics-informed neural networks for solving two-phase flow problems with moving interfaces

In this paper, a meshfree method using physics-informed neural networks (PINNs) is developed for solving two-phase flow problems with moving interfaces, where two immiscible fluids bearing different material properties, are separated by a dynamically evolving interface and interact with each other through interface conditions. Two kinds of distinct scenarios of interface motion are addressed: the prescribed interface motion whose moving velocity is explicitly given, and the solution-driven interface motion whose evolution is determined by the velocity field of two-phase flow. Based upon piecewise deep neural networks and spatiotemporal sampling points/training set in each fluid subdomain, the proposed PINNs framework reformulates the two-phase flow moving interface problem as a least-squares (LS) minimization problem, which involves all residuals of governing equations, interface conditions, boundary conditions and initial conditions. Furthermore, approximation properties of the proposed PINNs approach are analyzed rigorously for the presented two-phase flow model by employing the Reynolds transport theorem in evolving domains, moreover, a comprehensive error estimation is provided to account for additional complexities introduced by the moving interface and the coupling between fluid dynamics and interface evolution. Numerical experiments are carried out to illustrate the effectiveness of the proposed PINNs approach for various configurations of two-phase flow moving interface problems, and to validate the theoretical findings as well. A practical guidance is thus provided for an efficient training set distribution when applying the proposed PINNs approach to two-phase flow moving interface problems in practice.

math.NA↗
arXiv2026

Massively parallel Schwarz methods for the high frequency Helmholtz equation

We investigate the parallel one-level overlapping Schwarz method for solving finite element discretization of high-frequency Helmholtz equations. The resulting linear systems are large, indefinite, ill-conditioned, and complex-valued. We present a practical variant of the restricted additive Schwarz method with Perfectly Matched Layer transmission conditions (RAS-PML), which was originally analyzed in a theoretical setting in {\tt arXiv:2404.02156}, with some numerical experiments given in {\tt arXiv:2408.16580}. In our algorithm, the width of the overlap and the additional PML layer on each subdomain is allowed to decrease with $\mathcal{O}(k^{-1} \log(k))$, as the frequency $k \rightarrow \infty$, and this is observed to ensure good convergence while avoiding excessive communication. In experiments, the proposed method achieves $\mathcal{O}(k^d)$ parallel scalability under Cartesian domain decomposition and exhibits $\mathcal{O}(k)$ iteration counts and convergence time for $d$-dimensional Helmholtz problems ($d = 2,3$) as $k$ increases. In this preliminary note we restrict to experiments on 2D problems with constant wave speed. Details, analysis and extensions to variable wavespeed and 3D will be given in future work.

math.NA↗
arXiv2026

R-PINN: Recovery-type a-posteriori estimator enhanced adaptive PINN

In recent years, with the advancements in machine learning and neural networks, algorithms using physics-informed neural networks (PINNs) to solve PDEs have gained widespread applications. While these algorithms are well-suited for a wide range of equations, they often exhibit suboptimal performance when applied to equations with large local gradients, resulting in substantial localized errors. To address this issue, this paper proposes an adaptive PINN algorithm designed to improve accuracy in such cases. The core idea of the algorithm is to adaptively adjust the distribution of collocation points based on the recovery-type a-posterior error of the current numerical solution, enabling a better approximation of the true solution. This approach is inspired by the adaptive finite element method. By combining the recovery-type a-posteriori estimator, a gradient-recovery estimator commonly used in the adaptive finite element method (FEM) with PINNs, we introduce the Recovery-type a-posteriori estimator enhanced adaptive PINN (R-PINN) and compare its performance with a typical adaptive PINN algorithm, FI-PINN. Our results demonstrate that R-PINN achieves faster convergence with fewer adaptive points and significantly outperforms in the cases with multiple regions of large errors than FI-PINN. Notably, our method is a hybrid numerical approach for solving partial differential equations, integrating adaptive FEM with PINNs.

math.NA↗
arXiv2025

Recurrent Neural Operators: Stable Long-Term PDE Prediction

Neural operators have emerged as powerful tools for learning solution operators of partial differential equations. However, in time-dependent problems, standard training strategies such as teacher forcing introduce a mismatch between training and inference, leading to compounding errors in long-term autoregressive predictions. To address this issue, we propose Recurrent Neural Operators (RNOs)-a novel framework that integrates recurrent training into neural operator architectures. Instead of conditioning each training step on ground-truth inputs, RNOs recursively apply the operator to their own predictions over a temporal window, effectively simulating inference-time dynamics during training. This alignment mitigates exposure bias and enhances robustness to error accumulation. Theoretically, we show that recurrent training can reduce the worst-case exponential error growth typical of teacher forcing to linear growth. Empirically, we demonstrate that recurrently trained Multigrid Neural Operators significantly outperform their teacher-forced counterparts in long-term accuracy and stability on standard benchmarks. Our results underscore the importance of aligning training with inference dynamics for robust temporal generalization in neural operator learning.

cs.LG↗
arXiv2024

OpenCAEPoro: A Parallel Simulation Framework for Multiphase and Multicomponent Porous Media Flows

OpenCAEPoro is a parallel numerical simulation software developed in C++ for simulating multiphase and multicomponent flows in porous media. The software utilizes a set of general-purpose compositional model equations, enabling it to handle a diverse range of fluid dynamics, including the black oil model, compositional model, and thermal recovery models. OpenCAEPoro establishes a unified solving framework that integrates many widely used methods, such as IMPEC, FIM, and AIM. This framework allows dynamic collaboration between different methods. Specifically, based on this framework, we have developed an adaptively coupled domain decomposition method, which can provide initial solutions for global methods to accelerate the simulation. The reliability of OpenCAEPoro has been validated through benchmark testing with the SPE comparative solution project. Furthermore, its robust parallel efficiency has been tested in distributed parallel environments, demonstrating its suitability for large-scale simulation problems.

cs.MS↗
arXiv2024

A MgNO Method for Multiphase Flow in Porous Media

This research investigates the application of Multigrid Neural Operator (MgNO), a neural operator architecture inspired by multigrid methods, in the simulation for multiphase flow within porous media. The architecture is adjusted to manage a variety of crucial factors, such as permeability and porosity heterogeneity. The study extendes MgNO to time-dependent porous media flow problems and validate its accuracy in predicting essential aspects of multiphase flows. Furthermore, the research provides a detailed comparison between MgNO and Fourier Neural Opeartor (FNO), which is one of the most popular neural operator methods, on their performance regarding prediction error accumulation over time. This aspect provides valuable insights into the models' long-term predictive stability and reliability. The study demonstrates MgNO's capability to effectively simulate multiphase flow problems, offering considerable time savings compared to traditional simulation methods, marking an advancement in integrating data-driven methodologies in geoscience applications.

cs.CE↗
arXiv2023

A Survey on Intelligent Iterative Methods for Solving Sparse Linear Algebraic Equations

Efficiently solving sparse linear algebraic equations is an important research topic of numerical simulation. Commonly used approaches include direct methods and iterative methods. Compared with the direct methods, the iterative methods have lower computational complexity and memory consumption, and are thus often used to solve large-scale sparse linear equations. However, there are numerous iterative methods, parameters and components needed to be carefully chosen, and an inappropriate combination may eventually lead to an inefficient solution process in practice. With the development of deep learning, intelligent iterative methods become popular in these years, which can intelligently make a sufficiently good combination, optimize the parameters and components in accordance with the properties of the input matrix. This survey then reviews these intelligent iterative methods. To be clearer, we shall divide our discussion into three aspects: a method aspect, a component aspect and a parameter aspect. Moreover, we summarize the existing work and propose potential research directions that may deserve a deep investigation.

math.NA↗
arXiv2023

AutoAMG($θ$): An Auto-tuned AMG Method Based on Deep Learning for Strong Threshold

Algebraic Multigrid (AMG) is one of the most used iterative algorithms for solving large sparse linear equations $Ax=b$. In AMG, the coarse grid is a key component that affects the efficiency of the algorithm, the construction of which relies on the strong threshold parameter $θ$. This parameter is generally chosen empirically, with a default value in many current AMG solvers of 0.25 for 2D problems and 0.5 for 3D problems. However, for many practical problems, the quality of the coarse grid and the efficiency of the AMG algorithm are sensitive to $θ$; the default value is rarely optimal, and sometimes is far from it. Therefore, how to choose a better $θ$ is an important question. In this paper, we propose a deep learning based auto-tuning method, AutoAMG($θ$) for multiscale sparse linear equations, which are widely used in practical problems. The method uses Graph Neural Networks (GNNs) to extract matrix features, and a Multilayer Perceptron (MLP) to build the mapping between matrix features and the optimal $θ$, which can adaptively output $θ$ values for different matrices. Numerical experiments show that AutoAMG($θ$) can achieve significant speedup compared to the default $θ$ value.

math.NA↗
arXiv2022

An Improved Multi-Stage Preconditioner on GPUs for Compositional Reservoir Simulation

The compositional model is often used to describe multicomponent multiphase porous media flows in the petroleum industry. The fully implicit method with strong stability and weak constraints on time-step sizes is commonly used in the mainstream commercial reservoir simulators. In this paper, we develop an efficient multi-stage preconditioner for the fully implicit compositional flow simulation. The method employs an adaptive setup phase to improve the parallel efficiency on GPUs. Furthermore, a multi-color Gauss-Seidel algorithm based on the adjacency matrix is applied in the algebraic multigrid methods for the pressure part. Numerical results demonstrate that the proposed algorithm achieves good parallel speedup while yields the same convergence behavior as the corresponding sequential version.

math.NA↗
arXiv2022

A new analytical framework for the convergence of inexact two-grid methods

Two-grid methods with exact solution of the Galerkin coarse-grid system have been well studied by the multigrid community: an elegant identity has been established to characterize the convergence factor of exact two-grid methods. In practice, however, it is often too costly to solve the Galerkin coarse-grid system exactly, especially when its size is large. Instead, without essential loss of convergence speed, one may solve the coarse-grid system approximately. In this paper, we develop a new framework for analyzing the convergence of inexact two-grid methods: two-sided bounds for the energy norm of the error propagation matrix of inexact two-grid methods are presented. In the framework, a restricted smoother involved in the identity for exact two-grid convergence is used to measure how far the actual coarse-grid matrix deviates from the Galerkin one. As an application, we establish a unified convergence theory for multigrid methods.

math.NA↗
arXiv2021

Convergence analysis of inexact two-grid methods: A theoretical framework

Multigrid is one of the most efficient methods for solving large-scale linear systems that arise from discretized partial differential equations. As a foundation for multigrid analysis, two-grid theory plays an important role in motivating and analyzing multigrid algorithms. For symmetric positive definite problems, the convergence theory of two-grid methods with exact solution of the Galerkin coarse-grid system is mature, and the convergence factor of exact two-grid methods can be characterized by an identity. Compared with the exact case, the convergence theory of inexact two-grid methods (i.e., the coarse-grid system is solved approximately) is of more practical significance, while it is still less developed in the literature (one reason is that the error propagation matrix of inexact coarse-grid correction is not a projection). In this paper, we develop a theoretical framework for the convergence analysis of inexact two-grid methods. More specifically, we present two-sided bounds for the energy norm of the error propagation matrix of inexact two-grid methods, from which one can readily obtain the identity for exact two-grid convergence. As an application, we establish a unified convergence theory for multigrid methods, which allows the coarsest-grid system to be solved approximately.

math.NA↗
arXiv2021

A low-degree strictly conservative finite element method for incompressible flows

In this paper, a new $P_{2}-P_{1}$ finite element pair is proposed for incompressible fluid. For this pair, the discrete inf-sup condition and the discrete Korn's inequality hold on general triangulations. It yields exactly divergence-free velocity approximations when applied to models of incompressible flows. The robust capacity of the pair for incompressible flows are verified theoretically and numerically.

math.NA↗
arXiv2020

Lowest-degree robust finite element scheme for a fourth-order elliptic singular perturbation problem on rectangular grids

In this paper, a piecewise quadratic nonconforming finite element method on rectangular grids for a fourth-order elliptic singular perturbation problem is presented. This proposed method is robustly convergent with respect to the perturbation parameter. Numerical results are presented to verify the theoretical findings. The new method uses piecewise quadratic polynomials, and is of the lowest degree possible. Optimal order approximation property of the finite element space is proved by means of a locally-averaged interpolation operator newly constructed. This interpolator, however, is not a projection. Indeed, we establish a general theory and show that no locally defined interpolation associated with the locally supported basis functions can be projective for the finite element space in use. Particularly, the general theory gives an answer to a long-standing open problem presented in [Demko, J. Approx. Theory, $\bf{43}$(2):151--156, 1985].

math.NA↗
arXiv2017

On the Ideal Interpolation Operator in Algebraic Multigrid Methods

Various algebraic multigrid algorithms have been developed for solving problems in scientific and engineering computation over the past decades. They have been shown to be well-suited for solving discretized partial differential equations on unstructured girds in practice. One key ingredient of algebraic multigrid algorithms is a strategy for constructing an effective prolongation operator. Among many questions on constructing a prolongation, an important question is how to evaluate its quality. In this paper, we establish new characterizations (including sufficient condition, necessary condition, and equivalent condition) of the so-called ideal interpolation operator. Our result suggests that, compared with common wisdom, one has more room to construct an ideal interpolation, which can provide new insights for designing algebraic multigrid algorithms. Moreover, we derive a new expression for a class of ideal interpolation operators.

math.NA↗
arXiv2016

New perturbation bounds for the spectrum of a normal matrix

Let $A\in\mathbb{C}^{n\times n}$ and $\widetilde{A}\in\mathbb{C}^{n\times n}$ be two normal matrices with spectra $\{\lambda_{i}\}_{i=1}^{n}$ and $\{\widetilde{\lambda}_{i}\}_{i=1}^{n}$, respectively. The celebrated Hoffman--Wielandt theorem states that there exists a permutation $\pi$ of $\{1,\ldots,n\}$ such that $\left(\sum_{i=1}^{n}\big|\widetilde{\lambda}_{\pi(i)}-\lambda_{i}\big|^{2}\right)^{1\over 2}$ is no larger than the Frobenius norm of $\widetilde{A}-A$. However, if either $A$ or $\widetilde{A}$ is non-normal, this result does not hold in general. In this paper, we present several novel upper bounds for $\left(\sum_{i=1}^{n}\big|\widetilde{\lambda}_{\pi(i)}-\lambda_{i}\big|^{2}\right)^{1\over 2}$, provided that $A$ is normal and $\widetilde{A}$ is arbitrary. Some of these estimates involving the "departure from normality" of $\widetilde{A}$ have generalized the Hoffman--Wielandt theorem. Furthermore, we give new perturbation bounds for the spectrum of a Hermitian matrix.

math.NA↗
arXiv2014

An adaptive hybrid stress transition quadrilateral finite element method for linear elasticity

In this paper, we discuss an adaptive hybrid stress finite element method on quadrilateral meshes for linear elasticity problems. To deal with hanging nodes arising in the adaptive mesh refinement, we propose new transition types of hybrid stress quadrilateral elements with 5 to 7 nodes. In particular, we derive a priori error estimation for the 5-node transition hybrid stress element to show that it is free from Poisson-locking, in the sense that the error bound in the a priori estimate is independent of the Lame constant $\lambda$. We introduce, for quadrilateral meshes, refinement/coarsening algorithms, which do not require storing the refinement tree explicitly, and give an adaptive algorithm. Finally we provide some numerical results.

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