Chuanju Xu

Field reconstruction methods based on machine learning often face challenges when handling unstructured data and designing complex model structures. On the other hand, Reduced Order Models (ROM) struggle with robustness in the prerence of observation noise and sensor vibrations. Striking a balance between lightweight design, the ability to handle unstructured data, and robustness presents a significant challenge. In this paper, we introduce the EIM-NN algorithm, which leverages neural networks to determine the coefficients of the subspace found by the EIM algorithm. Furthermore, we present the EIM-TNN algorithm, which enhances robustness by designing a loss function incorporating Tikhonov regularization. Our neural network consists of only two fully connected layers, allowing it to handle unstructured data while maintaining a lightweight profile. Experimental results demonstrate the algorithm’s ability to significantly enhance robustness against noise and sensor vibrations without compromising its accuracy in fitting the original data. Additionally, the algorithm’s lightweight nature ensures that the added training time and memory requirements over EIM remain acceptable, making it adaptable to a range of industrial applications.

The numerical simulation of incompressible flows is challenging due to the tight coupling of velocity and pressure. Projection methods offer an effective solution by decoupling these variables, making them suitable for large-scale computations. This work focuses on reduced-order modeling using incremental projection schemes for the Stokes equations. We present both semi-discrete and fully discrete formulations, employing BDF2 in time and finite elements in space. A proper orthogonal decomposition (POD) approach is adopted to construct a reduced-order model for the Stokes problem. The method enables explicit computation of reduced velocity and pressure while preserving accuracy. We provide a detailed stability analysis and derive error estimates, showing second-order convergence in time. Numerical experiments are conducted to validate the theoretical results and demonstrate computational efficiency.

Spectral methods employing non-standard polynomial bases, such as Müntz polynomials, have proven effective for accurately solving problems with solutions exhibiting low regularity, notably including sub-diffusion equations. However, due to the absence of theoretical guidance, the key parameters controlling the exponents of Müntz polynomials are usually determined empirically through extensive numerical experiments, leading to a time-consuming tuning process. To address this issue, we propose a novel machine learning-based optimization framework for the Müntz spectral method. As an illustrative example, we optimize the parameter selection for solving time-fractional partial differential equations (PDEs). Specifically, an artificial neural network (ANN) is employed to predict optimal parameter values based solely on the time-fractional order as input. The ANN is trained by minimizing solution errors on a one-dimensional time-fractional convection-diffusion equation featuring manufactured exact solutions that manifest singularities of varying intensity, covering a comprehensive range of sampled fractional orders. Numerical results for time-fractional PDEs in both one and two dimensions demonstrate that the ANN-based parameter prediction significantly improves the accuracy of the Müntz spectral method. Moreover, the trained ANN generalizes effectively from one-dimensional to two-dimensional cases, highlighting its robustness across spatial dimensions. Additionally, we verify that the ANN substantially outperforms traditional function approximators, such as spline interpolation, in both prediction accuracy and training efficiency. The proposed optimization framework can be extended beyond fractional PDEs, offering a versatile and powerful approach for spectral methods applied to various low-regularity problems.

The Müntz Legendre polynomials are a family of generalized orthogonal polynomials, defined by contour integral associated with a complex sequence Λ = { λ 0 , λ 1 , λ 2 , ⋯ } \Lambda =\{\lambda _{0},\lambda _{1},\lambda _{2},\cdots \} . In this paper, we are interested in two subclasses of the Müntz Legendre polynomials. Precisely, we theoretically and numerically investigate the basic approximation properties of the Müntz Legendre polynomials for two sets of Λ \Lambda sequences: λ k = λ \lambda _{k}=\lambda , and λ k = k λ + q \lambda _k=k\lambda +q for some λ \lambda and q q . First, the projection and interpolation errors are analyzed and numerically tested for each of the two subclasses of polynomials, and some error estimates are derived for functions in non-uniformly weighted Sobolev spaces. Then, in order to demonstrate the applicability of the Müntz polynomials, a Galerkin spectral method based on the Müntz Legendre polynomials is proposed to solve the time-space fractional differential equation. The obtained numerical results show that the proposed method leads to an exponential convergence rate even if the exact solutions are not smooth. This is opposed to low order algebraic convergence if traditional orthogonal polynomials are used.