Machine learning-based parameter optimization for Müntz spectral methods
Wei Zeng, Chuanju Xu, Yiming Lu, Qian Wang
TL;DR
The paper tackles the lack of theoretical guidance for choosing the Müntz spectral method's exponent parameter $\lambda$ in low-regularity problems, notably time-fractional PDEs. It introduces an artificial neural network that learns a mapping from the time-fractional order $\mu$ to the optimal $\lambda$, trained offline on a 1D manufactured-solution dataset and then applied to 1D and 2D problems. The results show substantial accuracy gains, with exponential convergence in $N$ for NN-predicted $\lambda$, and strong generalization to higher dimensions, outperforming cubic spline interpolation in both prediction quality and training efficiency. This framework provides a flexible approach to parameter selection in spectral methods for low-regularity PDEs and can be extended to broader fractional-polynomial bases and problem classes.
Abstract
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.