Machine Learning-Based Parameter Optimization for Müntz Spectral Methods

Abstract. Spectral methods employing nonstandard polynomial bases, such as Müntz polynomials, have proven effective for accurately solving singular problems. However, due to the absence of theoretical guidance, the key parameter [Formula: see text] that controls the exponents of Müntz polynomials is usually determined empirically through extensive numerical experiments, leading to a time-consuming tuning process. To address this bottleneck, we propose a 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 [Formula: see text] based solely on the time-fractional order [Formula: see text]. The ANN is trained by minimizing the mean solution error on a set of sampled one-dimensional time-fractional convection-diffusion equations. Numerical results for time-fractional PDEs 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.