Xin Tai | Searcharxiv

arXiv2023

Augmented physics informed extreme learning machine to solve the biharmonic equations via Fourier expansions

To address the sensitivity of parameters and limited precision for physics-informed extreme learning machines (PIELM) with common activation functions, such as sigmoid, tangent, and Gaussian, in solving high-order partial differential equations (PDEs) relevant to scientific computation and engineering applications, this work develops a Fourier-induced PIELM (FPIELM) method. This approach aims to approximate solutions for a class of fourth-order biharmonic equations with two boundary conditions on both unitized and non-unitized domains. By carefully calculating the differential and boundary operators of the biharmonic equation on discretized collections, the solution for this high-order equation is reformulated as a linear least squares minimization problem. We further evaluate the FPIELM with varying hidden nodes and scaling factors for uniform distribution initialization, and then determine the optimal range for these two hyperparameters. Numerical experiments and comparative analyses demonstrate that the proposed FPIELM method is more stable, robust, precise, and efficient than other PIELM approaches in solving biharmonic equations across both regular and irregular domains.

math.NA↗

arXiv2023

Solving a class of multi-scale elliptic PDEs by means of Fourier-based mixed physics informed neural networks

Deep neural networks have garnered widespread attention due to their simplicity and flexibility in the fields of engineering and scientific calculation. In this study, we probe into solving a class of elliptic partial differential equations(PDEs) with multiple scales by utilizing Fourier-based mixed physics informed neural networks(dubbed FMPINN), its solver is configured as a multi-scale deep neural network. In contrast to the classical PINN method, a dual (flux) variable about the rough coefficient of PDEs is introduced to avoid the ill-condition of neural tangent kernel matrix caused by the oscillating coefficient of multi-scale PDEs. Therefore, apart from the physical conservation laws, the discrepancy between the auxiliary variables and the gradients of multi-scale coefficients is incorporated into the cost function, then obtaining a satisfactory solution of PDEs by minimizing the defined loss through some optimization methods. Additionally, a trigonometric activation function is introduced for FMPINN, which is suited for representing the derivatives of complex target functions. Handling the input data by Fourier feature mapping will effectively improve the capacity of deep neural networks to solve high-frequency problems. Finally, to validate the efficiency and robustness of the proposed FMPINN algorithm, we present several numerical examples of multi-scale problems in various dimensional Euclidean spaces. These examples cover both low-frequency and high-frequency oscillation cases, demonstrating the effectiveness of our approach. All code and data accompanying this manuscript will be made publicly available at \href{https://github.com/Blue-Giant/FMPINN}{https://github.com/Blue-Giant/FMPINN}.

math.NA↗