Accurate and computationally efficient basis function generation using physics informed neural networks

Basis functions that can accurately represent simulated or measured acoustic pressure fields with a small number of degrees of freedom is of great use across various applications, including finite element methods, model order reduction, and compressive sensing. In a previous work [B. M. Goldsberry, J. Acoust. Soc. Am. 153, A193 (2023)], basis functions were derived for an element in a given mesh using a combination of interpolation functions defined on the boundaries of the element and the Helmholtz-Kirchhoff (HK) integral. This forms a new interpolatory basis set that efficiently and accurately represents the interior of the element. However, the previous analysis was limited to a two-dimensional rectangular element. In this work, physics informed neural networks (PINN) are investigated as a means to generate HK basis functions for general element shapes. PINNs have been previously shown to accurately learn solutions to parameterized partial differential equations. The element geometry parameterization and the boundary interpolation functions are given as inputs to the PINN, and the output of the PINN consists of the physically accurate basis functions within the element. Details on the implementation and training requirements on the PINN to achieve a desired accuracy will be discussed. [Work supported by ONR.]