Integration of physics-informed operator learning and finite element method for parametric learning of partial differential equations

We present a method that employs physics-informed deep learning techniquesfor parametrically solving partial differential equations. The focus is on thesteady-state heat equations within heterogeneous solids exhibiting significantphase contrast. Similar equations manifest in diverse applications likechemical diffusion, electrostatics, and Darcy flow. The neural network aims toestablish the link between the complex thermal conductivity profiles andtemperature distributions, as well as heat flux components within themicrostructure, under fixed boundary conditions. A distinctive aspect is ourindependence from classical solvers like finite element methods for data. Anoteworthy contribution lies in our novel approach to defining the lossfunction, based on the discretized weak form of the governing equation. Thisnot only reduces the required order of derivatives but also eliminates the needfor automatic differentiation in the construction of loss terms, acceptingpotential numerical errors from the chosen discretization method. As a result,the loss function in this work is an algebraic equation that significantlyenhances training efficiency. We benchmark our methodology against the standardfinite element method, demonstrating accurate yet faster predictions using thetrained neural network for temperature and flux profiles. We also show higheraccuracy by using the proposed method compared to purely data-driven approachesfor unforeseen scenarios.