Data-driven and physics-informed deep learning operators for solution of heat conduction equation with parametric heat source

Deep neural networks as universal approximators of partial differential equations (PDEs) have attracted attention in numerous scientific and technical circles with the introduction of Physics-informed Neural Networks (PINNs). However, in most existing approaches, PINN can only provide solutions for defined input parameters, such as source terms, loads, boundaries, and initial conditions. Any modification in such parameters necessitates retraining or transfer learning. Classical numerical techniques are no exception, as each new input parameter value necessitates a new independent simulation. Unlike PINNs, which approximate solution functions, DeepONet approximates linear and nonlinear PDE solution operators by using parametric functions (infinite-dimensional objects) as inputs and mapping them to different PDE solution function output spaces. We devise, apply, and compare data-driven and physics-informed DeepONet models to solve the heat conduction (Poisson's) equation, one of the most common PDEs in science and engineering, using the variable and spatially multi-dimensional source term as its parameter. We provide novel computational insights into the DeepONet learning process of PDE solution with spatially multi-dimensional parametric input functions. We also show that, after being adequately trained, the proposed frameworks can reliably and almost instantly predict the parametric solution while being orders of magnitude faster than classical numerical solvers and without any additional training.