Reduced-PINN: An Integration-Based Physics-Informed Neural Networks for Stiff ODEs

Abstract

Physics-informed neural networks (PINNs) have recently received much attention due to their capabilities in solving both forward and inverse problems. For training a deep neural network associated with a PINN, one typically constructs a total loss function using a weighted sum of different loss terms and then tries to minimize that. This approach often becomes problematic for solving stiff equations since it cannot consider adaptive increments. Many studies reported the poor performance of the PINN and its challenges in simulating stiff chemical active issues with administering conditions of stiff ordinary differential conditions (ODEs). Studies show that stiffness is the primary cause of the failure of the PINN in simulating stiff kinetic systems. Here, we address this issue by proposing a reduced weak-form of the loss function, which led to a new PINN architecture, further named as Reduced-PINN, that utilizes a reduced-order integration method to enable the PINN to solve stiff chemical kinetics. The proposed Reduced-PINN can be applied to various reaction-diffusion systems involving stiff dynamics. To this end, we transform initial value problems (IVPs) to their equivalent integral forms and solve the resulting integral equations using physics-informed neural networks. In our derived integral-based optimization process, there is only one term without explicitly incorporating loss terms associated with ordinary differential equation (ODE) and initial conditions (ICs). To illustrate the capabilities of Reduced-PINN, we used it to simulate multiple stiff/mild second-order ODEs. We show that Reduced-PINN captures the solution accurately for a stiff scalar ODE. We also validated the Reduced-PINN against a stiff system of linear ODEs.