A robust Gated-PINN to resolve local minima issues in solving differential algebraic equations

Abstract

Physics-Informed Neural Network (PINN) has emerged as a promising tool for solving various physical problems with differential equations. However in practice, PINN often suffers from the local minima issue while solving problems with minimal initial conditions. In this paper, we propose a robust Gated-PINN to address this issue, which blends numerical methods and PINN. The Gated-PINN overcomes the local minima issue by controlling the flow of information of the neural network similar to a numerical method. We first investigate the use of conventional PINN to the Cartesian-coordinate simple pendulum problem with minimal initial conditions, as one of basic differential algebraic equation (DAE) problem. Our results show the advantage of PINN over existing numerical methods in that it does not require sophisticated mathematical techniques such as index reduction. But we also observe that conventional PINN can lead to inaccurate solutions; such solutions partially satisfy the differential equation requirement, but does not meet the given initial condition and fail to further improve. We demonstrate the effectiveness of the proposed Gated-PINN by showing that it yields accurate solutions, such that for a pendulum of length 1, the mean Euclidean error between the Gated-PINN model and the traditional numerical method model is less than 0.01 and pointwise maximum Euclidean error is less than 0.04. Moreover, Gated-PINN can operate without any complicated index reduction, and unlike conventional PINN, accurate solution can be obtained consistently without falling into a local minima. Overall, our study presents the potential of the Gated-PINN for solving DAE problems and provides a valuable insight into the challenges and limitations of using PINN for solving physical problems.