PINN-TI: Physical Information embedded in Neural Networks for solving ordinary differential equations with Time-varying Inputs

Abstract Modeling and predicting the dynamics of multiphysics and multiscale systems with hidden physics methods is often costly, requiring different formulations and complex computer codes. There has been significant progress in solving differential equations using physical information embedded in neural networks, but solving and generalizing for differential equations with time-varying inputs is still an open problem. In this paper, we design a PINN-TI network structure and a Step-by-Step Forward (SSF) algorithm. It enables the network to predict the results of ordinary differential equations under different initial values or boundary conditions and different time-varying inputs (including segmented functions) after one training. Our approach can extend the time domain of the solution to a larger range. We validated our model using a variant of Vander Pol’s equation, which contains one or two external inputs. The results show that our network can handle the prediction and generalization problem of ordinary differential equations under time-varying inputs well. Our method provides a new solution for solving and generalizing differential equations under time-varying inputs.