Constrained physics informed deep implicit neural network for ordinary and partial differential equations

This paper introduces a new physics-informed neural network architecture, the constrained physics-informed deep implicit neural network. This architecture utilizes an infinitely deep network and a constrained expression based on the theory of functional connections to solve differential equations. The main novelty of the proposed method is it helps to overcome the need to specify the number of hidden layers required in the network architecture by using infinitely deep network. Further, the usage of constrained expression helps satisfy the constraints of the problem analytically and thus reduces the problem to an unconstrained problem, which helps to increase accuracy. A fixed point solver handles the forward propagation, and the backpropagation uses phantom gradients. The proposed approach is tested on various standard ODEs and PDEs. The efficiency of the network in producing accurate solutions is verified using various error metrics and statistical analysis.