Variable separated physics-informed neural networks based on adaptive weighted loss functions for blood flow model

Physics-informed neural networks (PINN) architectures have been recently explored to accelerate hemodynamics simulations by leveraging mathematical models for blood flow and empirical data. In this paper, a variable separated physics-informed neural networks based on adaptive weighted loss functions (AW-vsPINN) is developed for blood flow model in arteries. In particular, sub-neural networks are proposed to separately predict the unknown scalar state variables by sharing the same input layer. The AW-vsPINN adaptively adjusts the weights of loss terms by the minmax algorithm, which will be updated synchronously along with the network parameters and can balance the contributions of different loss terms during training. The two-stage optimization is implemented to train the neural networks. Specifically, the Adam optimizer is iterated for initial steps with the learning rate generated by the inverse time decay scheduler, and then the L-BFGS optimizer continues to train until the loss converges. Numerical results illustrate that the AW-vsPINN can remarkably improve prediction accuracy and enhance the ability of generalization compared to the conventional PINN. The proposed AW-vsPINN framework has high potential in predicting the blood flow information in cardiovascular disease.