Self-adaptive weight balanced physics-informed neural networks for solving complex coupling equations

In order to enhance the accuracy of Physics-Informed Neural Networks (PINNs) in handling complex coupled partial differential equations and reduce the difficulty of manually adjusting weights, a Physics-Informed Neural Network with adaptive weight balancing is proposed. A weight adaptive adjustment algorithm was designed based on the maximum likelihood inference of a multivariate Gaussian distribution, which automatically balances the magnitudes of individual loss terms in the loss function. This algorithm enhances the convergence of the training process for PINNs training process, significantly reduces training time, and enhances the approximation accuracy. By comparing the convergence and accuracy of the Adaptive Weight-Balanced Physics-Informed Neural Network with the original PINN in solving complex Navier-Stokes and magnetohydrodynamics equations, the superiority of the Weight-Balanced Physics-Informed Neural Network in accuracy was fully validated. Furthermore, it exhibited faster convergence and a more stable training process, enabling more efficient learning of data features and rapid achievement of optimal solutions.