Self-adaptive loss balanced Physics-informed neural networks

Physics-informed neural networks (PINNs) have received significant attention as a representative deep learning-based technique for solving partial differential equations (PDEs). The loss function of PINNs is a weighted sum of multiple terms, including the mismatch of observed data, boundary and initial constraints, as well as PDE residuals. In this paper, we observe that the performance of PINNs is susceptible to the weighted combination of competitive multiple loss functions. Therefore, we establish Gaussian probabilistic models to define the self-adaptive loss function through the adaptive weights for each loss term. In particular, we propose a self-adaptive loss balanced method that automatically assigns the weights of losses by updating adaptive weights in each epoch based on the maximum likelihood estimation. Finally, we perform a series of numerical experiments with self-adaptive loss balanced physics-informed neural networks (lbPINNs), including solving Poisson, Burgers, Helmholtz, Navier–Stokes, and Allen–Cahn equations in regular and irregular areas. We also test the robustness of lbPINNs by varying the initial adaptive weights, numbers of observations, hidden layers, and neurons per layer. These experimental results demonstrate that lbPINNs consistently achieve better performance than PINNs, and reduce the relative L2 error by about two orders of magnitude.