  There have been several efforts to Physics-informed neural networks (PINNs)
in the solution of the incompressible Navier-Stokes fluid. The loss function in
PINNs is a weighted sum of multiple terms, including the mismatch in the
observed velocity and pressure data, the boundary and initial constraints, as
well as the residuals of the Navier-Stokes equations. In this paper, we observe
that the weighted combination of competitive multiple loss functions plays a
significant role in training PINNs effectively. We establish Gaussian
probabilistic models to define the loss terms, where the noise collection
describes the weight parameter for each loss term. We propose a self-adaptive
loss function method, which automatically assigns the weights of losses by
updating the noise parameters in each epoch based on the maximum likelihood
estimation. Subsequently, we employ the self-adaptive loss balanced
Physics-informed neural networks (lbPINNs) to solve the incompressible
Navier-Stokes equations,\hspace{-1pt} including\hspace{-1pt}
two-dimensional\hspace{-1pt} steady Kovasznay flow, two-dimensional unsteady
cylinder wake, and three-dimensional unsteady Beltrami flow. Our results
suggest that the accuracy of PINNs for effectively simulating complex
incompressible flows is improved by adaptively appropriate weights in the loss
terms. The outstanding adaptability of lbPINNs is not irrelevant to the
initialization choice of noise parameters, which illustrates the robustness.
The proposed method can also be employed in other problems where PINNs apply
besides fluid problems.
