Dynamic Weight Strategy of Physics-Informed Neural Networks for the 2D Navier–Stokes Equations

. 2022 Sep 6;24(9):1254. doi: 10.3390/e24091254

Algorithm 1: Dynamic weights strategy for PINNs

\begin{matrix} Step 1 : Set Training steps K, the learning rate η, η_{w}, initial values balance weights \\ w = \{w_{u}, w_{0}, w_{b}, w_{f}\} and neural network parameters \tilde{Θ} . \\ Step 2 : Consider a physics - informed neural network to define the weighted loss function \\ J (\tilde{Θ}, w_{u}, w_{0}, w_{b}, w_{f}) based on () . \\ Step 3 : Then use K steps of a gradient descent algorithm to update the parameters w \\ and \tilde{Θ} as : \\ for k = 1 to K do \\ if M S E_{0}^{k + 1} < M S E_{0}^{k} and M S E_{u}^{k + 1} < M S E_{u}^{k} and M S E_{b}^{k + 1} < M S E_{b}^{k} and M S E_{f}^{k + 1} < M S E_{f}^{k} \\ Tune the balance weights w via Adam to maximize the meeting constraints \\ w_{k + 1} \leftarrow Adam 1 (J (w_{k}; \tilde{Θ}; K); η; η_{w}) \\ Update the parameters \tilde{Θ} via Adam to minimize J \\ {\tilde{Θ}}_{k + 1} \leftarrow Adam 2 (J (w_{k}; {\tilde{Θ}}_{k}; K); η; η_{w}) \\ end for \end{matrix}