  This study introduces an accelerated training method for Vanilla
Physics-Informed-Neural-Networks (PINN) addressing three factors that imbalance
the loss function: initial weight state of a neural network, domain to boundary
points ratio, and loss weighting factor. We propose a novel two-stage training
method. During the initial stage, we create a unique loss function using a
subset of boundary conditions and partial differential equation terms.
Furthermore, we introduce preprocessing procedures that aim to decrease the
variance during initialization and choose domain points according to the
initial weight state of various neural networks. The second phase resembles
Vanilla-PINN training, but a portion of the random weights are substituted with
weights from the first phase. This implies that the neural network's structure
is designed to prioritize the boundary conditions, subsequently affecting the
overall convergence. Three benchmarks are utilized: two-dimensional flow over a
cylinder, an inverse problem of inlet velocity determination, and the Burger
equation. It is found that incorporating weights generated in the first
training phase into the structure of a neural network neutralizes the effects
of imbalance factors. For instance, in the first benchmark, as a result of our
process, the second phase of training is balanced across a wide range of ratios
and is not affected by the initial state of weights, while the Vanilla-PINN
failed to converge in most cases. Lastly, the initial training process not only
eliminates the need for hyperparameter tuning to balance the loss function, but
it also outperforms the Vanilla-PINN in terms of speed.
