  Physics-Informed Neural Networks (PINNs) have emerged recently as a promising
application of deep neural networks to the numerical solution of nonlinear
partial differential equations (PDEs). However, it has been recognized that
adaptive procedures are needed to force the neural network to fit accurately
the stubborn spots in the solution of "stiff" PDEs. In this paper, we propose a
fundamentally new way to train PINNs adaptively, where the adaptation weights
are fully trainable and applied to each training point individually, so the
neural network learns autonomously which regions of the solution are difficult
and is forced to focus on them. The self-adaptation weights specify a soft
multiplicative soft attention mask, which is reminiscent of similar mechanisms
used in computer vision. The basic idea behind these SA-PINNs is to make the
weights increase as the corresponding losses increase, which is accomplished by
training the network to simultaneously minimize the losses and maximize the
weights. In addition, we show how to build a continuous map of self-adaptive
weights using Gaussian Process regression, which allows the use of stochastic
gradient descent in problems where conventional gradient descent is not enough
to produce accurate solutions. Finally, we derive the Neural Tangent Kernel
matrix for SA-PINNs and use it to obtain a heuristic understanding of the
effect of the self-adaptive weights on the dynamics of training in the limiting
case of infinitely-wide PINNs, which suggests that SA-PINNs work by producing a
smooth equalization of the eigenvalues of the NTK matrix corresponding to the
different loss terms. In numerical experiments with several linear and
nonlinear benchmark problems, the SA-PINN outperformed other state-of-the-art
PINN algorithm in L2 error, while using a smaller number of training epochs.
