  Adaptive training methods for Physics-informed neural network (PINN) require
dedicated constructions of the distribution of weights assigned at each
training sample. To efficiently seek such an optimal weight distribution is not
a simple task and most existing methods choose the adaptive weights based on
approximating the full distribution or the maximum of residuals. In this paper,
we show that the bottleneck in the adaptive choice of samples for training
efficiency is the behavior of the tail distribution of the numerical residual.
Thus, we propose the Residual-Quantile Adjustment (RQA) method for a better
weight choice for each training sample. After initially setting the weights
proportional to the $p$-th power of the residual, our RQA method reassign all
weights above $q$-quantile ($90\%$ for example) to the median value, so that
the weight follows a quantile-adjusted distribution derived from the residuals.
This iterative reweighting technique, on the other hand, is also very easy to
implement. Experiment results show that the proposed method can outperform
several adaptive methods on various partial differential equation (PDE)
problems.
