Physics-Informed Neural Networks (PINNs) encounter challenges in dealing with imbalanced training losses, especially when there are sample points with extremely high losses. This can make the optimization process unstable, making it challenging to find the correct descent direction during training. In this paper, we propose a progressive learning approach based on anomaly points awareness to improve the optimization process of PINNs. Our approach comprises two primary steps: the awareness of anomaly data points and the update of training set. Anomaly points are identified by utilizing an upper bound calculated from the mean and standard deviation of the feedforward losses of all training data. In the absence of anomalies, the parameters of the PINN are optimized using the default training data; however, once anomalies are detected, a progressive exclusion method aligned with the network learning pattern is introduced to exclude potentially unfavorable data points from the training set. In addition, intermittent detection is employed, rather than performing anomaly detection in each iteration, to balance performance and efficiency. Extensive experimental results demonstrate that the proposed method leads to substantial improvement in approximation accuracy when solving typical benchmark partial differential equations. The code is accessible at https://github.com/JcLimath/Anomaly-Aware-PINN.