A Curriculum-Training-Based Strategy for Distributing Collocation Points during Physics-Informed Neural Network Training

Physics-informed Neural Networks (PINNs) often have, in their loss functions, terms based on physical equations and derivatives. In order to evaluate these terms, the output solution is sampled using a distribution of collocation points. However, density-based strategies, in which the number of collocation points over the domain increases throughout the training period, do not scale well to multiple spatial dimensions. To remedy this issue, we present here a curriculum-training-based method for lightweight collocation point distributions during network training. We apply this method to a PINN which recovers a full two-dimensional magnetohydrodynamic (MHD) solution from a partial sample taken from a baseline MHD simulation. We find that the curriculum collocation point strategy leads to a significant decrease in training time and simultaneously enhances the quality of the reconstructed solution.

Prediction of soft proton intensities in the near-Earth space using machine learning

The spatial distribution of energetic protons contributes towards the understanding of magnetospheric dynamics. Based upon 17 years of the Cluster/RAPID observations, we have derived machine learning-based models to predict the proton intensities at energies from 28 to 1,885 keV in the 3D terrestrial magnetosphere at radial distances between 6 and 22 RE. We used the satellite location and indices for solar, solar wind and geomagnetic activity as predictors. The results demonstrate that the neural network (multi-layer perceptron regressor) outperforms baseline models based on the k-Nearest Neighbors and historical binning on average by ~80% and ~33\%, respectively. The average correlation between the observed and predicted data is about 56%, which is reasonable in light of the complex dynamics of fast-moving energetic protons in the magnetosphere. In addition to a quantitative analysis of the prediction results, we also investigate parameter importance in our model. The most decisive parameters for predicting proton intensities are related to the location: ZGSE direction and the radial distance. Among the activity indices, the solar wind dynamic pressure is the most important. The results have a direct practical application, for instance, for assessing the contamination particle background in the X-Ray telescopes for X-ray astronomy orbiting above the radiation belts. To foster reproducible research and to enable the community to build upon our work we publish our complete code, the data, as well as weights of trained models. Further description can be found in the GitHub project at https://github.com/Tanveer81/deep_horizon.