Parameter estimation of high-altitude electromagnetic pulse based on physics and data co-supervised neural network

In this paper, we propose what we believe to be a novel physics and data co-supervised neural network (PaDCoSNN) for estimating the mathematical parameters of a double exponential function based on the physical parameters-rise time and pulse width-of high-altitude electromagnetic pulse (HEMP). Two key structural innovations improve the neural network's estimation accuracy and generalization ability. First, the input (physical parameters) and output (mathematical parameters) layers are structured to reflect their numerical characteristics. Second, nonlinear equations that interpret the relationship between the physical and mathematical parameters are embedded into the loss function to supervise the neural network's learning process. Additionally, a dual-scale probabilistic sampling strategy is proposed to address the issue that uniform sampling in the output space (traditional practice) will result in extremely uneven sample distribution in the input space, thereby generating more representative training samples. Comparative experiments, using 100 sparse training samples and 10000 test samples, demonstrate that PaDCoSNN reduces relative errors by an order of magnitude compared with conventional artificial neural networks (ANNs), with processing times remaining under 1.5 seconds. On out-of-scope test samples, PaDCoSNN demonstrates strong generalization, maintaining relative errors below 0.00001, compared to conventional ANNs, which reach up to 1.7. Moreover, PaDCoSNN has the potential to estimate parameters without sample-based training.