Physics Interpretable Shallow-Deep Neural Networks for Physical System Identification with Unobservability

The large amount of data collected in complex physical systems allows machine learning models to solve a variety of prediction problems. However, the directly applied learning approaches, especially deep neural networks (DNN), are difficult to balance between universal approximation to minimize error and the interpretability to reveal underlying physical law. Their performance drops even faster with system unobservability (of measurements) issues due to limited measurements. In this paper, we construct the novel physics interpretable shallow-deep neural networks to integrate exact physical interpretation and universal approximation to address the concerns in previous methods. We show that not only the shallow layer of the structural DNN extracts interpretable physical features but also the designed physical-input convex property of the DNN guarantees the true physical function recovery. While input convexity conditions are strict, the proposed model retains the representation capability to universally approximate for the unobservable system regions. We demonstrate its effectiveness by experiments on physical systems. In particular, we implement the proposed model on the forward kinematics and complex power flow reproduction tasks, with or without observability issues. We show that, besides the physical interpretability, our model provides consistently smaller or similar prediction error for system identification, compared to the state-of-art learning methods.