Physical Equation Discovery Using Physics-Consistent Neural Network (PCNN) under Incomplete Observability

Deep neural networks (DNNs) have been extensively applied to various fields, including physical-system monitoring and control. However, the requirement of a high confidence level in physical systems made system operators hard to trust black-box type DNNs. For example, while DNN can perform well at both training data and testing data, but when the physical system changes its operation points at a completely different range, never appeared in the history records, DNN can fail. To open the black box as much as possible, we propose a Physics-Consistent Neural Network (PCNN) for physical systems with the following properties: (1) PCNN can be shrunk to physical equations for sub-areas with full observability, (2) PCNN reduces unobservable areas into some virtual nodes, leading to a reduced network. Thus, for such a network, PCNN can also represent its underlying physical equation via a specifically designed deep-shallow hierarchy, and (3) PCNN is theoretically proved that the shallow NN in the PCNN is convex with respect to physical variables, leading to a set of convex optimizations to seek for the physics-consistent initial guess for the PCNN. We also develop a physical rule-based approach for initial guesses, significantly shortening the searching time for large systems. Comprehensive experiments on diversified systems are implemented to illustrate the outstanding performance of our PCNN.