Model Reconstruction Based on Physical-Informed Polynomial Neural ODEs

Neural networks are capable function approximators for data, but their lack of interpretability poses challenges. Physical-informed Polynomial Neural Ordinary Differential Equations (PIPNODEs) are proposed. They combine the constraints of prior knowledge of the data with interpretable Polynomial Neural Ordinary Differential Equations (PolyNODEs) to discover or approximate the governing equations. In this work, Neural Ordinary Differential Equations (NODEs) are categorized into two layers: a physical-informed term, which represents the priori knowledge of the data, and a discrepancy term, which is derived from the approximation of PolyNODEs. Pruning strategies are also used to remove redundant terms with coefficients lower than a certain threshold. We demonstrate the effectiveness and flexibility of this framework in a series of numerical algorithms with varying levels of prior knowledge. We found that having correct but incomplete physical information accelerates training and approaching thresholds compared to having incorrect physical information.