Inferring dynamical models from time-series biological data using an interpretable machine learning method based on weighted expression trees

The growing time-series data make it possible to glimpse the hidden dynamics in various fields. However, developing a computational toolbox with high interpretability to unveil the interaction dynamics from data remains a crucial challenge. Here, we propose a new computational approach called Automated Dynamical Model Inference based on Expression Trees (ADMIET), in which the machine learning algorithm, the numerical integration of ordinary differential equations and the interpretability from prior knowledge are embedded into the symbolic learning scheme to establish a general framework for revealing the hidden dynamics in time-series data. ADMIET takes full advantage of both machine learning algorithm and expression tree. Firstly, we translate the prior knowledge into constraints on the structure of expression tree, reducing the search space and enhancing the interpretability. Secondly, we utilize the proposed adaptive penalty function to ensure the convergence of gradient descent algorithm and the selection of the symbols. Compared to gene expression programming, ADMIET exhibits its remarkable capability in function fitting with higher accuracy and broader applicability. Moreover, ADMIET can better fit parameters in nonlinear forms compared to regression methods. Furthermore, we apply ADMIET to two typical biological systems and one real data with different prior knowledge to infer the dynamical equations. The results indicate that ADMIET can not only discover the interaction relationships but also provide accurate estimates of the parameters in the equations. These results demonstrate ADMIET's superiority in revealing interpretable dynamics from time-series biological data.