Yimin Hou

3 weeks ago | nature.com | Zeyu Li |Wang Han |Yimin Hou |Hongjue Li |Zhiguo Jiang |Lijun Yang | +2 more

Identifying governing equations from observational data is crucial for understanding nonlinear physical systems but remains challenging due to the risk of overfitting. Here we introduce the Bi-Level Identification of Equations (BILLIE) framework, which simultaneously discovers and validates equations using a hierarchical optimization strategy. The policy gradient algorithm of reinforcement learning is leveraged to achieve the bi-level optimization. We demonstrate BILLIE’s superior performance through comparisons with baseline methods in canonical nonlinear systems such as turbulent flows and three-body systems. Furthermore, we apply the BILLIE framework to discover RNA and protein velocity equations directly from single-cell sequencing data. The equations identified by BILLIE outperform empirical models in predicting cellular differentiation states, underscoring BILLIE’s potential to reveal fundamental physical laws across a wide range of scientific fields. In this study the authors introduce BILLIE, a bi-level framework for equation identification that decouples term selection and quantification, enhancing robustness and accuracy in modeling nonlinear systems. BILLIE outperforms existing methods in handling complex systems and imperfect data, as demonstrated through both simulations and real-world biological applications.