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<p>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.</p>

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Bi-level identification of governing equations for nonlinear physical systems

  • 期刊论文

  • 发表日期：

    2025年5月9日

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  • Zeyu Li,
  • Huining Yuan,
  • Wang Han,
  • Yimin Hou,
  • Hongjue Li,
  • Haidong Ding,
  • Zhiguo Jiang,
  • Lijun Yang

摘要

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.