A data-driven integrable BFGS algorithm (IBA-PDE) for discovering PDEs

doi:10.1016/j.physd.2026.135135

A data-driven integrable BFGS algorithm (IBA-PDE) for discovering PDEs

2026-01-28

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DOI：10.1016/j.physd.2026.135135

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Data-driven discovery of partial difference equations (PDEs) has become a hot topic, and scholars have proposed some excellent data-driven methods (PINNs,PDE-FIND,DLGA-PDE,SGA-PDE) and achieved good results in discovering PDEs. This paper proposes a new integrable BFGS algorithm (IBA-PDE) for PDE discovery, which solves two key problems: (1) To manage the complexity and redundancy of candidate PDE terms, it incorporates a weight balance condition tailored for partially integrable PDEs, along with a preliminary optimization strategy, we first solve the problem of narrowing down the range of PDEs candidates; (2) To accurately estimate unknown PDEs coefficients, the method employs the BFGS optimization algorithm, enhancing the precision of the identification process. Through systematic numerical experiments, IBA-PDE demonstrates superior capability that not only rediscovers fundamental PDEs but also resolves previously intractable systems with unprecedented precision. Specifically, IBA-PDE discovered several complex integrable PDEs (fifth-order KdV, Kaup Kupershmidt, Sawada Kotera, complex modified KdV, Hirota, and (2+1) dimensional Kadomtsev Petviashvili (KP) equations) and two non integrable PDEs (Burgers KdV and Chafee Infante equations), all of which have mean square errors (MSEs) of 10−9 and coefficient errors of almost zero. Moreover, IBA-PDE use fewer experimental data compared to other data-driven methods throughout the entire process of discovering complete PDEs, whether in the stage of determining PDEs candidate terms or coefficient determination. For non-integrable systems, IBA-PDE employs an adaptive discovery mechanism that not only successfully resolves the Burgers-KdV equation but also autonomously identifies a new PDE that better matches the data of the Chafee-Infante equation reducing MSE from 10−11 to 10−14. Robustness analysis confirms the method’s stability under noise conditions of 1 %, 3 % and 5 %, maintaining the same MSE levels. IBA-PDE establishes a new paradigm for data-driven PDEs discovery, with transformative potential for discovering new PDEs or matching known PDEs from experimental data in fields such as physics, engineering, mechanics, chemistry and biology.

A data-driven integrable BFGS algorithm (IBA-PDE) for discovering PDEs

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