Partial differential equations (PDEs) are commonly employed to model complex industrial systems characterized by multivariable dependence. Existing physics-informed neural networks (PINNs) excel in solving PDEs in a homogeneous medium. However, their feasibility is diminished when PDE parameters are unknown due to a lack of physical attributions and time-varying interface is unavailable arising from heterogeneous media. To this end, we propose a data-physics-hybrid method, physically informed synchronic-adaptive learning (PISAL), to solve PDEs for industrial systems modeling in heterogeneous media. First, Net1, Net2, and NetI, are constructed to approximate the solutions satisfying PDEs and the interface. Net1 and Net2 are utilized to synchronously learn each solution satisfying PDEs with diverse parameters, while NetI is employed to adaptively learn the unavailable time-varying interface. Then, a criterion combined with NetI is introduced to adaptively distinguish the attributions of measurements and collocation points. Furthermore, NetI is integrated into a data-physics-hybrid loss function. Accordingly, a synchronic-adaptive learning (SAL) strategy is proposed to decompose and optimize each subdomain. Besides, we theoretically prove the approximation capability of PISAL. Extensive experimental results verify that the proposed PISAL can be used for industrial systems modeling in heterogeneous media, which faces the challenges of lack of physical attributions and unavailable time-varying interface.

在本文中，我们提出了一种称为PISAL的数据-物理-混合方法，用于解决工业系统建模中的偏微分方程问题。该方法利用Net1、Net2和NetI来近似满足PDE和界面的解，并通过同步自适应学习来处理具有不同参数的PDE解和不可用的时变界面。通过引入与NetI相结合的准则，该方法能够自适应地区分测量和配准点的属性，并在数据-物理-混合损失函数中集成NetI。我们还在理论上证明了PISAL的逼近能力，并通过广泛的实验结果验证了该方法在面对缺乏物理属性和不可用的时变界面的异质媒体工业系统建模方面的有效性。

面向异质介质中不可用时变界面的工业系统建模的物理学驱动同步自适应学习