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.

翻译：偏微分方程（PDE）常用于刻画具有多变量依赖性的复杂工业系统。现有物理信息神经网络（PINNs）在求解均匀介质中的PDE时表现优异，但在缺乏物理属性导致PDE参数未知、以及异构介质导致时变界面不可得时，其可行性将显著降低。为此，我们提出一种数据-物理混合方法——物理信息同步自适应学习（PISAL），用于求解异构介质中工业系统建模的PDE。首先，构建Net1、Net2和NetI三个网络，分别逼近满足PDE和界面条件的解。Net1与Net2用于同步学习满足不同参数PDE的解，NetI则用于自适应学习不可得的时变界面。其次，引入基于NetI的判别准则，自适应区分测量点与配置点的物理属性。进一步将NetI集成到数据-物理混合损失函数中，并据此提出同步自适应学习（SAL）策略以实现各子域的解耦与优化。此外，我们从理论上证明了PISAL的逼近能力。大量实验结果表明，所提PISAL能够有效应用于面临物理属性缺失和时变界面不可得挑战的异构介质工业系统建模。