Physically Informed Synchronic-adaptive Learning for Industrial Systems Modeling in Heterogeneous Media with Unavailable Time-varying Interface

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.