  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.
