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. However, existing physics-informed neural networks (PINNs) barely perform well in heterogeneous media modeling compared to their achievement for a homogeneous medium. Unknown PDE parameters due to insufficient prior knowledge with respect to physical attributes and unavailable time-varying interface caused by heterogeneous media may weaken PINNs feasibility. To this end, physically informed synchronic-adaptive learning (PISAL) is proposed. First, PISAL is proposed for learning the solutions and interface satisfying PDEs, in which $Net_{1}$ , $Net_{2}$ , and $Net_{I}$ are constructed. $Net_{1}$ and $Net_{2}$ are for synchronically learning the solutions satisfying PDEs with diverse parameters; $Net_{I}$ is for adaptively learning the interface to decompose the domain with heterogeneous media. Then, a criterion combined with the output of neural networks is introduced to adaptively distinguish the attributes of measurements and collocation points. Furthermore, $Net_{1}$ , $Net_{2}$ , and $Net_{I}$ are integrated into a data-physics-hybrid loss function. Accordingly, a synchronic-adaptive learning (SAL) strategy is proposed to decompose the domain and optimize the three networks by iteratively erasing the training errors. Besides, we theoretically prove the proposed PISAL can iteratively approximate the fields with diverse physical attributes. Finally, extensive experimental results and comparisons with relevant state-of-the-art methods verify the feasibility and effectiveness of PISAL for industrial systems modeling in heterogeneous media. Note to Practitioners—The motivation behind this paper is to devise a method for industrial systems modeling, even in cases where unknown PDE parameters are caused by a lack of prior knowledge with respect to physical attributes and the unavailable time-varying interface is caused by heterogeneous media. Existing methods apply the domain decomposition technique under the assumption that the interface is available. To this end, a data-physics-hybrid method, PISAL, in which $Net_{1}$ , $Net_{2}$ , and $Net_{I}$ with SAL strategy are proposed. $Net_{1}$ , $Net_{2}$ , and $Net_{I}$ are first constructed. $Net_{1}$ and $Net_{2}$ are for synchronically learning the solutions satisfying PDEs with diverse parameters; $Net_{I}$ is for adaptively learning the unavailable time-varying interface. Subsequently, a criterion combined with the output of neural networks is introduced, which is to adaptively distinguish different physical attributes of measurements and collocation points. Additionally, the three neural networks are integrated into a data-physics-hybrid loss function. Accordingly, a SAL strategy is proposed to decompose the domain and optimize the three neural networks. Besides, the approximation capability of PISAL is theoretically proved based on well-posedness and denseness. To validate the efficacy of the proposed PISAL, the two-phase Stefan problem and the mixed Navier-Stokes problem are employed. Meanwhile, some comparisons with relevant state-of-the-art approaches are given. Results highlight the feasibility of our method in industrial systems modeling with the above-mentioned challenges. Thus, the proposed method is suitable for industrial automation applications. In future research, we intend to conduct the proposed PISAL on an experimental platform to further verify the feasibility in practical scenarios and more complex applications.