Adaptive physics-informed neural network with unknown parameter identification for heat conduction in heterogeneous materials

Predicting temperature fields and determining effective thermal conductivity (ETC) in heterogeneous materials with inclusions are critical for understanding their heat transfer behavior. Traditional ETC models, such as the series and parallel formulations, require subjective selection and fail to capture the coupling between temperature distribution, boundary conditions, and material properties. This study proposes an adaptive physics-informed neural network (PINN) framework that overcomes these limitations. The key work is to embed both series and parallel ETC formulas into the loss function via a trainable adaptive coefficient λ, allowing the network to dynamically select or combine them during training. A transfer learning strategy is further introduced, enabling the learned λ to be directly applied to similar material configurations without additional measurements. The framework simultaneously predicts the full temperature field and identifies unknown parameters-including the convective heat transfer coefficient and inclusion volume fraction-using only sparse boundary measurements. Numerical examples show that the adaptive ETC model consistently outperforms both pure series and pure parallel models in accuracy. The transfer learning variant achieves comparable precision to the adaptive model without requiring extra measurement points. This work provides a robust, accurate, and generalizable tool for thermal analysis of heterogeneous materials, unifying forward prediction and inverse characterization within a single solving framework.