An adaptive collocation physics-informed neural networks for multidimensional heat conduction problems in complex geometries

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as an effective framework for solving partial differential equations (PDEs) by embedding physical laws directly into the learning process. However, variable thermal conductivity and discontinuous solutions introduce heterogeneous coefficients, sharp gradients and non-smooth solution behaviors that challenge the approximation accuracy and convergence of conventional PINNs. To address these challenges, we develop a constructive solid geometry (CSG)-based framework for solving heat equations with variable thermal conductivity on complex computational domains. Within this framework, adaptive weighted loss and gradient-enhanced PINNs (AWL-gPINNs) employ adaptive loss weighting together with gradient information from the governing equations to improve solution accuracy, training stability and convergence. Adaptive collocation PINNs (AC-PINNs) complement this approach by dynamically redistributing collocation points toward regions with large residuals, thereby improving the resolution of steep solution variations and enhancing local approximation accuracy. The proposed methodology is applied to heat conduction problems involving variable thermal conductivity, nonlinear source terms, discontinuous solutions and irregular geometries. Numerical experiments demonstrate that the proposed framework consistently outperforms standard PINNs and existing variants by achieving lower relative L2 , L∞ and mean squared errors, together with improved training convergence and generalization performance. These results demonstrate the effectiveness of combining CSG-based geometric modeling with the complementary AWL-gPINN and AC-PINN strategies for solving challenging heat conduction problems and highlight the potential of the proposed framework for a broad class of PDEs arising in scientific and engineering applications.