Analytical and Neural Network Approaches for Solving Two-Dimensional Nonlinear Transient Heat Conduction

Accurately predicting nonlinear transient thermal fields in two-dimensional domains is a significant challenge in various engineering fields, where conventional analytical and numerical methods struggle to balance physical fidelity with computational efficiency when dealing with strong material nonlinearities and evolving multiphysics boundary conditions. To address this challenge, we propose a novel cross-disciplinary approach integrating Green's function formulations with adaptive neural operators, enabling a new paradigm for multiphysics thermal analysis. Our methodology combines rigorous analytical derivations with a physics-informed neural architecture consisting of five adaptive hidden layers (64 neurons per layer) that incorporates solutions as physical constraints, optimizing learning rates to balance convergence stability and computational speed. Extensive validation demonstrates superior performance in handling rapid thermal transients and strongly coupled nonlinear responses, which significantly improves computational efficiency while maintaining high agreement with analytical benchmarks across a range of material configurations and boundary conditions.