WarpagePINN: Thermal Warpage Prediction in Advanced Packaging via a Two-Stage Physics-Informed Neural Networks
Thermal warpage has become a critical issue in advanced packaging, primarily caused by the mismatch in coefficients of thermal expansion (CTE) among heterogeneously integrated materials. However, only a limited number of studies have focused on developing computational methods for coupled thermal-warpage prediction in the chiplet. This paper proposes a two-stage physics-informed neural network (WarpagePINN) framework to compute both temperature profile and warpage deformation of chiplets. The neural networks are trained without relying on labeled datasets generated by conventional simulators. In the first stage, the temperature field is modeled using a Fourier series representation that inherently satisfies boundary conditions, and the network is trained solely through a loss function derived from the governing equation. In the second stage, a multilayer perceptron (MLP) is employed for warpage prediction, utilizing a novel hybrid supervisory strategy to optimize the energy-based loss function instead of residual loss. A parametric WarpagePINN is also developed to quantify uncertainties associated with the CTE. Numerical results show that the proposed WarpagePINN framework achieves excellent agreement with conventional finite element methods, with a mean absolute error (MAE) of 0.2 μm, while achieving a speedup of approximately 1000 {\times} in CTE parameterization studies.