Enforcing hard constraints in physics-informed learning for transient TSV electromigration analysis

Due to the high local current densities, Through Silicon Vias (TSVs) are susceptible to electromigration (EM) degradation, which reduces the reliability of integrated circuits. Unlike traditional methods for TSV modeling and simulation, this paper introduces a unified hard constraint physics-informed learning neural network approach, called HCPINN, for the transient analysis of electromigration in TSVs by solving the conventional mass balance equation. The proposed method allows simultaneous consideration of atomic depletion and accumulation, effective resistance degradation, electric current evolution, and stress distribution. Specifically, we propose a hard constraint method for solving partial differential equations (PDEs) with general boundary conditions (BCs) for transient TSV electromigration analysis. By using the extra fields derived from the mixed finite element method, we reconstruct the corresponding PDEs by transforming general BCs into linear forms. Based on this derivation, we embed general BCs of mass balance equation into the proposed ansatz and employ sub-networks for the approximation on general BCs. The main neural network is responsible for training the internal part of the problem domain without adding loss terms with BCs, overcoming the convergence issue due to unbalanced gradients among different loss terms. Besides, we theoretically demonstrate that this reformulation of general BCs can stabilize the training process. Experimental results indicate that the proposed HCPINN exhibits superior performance and reduces boundary error in TSV electromigration analysis. Compared to the finite element method, the proposed network achieves approximately 100 times faster inference with a minimal mean squared error increase of less than 0.1%.