PostPINN-EM: Fast Post-Voiding Electromigration Analysis Using Two-Stage Physics-Informed Neural Networks

In this paper, we propose a novel machine learning-based approach, called PostPINN-EM, for solving the partial differential equations for stress evolution in a confined metal interconnect multi-segment trees during the post-voiding stage for fast electromigration (EM) check for interconnects. The new approach is based on an enhanced two-stage Physics-Informed Neural Networks (PINN) framework in which the physics law for a single wire is enforced first and then atomic flux conservation and stress continuity at the inter-segment junctions of wire segments are then fulfilled to reduce the number of variables of loss functions for the fast training process. Existing two-stage PINN method uses supervised learning method for modeling a single wire under various atomic flux conditions for the first stage, which turns out to be much more difficult for post-voiding phase due to arbitrary non-zero initial conditions. To mitigate this problem, we propose a new closed-form parameterized formula for stress solution of single wires with variable boundary conditions based on the Laplace transformation methods. Furthermore, we derive the analytic solutions for wire segment with and without voiding as not all the wire segments will have voids during the post-voiding phase. Numerical results on some synthesized multi-segment interconnects show that the proposed PostPINN-EM can achieve more than 100X speedup compared to FEM based tool COMSOL with the expense of less than 1% accuracy. Compared to the state of the art tool EMspice v1.0 [1], this method can achieve more than 25X speedup with similar accuracy compared to golden results from COMSOL.