Linear Time Electromigration Analysis Based on Physics-Informed Sparse Regression

In this work, we propose a novel physics-informed sparse regression (PISR) framework to solve stress evolution (described by Korhonen’s equations) in general multisegment wires using an unsupervised learning scheme. Unlike the existing physics-informed neural network (PINN) framework, the PISR method trains the trainable weights through the Moore–Penrose generalized inverse algorithm used in extreme learning machine (ELM), which is extremely faster than the backpropagation algorithm. To improve the accuracy of PISR for complex multisegment interconnects, we employ domain decomposition schemes in both space and time. For each subdomain, we use different trainable weights but the same shared neural network to represent each subsolution, which leads to more efficient memory usage. Furthermore, we propose to use sparse matrix techniques to accelerate the training speed of the PISR method and prove that the resulting PISR has linear time complexity for analyzing tree-structured interconnects. Finally, we divide the time into many time intervals and apply an autoregressive model to simulate each time interval in sequence to further improve scalability and reduce memory cost so that the PISR method can perform EM analysis for large-scale multisegment interconnects. Experimental results on different kinds of interconnect structures show that the proposed PISR method has the same accuracy level as the numerical methods. The results on $N_{T}$ T-junctions interconnect trees show that the proposed PISR method indeed demonstrates true linear time complexity. Furthermore, PISR can deliver $8.9\times $ , $20.6\times $ , and $1284\times $ speedups over the recently proposed semi-analytic method (ASOV), finite difference method accelerated with model order reduction (FDM-MOR), FDM for the interconnect with $N_{T}= 5000$ , respectively. Furthermore, we show that PISR also achieves an $818\times $ speedup in training over the plain PINN method based on the traditional backpropagation algorithm.