Electromigration (EM) induced aging and degradation in interconnect wires is inherently a stochastic process, with lifetime typically measured in terms of mean time to failure at both wire and circuit levels. However, existing approaches still incur high computing costs, as computing both means and variances is generally expensive. In this work, we propose a novel fast variational analysis framework to tackle the challenges of stochastic estimation of EM stress evolution in multi-segment interconnect wires. We utilize Bayesian networks in conjunction with the recently introduced hierarchical (two-step) physics-informed neural networks (PINN). The resulting method, termed BPINN-EM, enables rapid variational stress analysis of metal wires by leveraging the robust uncertainty quantification capability of Bayesian networks with expedited training over small dataset. Moreover, we devise BPINN-EM to incorporate Bayesian networks only in the first stage of the hierarchical PINN, thereby circumventing the need for sampling across the entire PINN level during training and significantly reducing training costs. Our results on several general multi-segment interconnect structure demonstrate that the proposed BPINN-EM approach is much more efficient than conventional baselines and state-of-the-art algorithms. Compared to a Monte Carlo simulator implemented in COMSOL, BPINN-EM offers a 240× speedup. Moreover, compared to the recently proposed EMSpice simulated by the Monte Carlo method, the new method provides more than an 85× speedup with almost no loss of accuracy.

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 bound-ary 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.

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.

Electromigration (EM) becomes a major concern for VLSI circuits as the technology advances in the nanometer regime. The crux of problem is to solve the partial differential Korhonen equations, which remains challenging due to the increasing integrated density. Recently, scientific machine learning has been explored to solve partial differential equations (PDE) due to breakthrough success in deep neural networks and existing approach such as physics-informed neural networks (PINN) shows promising results for some small PDE problems. However, for large engineering problems like EM analysis for large interconnect trees, it was shown that the plain PINN does not work well due the to large number of variables. In this work, we propose a novel hierarchical PINN approach, HierPINN-EM for fast EM induced stress analysis for multi-segment interconnects. Instead of solving the interconnect tree as a whole, we first solve EM problem for one wire segment under different boundary and geometrical parameters using supervised learning. Then we apply unsupervised PINN concept to solve the whole interconnects by enforcing the physics laws in the boundaries for all wire segments. In this way, HierPINN-EM can significantly reduce the number of variables at plain PINN solver. Numerical results on a number of synthetic interconnect trees show that HierPINN-EM can lead to orders of magnitude speedup in training and more than 79× better accuracy over the plain PINN method. Furthermore, HierPINN-EM yields 19% better accuracy with 99% reduction in training cost over recently proposed Graph Neural Network-based EM solver, EMGraph.