In contrast to the assumptions of most existing Electromigration (EM) analysis tools, the evolution of EM-induced stress is inherently non-deterministic, influenced by factors such as input current fluctuations and manufacturing non-idealities. Traditional approaches for estimating stress variations typically involve computationally expensive and inefficient Monte Carlo simulations with industrial solvers, which quantify variations using mean and variance metrics. In this work, we introduce a novel machine learning-based framework, termed BPINN-EM- Post, for efficient stochastic analysis of EM-induced post-voiding aging processes. For the first time, our new approach integrates closed-form analytical solutions with a Bayesian Physics- Informed Neural Network (BPINN) framework to accelerate the analysis. The closed-form solutions enforce physical laws at the individual wire segment level, while the BPINN ensures that physics constraints at inter-segment junctions are satisfied and stochastic behaviors are accurately modeled. By reducing the number of variables in the loss functions through utilizing analytical solutions, our method significantly improves training efficiency without accuracy loss and naturally incorporates variational effects. Additionally, the analytical solutions effectively address the challenge of incorporating initial stress distributions in interconnect structures during post-void stress calculations. Numerical results demonstrate that BPINN-EM-Post achieves over 240x and more than 67x speedup compared to Monte Carlo simulations using the FEM-based COMSOL solver and FDM-based EMSpice, respectively, with marginal accuracy loss.

翻译：与大多数现有电迁移分析工具的假设相反，电迁移引起的应力演化本质上是非确定性的，受到输入电流波动和制造非理想性等因素的影响。传统的应力变化估计方法通常涉及计算成本高昂且效率低下的蒙特卡洛模拟，这些模拟使用工业求解器并通过均值和方差指标来量化变化。在本工作中，我们提出了一种新颖的基于机器学习的框架，称为BPINN-EM-Post，用于高效地对电迁移引起的后空洞老化过程进行随机分析。我们的新方法首次将闭式解析解与贝叶斯物理信息神经网络框架相结合，以加速分析过程。闭式解析解在单个导线线段级别强制执行物理定律，而BPINN则确保线段间连接处的物理约束得到满足，并能精确建模随机行为。通过利用解析解减少损失函数中的变量数量，我们的方法在保证精度不损失的前提下显著提高了训练效率，并自然地纳入了变分效应。此外，解析解有效解决了在后空洞应力计算中纳入互连结构初始应力分布的挑战。数值结果表明，与使用基于有限元的COMSOL求解器和基于有限差分的EMSpice进行的蒙特卡洛模拟相比，BPINN-EM-Post分别实现了超过240倍和67倍的加速，且精度损失微小。