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BPINN-EM-Post: Bayesian Physics-Informed Neural Network based Stochastic Electromigration Damage Analysis in the Post-void Phase

Subed Lamichhane, Haotian Lu, Sheldon X. -D. Tan

TL;DR

The paper tackles stochastic electromigration stress in the post-void phase and proposes BPINN-EM-Post, a hierarchical framework that fuses closed-form analytical solutions for single wire segments with a Bayesian Physics-Informed Neural Network to enforce stress continuity and atomic flux conservation at inter-segment junctions. It uses Hamiltonian Monte Carlo to infer a posterior over neural network weights, enabling rapid, uncertainty-quantified stress predictions across multi-segment interconnects. The approach delivers substantial speedups (e.g., ~240x- vs COMSOL, ~67x vs EMSpice) with minimal accuracy loss (RMSE on the order of 0.1–1% of baseline results) and scalable training/inference as the interconnect size grows. This framework provides a practical, scalable method for variational EM reliability analysis in large integrated circuits.

Abstract

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.

BPINN-EM-Post: Bayesian Physics-Informed Neural Network based Stochastic Electromigration Damage Analysis in the Post-void Phase

TL;DR

The paper tackles stochastic electromigration stress in the post-void phase and proposes BPINN-EM-Post, a hierarchical framework that fuses closed-form analytical solutions for single wire segments with a Bayesian Physics-Informed Neural Network to enforce stress continuity and atomic flux conservation at inter-segment junctions. It uses Hamiltonian Monte Carlo to infer a posterior over neural network weights, enabling rapid, uncertainty-quantified stress predictions across multi-segment interconnects. The approach delivers substantial speedups (e.g., ~240x- vs COMSOL, ~67x vs EMSpice) with minimal accuracy loss (RMSE on the order of 0.1–1% of baseline results) and scalable training/inference as the interconnect size grows. This framework provides a practical, scalable method for variational EM reliability analysis in large integrated circuits.

Abstract

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.

Paper Structure

This paper contains 15 sections, 20 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. EM Stress Evolution in Post-voiding Phase
  4. Estimation of Variations in EM Stress Evolution
  5. Proposed BPINN-EM-Post Framework for Stochastic EM Analysis
  6. Estimation of Prior for BPINN
  7. Generating Observations with Analytical Solutions
  8. Evaluation of Posterior for BPINN
  9. Estimation of Variations in EM Stress
  10. Experimental Results
  11. Experimental Setup
  12. Data Preparation and Scaling
  13. Accuracy and Performance of Analytical Solutions
  14. Overall Accuracy and Performance Analysis
  15. Conclusion

Figures (4)

  • Figure 1: Framework of the proposed BPINN-EM-Post variational EM simulator
  • Figure 2: Single interconnect segments that are part of a multi-segment interconnect tree in the post-voiding phase.
  • Figure 3: Comparison of the stress distribution obtained from analytical solution with the stress distribution from FEM-based COMSOL for single interconnect segment for three different cases.
  • Figure 4: Illustration of stress variations at junctions of a multi-segment interconnect structure. (a): Example of a ten-segment interconnect. (b), (c), (d), (e): Comparisons of variations estimation between the proposed framework, COMSOL and EMSpice at different junctions.