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A Physics-Guided Transformer Framework for Electromigration Analysis in Multi-Segment Interconnects

Pavlos Stoikos, Anuj Pathania, George Floros

arXiv:2610.06464v1cs.LGcs.AI

TL;DR

Transient EM stress analysis across many interconnect lines is computationally expensive, while predictions must account for both local segment properties and line context. The paper uses a physics-guided transformer with segment tokens and query-based stress prediction; on IBM benchmarks, test relative-L2 error remains below 8% and average speedup is 1054.84× over the matrix exponential solver.

  • Problem

    Repeated physics-based transient stress analysis across many interconnect lines and reliability times is computationally expensive, and stress depends on both local segment properties and wider line context.

    Source: §I · p. 1, §II · p. 2

  • Method

    A physics-guided transformer encodes geometry- and DC-aware segment tokens, uses line-context attention and a query decoder, and trains with supervised and physics-guided terms.

    Source: §I · p. 1, C. Physics-guided objective · pp. 2–3

  • Results

    Test relative-L2 error is below 8%, with an average speedup of 1054.84× over the matrix exponential solver on IBM power-grid benchmarks.

    Source: B. Representative-subset results · p. 3, §V · p. 4

  • Takeaways & Limitations

    The framework predicts stress at selected locations and times with low test error and substantially faster inference on the evaluated IBM benchmarks.

    Source: §V · p. 4

Abstract

from arXiv · show

As technology scales to smaller nodes, increasing current densities make electromigration (EM) one of the dominant reliability challenges in on-chip interconnects. Accurate transient stress analysis is needed to identify wires susceptible to EM degradation, but applying physics-based solvers across many interconnects remains computationally expensive. This paper proposes a physics-guided transformer framework for fast EM stress prediction in multi-segment interconnect lines. The framework converts each line into geometry- and DC-aware segment tokens and uses transformer attention to capture line-level context. A lightweight query decoder then predicts stress at selected locations and time instants. The model is trained with an objective that combines normalized supervised regression, linewise relative-$L_2$ loss, and physics-guided continuity and terminal-flux terms. Experiments on IBM power grid benchmarks show that the proposed model achieves relative-$L_2$ error below 8\% and reaches up to 2459.68$\times$ speedup compared with the matrix exponential~solver.

I. INTRODUCTION

Electromigration threatens scaled interconnects, and transient physics-based stress analysis is costly across many lines. The paper motivates prediction that captures both segment properties and whole-line context.

I. INTRODUCTION

  • Physics-based transient stress solvers provide more accurate solutions than lifetime guidelines, but repeatedly evaluating many lines and reliability times is computationally expensive.Korhonen-type models describe stress evolution through the balance of EM driving and opposing back-stress contributions.Source: §I · p. 1, §II · p. 1

II. BACKGROUND

  • Learning-based predictors have shown speedups, yet EM stress depends on local segment properties and broader interconnect-line context.The background describes layer-wise analysis of multi-segment lines composed of wire segments and vias, each with locally defined current density.Source: §I · p. 1, §II · pp. 1–2

I. INTRODUCTION

  • The proposed framework uses a transformer for variable-length lines, DC-aware segment representations, a query decoder, and continuity and terminal-flux regularization.It is evaluated on IBM power-grid benchmarks against the matrix exponential solver.Source: §I · p. 1

III. PHYSICS-GUIDED TRANSFORMER FRAMEWORK

The framework turns each interconnect line into segment-level inputs, trains against selected reference stress labels, and combines contextual encoding with query-based stress prediction.

III. PHYSICS-GUIDED TRANSFORMER FRAMEWORK

  • The framework represents each extracted line as a sequence of segment-level physical and electrical features.Source: §III · p. 2
  • Reference stress labels are generated at selected space-time query points, and a transformer encoder learns contextual representations across the line.Source: §III · p. 2
  • A query-based decoder predicts EM stress using a combined supervised and physics-guided training objective.Source: §III · p. 2

A. Feature construction and reference labels

Each line is encoded as an ordered sequence of segment tokens containing normalized geometry, line position, and DC electrical quantities; reference labels come from the matrix exponential solver.

A. Feature construction and reference labels

B. Transformer architecture for EM stress prediction

The transformer encoder builds line-contextual segment embeddings, and a local query decoder combines an embedding with location and time to predict stress at sampled points.

B. Transformer architecture for EM stress prediction

  • Self-attention lets each segment aggregate information from all segments, capturing context relevant to stress beyond local current.Encoder blocks use multi-head self-attention and a feed-forward network with residual connections and layer normalization.Source: B. Transformer architecture for EM… · p. 2
  • After L encoder layers, each contextual segment embedding summarizes local segment features together with line-level context.Source: B. Transformer architecture for EM… · p. 2
  • For a query on segment k, the decoder combines its contextual embedding with a vector encoding local coordinate and aging time to produce scalar stress.The concatenated representation passes through a two-hidden-layer MLP with GELU activations and a final linear layer.Source: B. Transformer architecture for EM… · p. 2
  • The architecture separates line-level context extraction from pointwise stress evaluation at sampled locations and times.Source: B. Transformer architecture for EM… · p. 2

C. Physics-guided objective

The objective combines normalized supervised regression and linewise relative-L2 fitting with physical regularizers for interface continuity and terminal flux.

C. Physics-guided objective

  • The training objective combines normalized MSE, linewise relative-L2, continuity, and terminal-flux terms.The compound loss balances supervised fitting with line-level and boundary-aware physical regularization.Source: C. Physics-guided objective · p. 2
  • Stress normalization uses the training-set offset and scale to improve training stability across benchmarks with different stress magnitudes.The supervised regression term is computed on normalized stresses.Source: C. Physics-guided objective · pp. 2–3
  • The linewise relative-L2 term optimizes full stress profiles as line-level objects, while σfloor prevents near-zero-stress lines from dominating the objective.The linewise term complements pointwise fitting.Source: C. Physics-guided objective · p. 3
  • Continuity regularization penalizes stress mismatch at internal segment interfaces, and terminal-flux regularization constrains predicted stress slopes at blocking terminals.The terminal derivative is approximated with a short finite difference near each endpoint.Source: C. Physics-guided objective · p. 3

D. Hyperparameter search

Hyperparameters are selected using validation relative-L2 error, then the best configuration is evaluated on benchmark lines withheld from supervised training.

D. Hyperparameter search

  • Validation relative-L2 error is the Optuna hyperparameter-search objective.The metric is defined for an evaluation set D.Source: D. Hyperparameter search · p. 3
  • The search tunes transformer width, encoder layers, decoder hidden dimension, learning rate, batch size, and physics-loss weights.Promising trials train for up to 600 epochs with periodic validation and pruning of unpromising configurations.Source: D. Hyperparameter search · p. 3
  • The selected best configuration is evaluated on benchmark lines not used for supervised training.Source: D. Hyperparameter search · p. 3

A. Experimental setup

The experiments use extracted lines from IBM power-grid benchmarks, with reference-solver stress labels and held-out lines for evaluation.

A. Experimental setup

  • The model is evaluated on IBM power-grid benchmark lines with stress labels generated by the reference solver.Accuracy metrics are averaged over t = 5, 10, and 20 years.Source: A. Experimental setup · p. 3
  • Experiments ran on a Linux Py-Torch server with an NVIDIA RTX 6000 Ada Generation GPU, and Optuna selected benchmark-specific model and training settings.Selected encoder dimensions were 96, 128, or 160, with 2–4 MHA/FFN layers; decoder dimensions were 96, 128, or 192.Source: A. Experimental setup · p. 3
  • A compact representative subset of each benchmark's extracted lines is used for supervised training, while remaining lines are held out for evaluation.Table I summarizes extracted-line statistics; its Train/all column gives the percentage used for supervised training.Source: A. Experimental setup · p. 3, Table I · p. 3

B. Representative-subset results

Across six IBM benchmarks, test relative-L2 error remains below 8%, while the accompanying figures show stress predictions tracking reference values across benchmarks and over time. IBM2 and IBM3 are the most challenging cases, with larger structural variation and wider stress-scale ranges.

B. Representative-subset results

  • Test relative-L2 error remains below 8% across all six IBM benchmarks, with the strongest relative accuracy on IBM5 and IBM6 and the greatest challenges on IBM2 and IBM3.IBM2 and IBM3 have greater structural variation and wider stress-scale ranges; IBM3 also has larger RMSE and MAE despite relative error comparable to other benchmarks.Source: B. Representative-subset results · p. 3
  • Predicted-versus-ground-truth comparisons show close alignment with the ideal diagonal, preserving stress scale across IBM benchmarks with different line lengths and stress ranges.The comparison is shown in Figure 2.Source: B. Representative-subset results · p. 4, Figure 2 · p. 4
  • For a representative 110-segment interconnect, predicted stress curves track reference solutions across multiple time points and reproduce dominant peaks, including localized variations near segment transitions.The profile-level comparison is shown in Figure 3.Source: B. Representative-subset results · p. 4, Figure 3 · p. 4

C. Runtime comparison

On IBM power-grid benchmarks, the model achieves test relative-L2 error below 8% and an average 1054.84× speedup over the matrix exponential solver. Runtime comparisons average measurements over each benchmark’s 10 largest lines, with the largest speedup on IBM2.

C. Runtime comparison

  • Speedup reaches 2459.68× on IBM2 and remains above two orders of magnitude on IBM1, the least favorable case.Runtime is averaged over the 10 largest extracted lines per benchmark; the model measurement is a forward pass and the reference measurement is the solver’s stress solution for the same line, excluding offline training and hyperparameter search.Source: C. Runtime comparison · p. 4, Table III · p. 4

V. CONCLUSION

  • Test relative-L2 error below 8% and average speedup of 1054.84× over the matrix exponential solver summarize the IBM power-grid benchmark results.The conclusion identifies future work on interconnect-tree topologies and adaptive selection of samples for label generation.Source: §V · p. 4
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