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WarPGNN: A Parametric Thermal Warpage Analysis Framework with Physics-aware Graph Neural Network

Haotian Lu, Jincong Lu, Sachin Sachdeva, Sheldon X. -D. Tan

Abstract

With the advent of system-in-package (SiP) chiplet-based design and heterogeneous 2.5D/3D integration, thermal-induced warpage has become a critical reliability concern. While conventional numerical approaches can deliver highly accurate results, they often incur prohib- itively high computational costs, limiting their scalability for complex chiplet-package systems. In this paper, we present WarPGNN, an ef- ficient and accurate parametric thermal warpage analysis framework powered by Graph Neural Networks (GNNs). By operating directly on graphs constructed from the floorplans, WarPGNN enables fast warpage-aware floorplan exploration and exhibits strong transfer- ability across diverse package configurations. Our method first en- codes multi-die floorplans into reduced Transitive Closure Graphs (rTCGs), then a Graph Convolution Network (GCN)-based encoder extracts hierarchical structural features, followed by a U-Net inspired decoder that reconstructs warpage maps from graph feature embed- dings. Furthermore, to address the long-tailed pattern of warpage data distribution, we developed a physics-informed loss and revised a message-passing encoder based on Graph Isomorphic Network (GIN) that further enhance learning performance for extreme cases and expressiveness of graph embeddings. Numerical results show that WarPGNN achieves more than 205.91x speedup compared with the 2-D efficient FEM-based method and over 119766.64x acceleration with 3-D FEM method COMSOL, respectively, while maintaining comparable accuracy at only 1.26% full-scale normalized RMSE and 2.21% warpage value error. Compared with recent DeepONet-based model, our method achieved comparable prediction accuracy and in- ference speedup with 3.4x lower training time. In addition, WarPGNN demonstrates remarkable transferability on unseen datasets with up to 3.69% normalized RMSE and similar runtime.

WarPGNN: A Parametric Thermal Warpage Analysis Framework with Physics-aware Graph Neural Network

Abstract

With the advent of system-in-package (SiP) chiplet-based design and heterogeneous 2.5D/3D integration, thermal-induced warpage has become a critical reliability concern. While conventional numerical approaches can deliver highly accurate results, they often incur prohib- itively high computational costs, limiting their scalability for complex chiplet-package systems. In this paper, we present WarPGNN, an ef- ficient and accurate parametric thermal warpage analysis framework powered by Graph Neural Networks (GNNs). By operating directly on graphs constructed from the floorplans, WarPGNN enables fast warpage-aware floorplan exploration and exhibits strong transfer- ability across diverse package configurations. Our method first en- codes multi-die floorplans into reduced Transitive Closure Graphs (rTCGs), then a Graph Convolution Network (GCN)-based encoder extracts hierarchical structural features, followed by a U-Net inspired decoder that reconstructs warpage maps from graph feature embed- dings. Furthermore, to address the long-tailed pattern of warpage data distribution, we developed a physics-informed loss and revised a message-passing encoder based on Graph Isomorphic Network (GIN) that further enhance learning performance for extreme cases and expressiveness of graph embeddings. Numerical results show that WarPGNN achieves more than 205.91x speedup compared with the 2-D efficient FEM-based method and over 119766.64x acceleration with 3-D FEM method COMSOL, respectively, while maintaining comparable accuracy at only 1.26% full-scale normalized RMSE and 2.21% warpage value error. Compared with recent DeepONet-based model, our method achieved comparable prediction accuracy and in- ference speedup with 3.4x lower training time. In addition, WarPGNN demonstrates remarkable transferability on unseen datasets with up to 3.69% normalized RMSE and similar runtime.
Paper Structure (17 sections, 18 equations, 6 figures, 5 tables, 1 algorithm)

This paper contains 17 sections, 18 equations, 6 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Thermal Warpage Problem for Package Design
  4. 3-D FEM Numerical Analysis Method
  5. 2-D FEM analysis based on laminate theory
  6. Proposed WarPGNN for Parametric Thermal Warpage Analysis
  7. Graph Encoding for Multi-die Placement
  8. GCN-based WarPGNN Architecture for Parametric Thermal Warpage Prediction
  9. Physics-aware Loss and Graph Representation Enhancement for Learning Improvement
  10. Overall Training Framework of WarPGNN
  11. Experimental Results
  12. Experimental Setup
  13. Dataset Generation
  14. Ablation Study: Performance of the Proposed GCN-based WarPGNN on Seen Datasets
  15. Overall Performance of the proposed GIN-based WarPGNN
  16. ...and 2 more sections

Figures (6)

  • Figure 1: (a) Illustration of thermal warpage formation. (b) TCG representation for a 6-die floorplan. Closure edges have been eliminated.
  • Figure 2: Illustration of node embedding update within a single $l$-th layer of our proposed GCN-based encoder. $\textbf{h}_i^{(l)}$ are node embeddings.
  • Figure 3: (a) Distribution statistics and model performance on normalized ground-truth results. (b) An example of the error map between GCN-based WarPGNN prediction and ground-truth deformation map.
  • Figure 4: The overall framework of our proposed WarPGNN for efficient and accurate thermal warpage analysis
  • Figure 5: Comparison of the predicted deformation map obtained from our GIN-based WarPGNN and COMSOL. All values are in $\mu$m.
  • ...and 1 more figures