Technical Summary: 2D-ThermAl: Physics-Informed Framework for Thermal Analysis of Circuits using Generative AI

1. Problem Statement

Modern integrated circuits (ICs) face escalating thermal challenges due to non-uniform power dissipation and high transistor densities, which can lead to rapid temperature spikes and reliability failures. Traditional thermal analysis relies on Finite Element Method (FEM) simulations, which offer high fidelity but are computationally prohibitive for early-stage design exploration, often requiring iterative redesigns to resolve late-stage thermal failures. Conversely, existing coarse-grain models (e.g., analytical RC networks) and inverse thermal mapping techniques lack the spatial resolution to detect fine-grained hotspots or struggle to generalize across diverse circuit layouts and boundary conditions. There is a critical need for a predictive method that balances circuit-level accuracy, computational efficiency, and seamless integration into early Electronic Design Automation (EDA) workflows.

2. Methodology

The authors propose ThermAl, a physics-informed generative AI framework that reframes thermal analysis as an image-to-image translation task. The methodology integrates deep learning with physical constraints to predict full-chip transient and steady-state thermal distributions directly from input activity profiles.

A. Dataset Generation

The model is trained on an extensive synthetic dataset comprising over 200 circuit configurations, ranging from simple logic gates (inverters, NAND, XOR) to complex functional blocks.

• Generation Pipeline: Circuit designs and power distribution data are generated using standard EDA tools (Cadence) and standard cell libraries (TSMC 65nm). These are coupled with thermal parameters and simulated using COMSOL Multiphysics® to generate high-fidelity 2D thermal maps.
• Resolution: Thermal maps are represented as $256 \times 256$ pixel grids, where each pixel corresponds to approximately $3.9\mu m \times 3.9\mu m$ of silicon, aligning with the characteristic thermal diffusion length in silicon over microsecond timescales.
• Scope: The primary dataset covers a temperature range of $25^\circ\text{C}$ to $55^\circ\text{C}$ (typical early-design regimes), with an extended cross-validation set spanning $25^\circ\text{C}$ to $95^\circ\text{C}$ to test robustness under stress.

B. Network Architecture

ThermAl employs a hybrid U-Net architecture enhanced with specific mechanisms to handle thermal dynamics:

1. Input Encoding: The model accepts a query image ($I$, current thermal state) and a source-target pair ($E, E'$, reference thermal maps from different time steps/workloads).
2. Feature Concatenation & Positional Encoding: Unlike pixel-level concatenation, the framework uses feature-level concatenation of the encoded source, target, and query maps. This is augmented with positional encoding (sinusoidal functions of pixel coordinates) to explicitly embed spatial relationships, crucial for modeling heat diffusion across the chip.
3. Decoder: A decoder with transposed convolutions and skip connections reconstructs the thermal map at the original resolution, preserving fine-grained spatial details.

C. Physics-Informed Regularization

To ensure the model adheres to thermodynamic laws rather than just learning data correlations, a physics-aware regularizer is integrated into the loss function.

• Loss Function: The total loss $L$ combines a pixel-wise Root Mean Squared Error ($L_{RMSE}$) with a physics-based residual term ($L_{physics}$).
• Physical Constraint: The physics term is derived from the Cattaneo–Vernotte (CV) form of the hyperbolic heat equation (a simplified Boltzmann Transport Equation). This formulation accounts for finite thermal propagation speeds and relaxation effects ($\tau$), which classical Fourier diffusion assumes to be instantaneous.
  $$L = L_{RMSE} + \lambda L_{physics}$$
  where $L_{physics}$ penalizes deviations from the heat conduction model: $\frac{\partial T}{\partial t} + \tau \frac{\partial^2 T}{\partial t^2} - \alpha \nabla^2 T$.

3. Key Contributions

1. Generative Framework for Thermal Analysis: ThermAl introduces a conditional generative model that predicts full-chip thermal maps with high accuracy, achieving an inference speedup of approximately 200$\times$ compared to commercial FEM tools (COMSOL).
2. Comprehensive Dataset: The authors assembled a diverse dataset of heat dissipation maps for 200+ circuit configurations using commercial EDA and FEA flows, covering simple gates to complex workloads under varied boundary conditions.
3. Hybrid Architecture: The paper details a novel U-Net architecture augmented with positional embeddings and feature-level concatenation, which significantly improves spatial consistency and prediction accuracy for both transient and steady-state thermal maps.
4. Physics-Aware Learning: By embedding the Cattaneo–Vernotte heat equation into the training loss, the framework ensures physical fidelity, particularly in high-gradient regions and transient regimes, outperforming purely data-driven approaches.
5. Extended Range Validation: The model demonstrates robustness beyond nominal operating conditions, maintaining high accuracy (full-scale RMSE $< 2.2\%$) across an extended temperature range of $25^\circ\text{C}$ to $95^\circ\text{C}$.

4. Experimental Results

• Accuracy: On a $256 \times 256$ grid, ThermAl achieves a Root Mean Squared Error (RMSE) of $0.71^\circ\text{C}$ (standard deviation $0.26^\circ\text{C}$) and a Structural Similarity Index (SSIM) of $0.97$. For higher-resolution $512 \times 512$ inputs, the RMSE is $1.18^\circ\text{C}$.
• Speed: Inference time is 7–10 milliseconds per sample (exceeding 100 maps/second), compared to 3–4 seconds for commercial FEM solvers under comparable conditions.
• Ablation Studies:
  • Removing the physics regularizer increases RMSE from $0.71^\circ\text{C}$ to $1.12^\circ\text{C}$ and reduces SSIM from $0.97$ to $0.91$.
  • Using feature-level concatenation instead of pixel-level concatenation improves RMSE from $1.18^\circ\text{C}$ to $0.71^\circ\text{C}$.
  • Incorporating source-target image pairs reduces RMSE from $1.12^\circ\text{C}$ to $0.84^\circ\text{C}$.
• Generalization: The model maintains low error rates across diverse circuit layouts and workloads, confirming its ability to generalize beyond the specific training configurations.

5. Significance and Claims

The paper positions ThermAl as a practical solution for early-stage thermal hotspot detection and pattern learning within EDA workflows. Its primary significance lies in bridging the gap between the high accuracy of FEM tools and the speed required for iterative design exploration.

• Design Impact: By enabling rapid, layout-aware thermal estimation, ThermAl allows designers to identify and correct thermal issues during placement and floorplanning, potentially preventing costly late-stage redesigns.
• Physical Consistency: Unlike purely data-driven models that may fail under new boundary conditions, ThermAl's physics-informed approach ensures that predictions adhere to fundamental heat transfer laws, enhancing reliability in transient and high-gradient scenarios.
• Limitations and Future Work: The authors modestly acknowledge current limitations, including the restriction to 2D modeling (ignoring vertical heat transfer in stacked 3D ICs), the assumption of constant thermal resistance at the die-ambient interface, and the lack of real-world experimental validation (relying on synthetic FEM data). Future directions include extending the framework to 3D, generalizing across advanced technology nodes (e.g., sub-7nm), and integrating transfer learning strategies.

The code and dataset are made publicly available to facilitate further research in physics-informed thermal analysis.