Poisson Energy Formulation for Floorplanning: Variational Analysis and Mathematical Foundations

2025-10-09

AI Summary

This work addresses the module placement problem in VLSI physical design—a discrete geometric optimization task over a bounded domain with non-overlapping constraints. We propose a variational-spectral framework grounded in Poisson energy. First, we establish the equivalence between Poisson energy and the Sobolev $H^{-1}$ norm, revealing its intrinsic low-pass filtering property and deriving a quantitative lower bound linking energy to overlap area—thereby providing a rigorous mathematical foundation for PDE-based regularization. Integrating the Neumann Poisson equation, spectral decomposition, projected gradient descent, and the Wasserstein-2 gradient flow, our method enables non-local, global optimization for high-dimensional placements. We prove global convergence of the algorithm with local linear convergence rate. Experiments demonstrate significant overlap suppression, enhanced layout stability, and improved interpretability.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

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• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Arranging many modules within a bounded domain without overlap, central to the Electronic Design Automation (EDA) of very large-scale integrated (VLSI) circuits, represents a broad class of discrete geometric optimization problems with physical constraints. This paper develops a variational and spectral framework for Poisson energy-based floorplanning and placement in physical design. We show that the Poisson energy, defined via a Neumann Poisson equation, is exactly the squared H^{-1} Sobolev norm of the density residual, providing a functional-analytic interpretation of the classical electrostatic analogy. Through spectral analysis, we demonstrate that the energy acts as an intrinsic low-pass filter, suppressing high-frequency fluctuations while enforcing large-scale uniformity. Under a mild low-frequency dominance assumption, we establish a quantitative linear lower bound relating the Poisson energy to the geometric overlap area, thereby justifying its use as a smooth surrogate for the hard nonoverlap constraint. We further show that projected gradient descent converges globally to stationary points and exhibits local linear convergence near regular minima. Finally, we interpret the continuous-time dynamics as a Wasserstein-2 gradient flow, revealing the intrinsic nonlocality and global balancing behavior of the model. These results provide a mathematically principled foundation for PDE-regularized optimization in large-scale floorplanning and related geometric layout problems.

Problem

• Developing variational framework for Poisson energy-based VLSI floorplanning
• Establishing mathematical link between Poisson energy and geometric overlap
• Analyzing convergence properties of gradient descent in floorplan optimization

Innovation

• Poisson energy as smooth surrogate for nonoverlap constraints
• Spectral analysis reveals intrinsic low-pass filtering behavior
• Wasserstein-2 gradient flow interpretation for global balancing