Hierarchical Geometric Accelerated Optimization, Collision-based Constraint Satisfaction, and Sensitivity Analysis for VLSI Chip Design

现代片上系统（SoC）微处理器的高效设计具有提高性能、降低功耗和降低成本的潜力。这种复杂的工程系统具有层次和相互联系的结构，可能有超过一千亿个组件。该项目将利用该分层结构获得精确有效的最佳芯片布局数值方法，并提供改进半导体晶圆制造所需的灵敏度工具。由此产生的方法将降低设计和制造高性能、节能微处理器的成本和时间。与工业界的合作将确保数值方法的发展，以应对先进半导体设计的现实挑战，同时也确保由此产生的数值工具将广泛传播到工程实践中。此外，这些工程问题对传统的机器学习算法提出了独特的挑战，因为这些问题的数据通常非常昂贵，这就需要构建和训练新的深度神经网络架构，以更好地尊重物理和几何约束，从而减少必要的训练数据，提高复杂物理系统的神经网络表示的泛化性。计算机与电气工程、半导体工业、应用与计算数学之间的深度跨学科合作为交叉培养研究生提供了独特的机会。本项目将开发的理论和计算工具将基于流形上离散狄拉克力学的内在公式，用广义能量表示，Hamilton-Dirac变分积分器及其相互联系，以及辛加速优化，满足不等式约束的变分碰撞算法，以及常微分方程和微分代数方程的几何伴随灵敏度分析。这种方法有望在流形上提供一类内在的、鲁棒的、高效的几何加速优化和伴随设计工具，这些工具适用于复杂的、分层的、相互关联的系统，如现代VLSI芯片，以及基于神经微分方程和群等变神经网络的对称性的深度神经网络的鲁棒和高效训练。通过利用复杂工程系统的分层和互联结构，研究者和合作者将开发准确有效的辛伴随灵敏度分析工具，以促进复杂工程系统的仿真驱动设计。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The efficient design of modern system-on-a-chip (SoC) microprocessors has the potential to improve performance, reduce power consumption, and decrease cost. Such complex engineering systems have a hierarchical and interconnected structure and may have over one hundred billion components. This project will leverage that hierarchical structure to obtain accurate and efficient numerical methods for optimal chip layout and provide the sensitivity tools necessary to improve semiconductor wafer fabrication. The resulting methods will decrease the cost and time to design and manufacture high-performance, power-efficient microprocessors. Collaborations with industry will ensure the development of numerical methods that are responsive to the realistic, real-world challenges of advanced semiconductor design, while also ensuring that the resulting numerical tools will be broadly disseminated into engineering practice. In addition, such engineering problems pose unique challenges for traditional machine learning algorithms, as data for such problems are often prohibitively expensive, which necessitates the construction and training of novel deep neural network architectures that better respect the physical and geometric constraints, thereby reducing the training data necessary and improving the generalizability of such neural network representations of complex physical systems. The deep interdisciplinary collaboration between computer and electrical engineering, the semiconductor industry, and applied and computational mathematics provides unique opportunities for cross-training graduate students.The theoretical and computational tools to be developed in this project will be based on intrinsic formulations of discrete Dirac mechanics on manifolds, expressed in terms of the generalized energy, Hamilton-Dirac variational integrators and their interconnections, together with symplectic accelerated optimization, variational collision algorithms for the satisfaction of inequality constraints, and geometric adjoint sensitivity analysis for ordinary differential equations and differential-algebraic equations. Such an approach is expected to provide a class of intrinsic, robust, and efficient geometric accelerated optimization and adjoint design tools on manifolds that apply to complex, hierarchical, interconnected systems, such as modern VLSI chips, and the robust and efficient training of deep neural networks with symmetries based on neural differential equations and group-equivariant neural networks. By leveraging a complex engineering system's hierarchical and interconnected structure, the investigator and collaborators will develop accurate and efficient symplectic adjoint sensitivity analysis tools to facilitate the simulation-driven design of complex engineering systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.