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

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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.