Data-Driven Electrostatics Analysis based on Physics-Constrained Deep learning

Computing the electric potential and electric field is important for modeling and analysis of VLSI chip and high speed circuits. For instance, it is an important step for DC analysis for high speed circuits as well as dielectric reliability and capacitance extraction for VLSI interconnects. In this paper, we propose a new data-driven meshless 2D analysis method, called PCEsolve, of electric potential and electric fields based on the physics-constrained deep learning scheme. We show how to formulate the differential loss functions to consider the Laplace differential equations with voltage boundary conditions for typical electrostatic analysis problem so that the supervised learning process can be carried out. We apply the resulting PCEsolve solver to calculate electric potential and electric field for VLSI interconnects with complicated boundaries. We show the potential and limitations of physics-constrained deep learning for practical electrostatics analysis. Our study for purely label-free training (in which no information from FEM solver is provided) shows that PCEsolve can get accurate results around the boundaries, but the accuracy degenerates in regions far away from the boundaries. To mitigate this problem, we explore to add some simulation data or labels at collocation points derived from FEM analysis and resulting PCEsolve can be much more accurate across all the solution domain. Numerical results demonstrate that the PCEsolve achieves an average error rate of 3.6% on 64 cases with random boundary conditions and it is 27.5× faster than COMSOL on test cases. The speedup can be further boosted to ~ 38000× in single-point estimations. We also study the impacts of weights on different components of loss functions to improve the model accuracy for both voltage and electric field.