LEARNING TO SOLVE NONLINEAR PARTIAL DIFFER-ENTIAL EQUATION SYSTEMS TO ACCELERATE MOS-FET SIMULATION

Abstract

Semiconductor device simulation uses numerical analysis, where a set of coupled nonlinear partial differential equations is solved with the iterative Newton-Raphson method. Since an appropriate initial guess to start the Newton-Raphson method is not available, a solution of practical importance with desired boundary conditions cannot be trivially achieved. Instead, several solutions with intermediate boundary conditions should be calculated to address the nonlinearity and introducing intermediate boundary conditions significantly increases the computation time. In order to accelerate the semiconductor device simulation, we propose to use a neural network to learn an approximate solution for desired boundary conditions. With an initial solution sufficiently close to the final one by a trained neural network, computational cost to calculate several unnecessary solutions is significantly reduced. Specifically, a convolutional neural network for MOSFET (Metal-Oxide-Semiconductor Field-Effect Transistor), the most widely used semiconductor device, are trained in a supervised manner to compute the initial solution. Particularly, we propose to consider device grids with varying size and spacing and derive a compact expression of the solution based upon the electrostatic potential. We empirically show that the proposed method accelerates the simulation by more than 12 times. Results from the local linear regression and a fully-connected network are compared and extension to a complex twodimensional domain is sketched.

1. INTRODUCTION

Nonlinear partial differential equations (PDEs) appear frequently in many science and engineering problems including transport equations for certain quantities like heat, mass, momentum, and energy (Fischetti & Vandenberghe, 2016) . The Maxwell equations for the electromagnetic fields (Jackson, 1999) , which govern one of the fundamental forces in the physical world, is one of the examples. By calculating the solution of those equations, the status of system-under-consideration can be characterized. In the machine learning society, solving a set of coupled partial differential equations has become an important emerging application field. (de Avila Belbute-Peres et al., 2020; Sanchez-Gonzalez et al., 2020) Among many nonlinear partial differential equations, we consider the semiconductor device simulation (Grasser et al., 2003) . The simulation is a pivotal application to foster next-generation semiconductor device technology at scale. Since the technology development heavily relies on the device simulation results, if the simulation time reduces, the turnaround time also significantly reduce. In order to reduce the simulation time, acceleration techniques based upon the multi-core computing have been successfully applied (Rupp et al., 2011; Sho & Odanaka, 2017) . However, the number of cores cannot be exponentially increased and the cost also increases drastically as the number of cores involved increases. Moreover, as engineers submit many simulation jobs for a group of semiconductor devices, computing resources available to each simulation job is limited. As an alternative, we propose to improve the efficiency of the simulation per se. In the semiconductor device simulation, a solution of a system of partial differential equations is numerically calculated with a certain boundary condition. Those differential equations are coupled 1

