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 together and the overall system is highly nonlinear. The Newton-Raphson method (Stoer & Bulirsch, 2002) is known to be one of the most robust methods to solve a set of coupled nonlinear equations. When the method converges to the solution, the error decreases rapidly as the Newton iterations proceed. To achieve a rapid convergence, it is crucial that initial guess for the solution needs to be close enough to the real solution; otherwise, the method converges very slowly or may even diverge. Although we are interested in obtaining a solution at a specific boundary condition which is determined by an applied voltage, even obtaining an approximated solution to initiate the Newton-Raphson method successfully is challenging. In literature, in order to prepare an initial guess for the target boundary condition, several intermediate boundary conditions are introduced and solutions with those boundary conditions are computed sequentially (Synopsys, 2014) . It, however, increases the overall computation time significantly. If the initial solution that is sufficiently close to the final one is provided by any means, we can save huge computational cost of calculating several unnecessary solutions. Instead, we propose to learn an approximate initial solution of a set of coupled PDE for a target boundary condition by an artificial neural network. Specifically, when a set of labeled images is available, a neural network can be trained to generate a similar image for a given label. The trained model generates a numerical solution for a target boundary condition. We show that the proposed initial solution by our method can speed up the device simulation significantly by providing a better initial guess. We summarize our contributions as follows: • We derive a compact solution for PDE systems based on the electrostatic potential. As a result, the network size is reduced by a factor of three. Since the electrostatic potential is well bounded, the normalization issue can be avoided. • For addressing various semiconductor devices, we propose a device template that can address various device structures with a set of hyper-parameters. Since the electrical characteristics of semiconductor devices are largely determined by the physical sizes of internal components, handling grids with varying size and spacing is particularly important. • We propose a convolutional neural network (CNN) which generates the electrostatic potential from the device parameters. It can be used to accelerate the device simulation. • Compared with the conventional method, the simulation time is significantly reduced (at least 12 times) with the proposed method while the numerical stability is not hampered. • Results from the convolutional neural network are compared with other methods such as an alternative architecture (the fully-connected network) and the local linear regression. • Our approach can be extended to the complex two-dimensional domains. Preliminary results are shown.

2.1. NEURAL NETWORKS FOR SOLVING DIFFERENTIAL EQUATIONS

Recently, there have been many attempts to build a neural network to solve a differential equation (Han et al., 2018; Long et al., 2018; Piscopo et al., 2019; Raissi et al., 2019; Winovich et al., 2019; Zhu et al., 2019; Obiols-Sales et al., 2020; Lu et al., 2020; Xiao et al., 2020) 



. Among them, the Poisson equation is particularly of importance in the semiconductor device simulation. The Poisson equation plays a fundamental role in the semiconductor device simulation, by connecting the electrostatic potential and other physical quantities. In Magill et al. (2018), the Laplace equation (the Poisson equation with a vanishing source term) is considered for a nanofluidic device. A fully-connected neural network is trained to minimize the loss function, which combines the residue vector and the boundary condition. The authors assume a specific two-dimensional structure and the mixed boundary condition is applied. Another attempt to solve the Poisson equation is suggested in Özbay et al. (2019). The Poisson equation with the Dirichlet boundary condition is considered. It is decomposed into two equations. One is the Poisson equation with the homogeneous Neumann boundary condition. The other one is the Laplace equation with the Dirichlet boundary condition. A convolutional neural network architecture is adopted and the entire source term and the grid spacing are used as the input parameters. The network is trained with randomly generated source terms and

