Learning to Solve Nonlinear Partial Differential Equation Systems To Accelerate MOSFET Simulation

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 can not 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 NewtonRaphson 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 RELATED WORK . 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 a predicted solution shows a good agreement with the numerical solution . In order to consider a complex two-dimensional geometry , a rectangular domain covering the original geometry is introduced in Winovich et al . ( 2019 ) and a convolutional network is used . A neural network is used to solve the Poisson equation in a L-shape domain in Lu et al . ( 2020 ) . Quite recently , it has been independently proposed to accelerate the computational fluid dynamics simulations by predicting an appropriate initial guess from a convolutional neural network ( ObiolsSales et al. , 2020 ) . The basic idea is quite similar with this work and such a coincidence strongly demonstrates that the proposed approach is not restricted to a relatively narrow application domain . 2.2 NEURAL NETWORKS IN SEMICONDUCTOR DEVICE SIMULATION . Researchers are much interested with application of machine learning technique to the semiconductor device simulation ( Carrillo-Nuñez et al. , 2019 ; Bankapalli & Wong , 2019 ; Han & Hong , 2019 ; Souma & Ogawa , 2020 ) . However , the actual research activities are quite diverse . In many cases , the neural networks are merely used as an efficient descriptor of the input-output relation without considering the internal physical quantities ( Carrillo-Nuñez et al. , 2019 ; Bankapalli & Wong , 2019 ) . On the other hand , in their quantum transport simulation , Souma & Ogawa ( 2020 ) try to predict the electron density profile from the electrostatic potential . Although application of neural networks on the semiconductor device simulation is in its infancy , it has a huge potential . 3 PRELIMINARY : SEMICONDUCTOR DEVICE SIMULATION . We briefly introduce the semiconductor device simulation as a preliminary . As shown in Figure 1a , a multi-dimensional device structure has multiple terminals ( yellow bars in the figure ) whose voltages are controllable . Since the planar MOSFET has the translational symmetry along the direction perpendicular to the plane in Figure 1a , the three-dimensional device structure can be treated as a two-dimensional one . The terminal currents ( in Ampere ) under the determined terminal voltages ( in Volt ) are the output of the simulator , as shown in Figure 1b . As the motion of electrons yields the current , the terminal current itself can be only calculated from the internal physical quantities . ( See Figure 1c . ) Since the electrons are charged particles , the electric field due to the net charge density is also affected by the electronic motion . Therefore , we need to consider both effects of electronic motion and electric field by solving two equations , to calculate the terminal currents . The first one is the continuity equation for electrons : ∇ · Jn = 0 , ( 1 ) where the electron current density vector , Jn , is given by Jn = qµnnE+ qDn∇n , ( 2 ) where the elementary charge ( q ) , the electron mobility ( µn ) , and the electron diffusion constant ( Dn ) are scalar parameters and the electron density ( n ) is a position-dependent unknown variable . Note that the current density ( Jn ) also depends on the electric field vector ( E ) . Similar relations hold for holes with minor modification . The second equation is the Gauss law , also called as the Poisson equation : ∇ · ( E ) = q ( p− n+N+dop ) , ( 3 ) where p is the hole density , is the permittivity ( Jackson , 1999 ) and N+dop is the positively charged impurity density . Under the electrostatic approximation , the electric field vector can be expressed in terms of the electrostatic potential ( φ ) as E = −∇φ . ( 4 ) With the electrostatic potential , the above set of equations has two unknown variables ; n and φ . Note that nonlinearity is originated from ( 2 ) . The first term in the right-hand-side of ( 2 ) is proportional to nE , which is nonlinear with respect to unknown variables of n and φ . Therefore , when φ is fixed , the continuity equation becomes linear . For numerical analysis , we need to discretize the equations . In general , the i-th component of the residual vector can be written as ri = fi ( φ1 , ... φN , n1 , ... , nN , p1 , ... , pN ) − si = 0 , ( 5 ) where N is the number of grid points , fi ( · ) is a nonlinear function , and si is a constant . si becomes nonzero only at the boundary nodes and depends on the voltages applied to the device terminals . Since the system is nonlinar , the solution at each boundary condition must be computed . Detailed discussion can be found in Appendix A . The set of discretized equations in ( 5 ) is highly nonlinear and the iterative Newton-Raphson method is used to solve the system . Typically , a user wants to know the drain current at VG = VD = VDD , where VDD is the maximum allowed voltage . However , an appropriate initial guess to start the Newton-Raphson method at that condition is not directly available . Only the solution at VG = VD = 0 V can be easily calculated . Starting from the solution with all zero terminal voltages , many intermediate steps toward the target terminal voltages are introduced . Based upon the converged solution at the previous condition as an initial guess , the next step is solved sequentially . Detailed discussion can be found in Appendix B . Therefore , computing several intermediate boundary values is the main source of increase of computation . Let us denote the number of boundary values simulated during the solution procedure as Nstep . To evaluate the acceleration quantitatively , we define the reduction factor of the simulation time , α , and it is well approximated as α ≡ τconv τnn = Nstep ×Nnewton Ndirectnewton ≈ Nstep , ( 6 ) where τconv is the simulation time with the conventional method while τnn is the one with the proposed method . Moreover , Nnewton is the average number of the Newton iterations per a bias condition and Ndirectnewton is the number of the Newton iterations at the target bias condition . Since Nstep is typically larger than 10 , we expect that the reduction factor can be larger than 10 .

Summary: Contribution: Authors propose using a neural network to learn an approximate solution for desired boundary conditions in order to accelerate the semiconductor device simulation. They significately reduce the computational cost to calculate several unnecessary solutions when considering an initial solution sufficiently close to the final one by a convolutional neural network (CNN). To compute this initial solution, authors authors train a MOSFET based convolutional neural network. They empirically show that their proposed method accelerates the simulation by more than 12 times.