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: The paper proposes to use a CNN to compute an initial guess for the iterative Newton-Rhapson solution of a coupled PDE system used for semiconductor device simulation. To do so, the authors construct a "device template" which parametrizes the design space. The CNN then maps a device configuration in this 6-dim space to the predicted electrostatic potential in the form of a 64x64 grid. The authors provide an analysis showing why predicting the electrostatic potential alone is sufficient. Overall, this approach can provide a simulation speedup of 12x or more. The authors are also planning to publish the dataset generated for the paper.