A neural network framework for learning Green's function

1 INTRODUCTION . Green ’ s function plays an important role in the theoretical research and engineering application of many important partial differential equations ( PDEs ) , such as the Poisson equation , Helmholtz equation , Maxwell equation , and so on . For one thing , Green ’ s function can help solve PDE problems . When Green ’ s function is known , the solutions of a class of PDE problems can be written explicitly in an integral form where Green ’ s function serves as the integral kernel . Therefore , Green ’ s functions can be regarded as the solution operators of PDEs mapping PDE data to the solution . For another , this integral representation makes Green ’ s function a powerful tool to help study the analytical properties of the PDEs in theoretical research . Therefore , numerical methods to compute the Green ’ s function have drawn great attention over the years . However , there are mainly three problems in computing Green ’ s functions . Firstly , solving Green ’ s functions is a high-dimensional problem . Compared with the original PDE , its dimension is doubled , which limits the application of traditional methods to solve Green ’ s function directly . At the same time , the computation cost is much higher than that of solving the original equation . Secondly , the derivative of Green ’ s function is a Dirac delta function . Therefore , singularity exists in Green ’ s functions , making it difficult to obtain a high-precision solution directly . Thirdly , the PDE may be defined on a very complex domain . In addition to the complex boundary , the domain may also be unbounded or has an interface , which also further increases the difficulties in the computation of Green ’ s function . Fortunately , with the rapid development of neural network and deep learning in recent years , machine learning has made important progress in many fields such as image recognition , natural language processing and so on ( Goodfellow et al. , 2016 ) . At the same time , due to the universal approximation ability of neural networks and its great generalization ability for high-dimensional functions , there are numerous works attempting to solve problems in scientific computation with neural networks . Among them , many works focusing on solving parametric equations and learning solution operators are closely related to Green ’ s function , which can also be seen as a solution operator , such as FNO ( Li et al. , 2020 ) , DeepONet ( Lu et al. , 2019 ) , Deep Green ( Gin et al. , 2020 ) . In addition , there are also works , like MOD-net ( Zhang et al. , 2021 ) , using neural network to compute Green ’ s function directly . However , due to the complexity of Green ’ s function and the learning of solution operators , there are three main issues in most of the existing work to be solved . First , most of these works of learning PDE solution operators or Green ’ s function are based on supervised learning , which requires a large amount of solution data as the supervisory signal . The data needed is often obtained by solving a large number of PDEs through traditional methods or via actual measurements . However , in practice , the accurate solution data set may be difficult to obtain , and the computation cost for the traditional method to solve the equation is relatively high . Second , the generalization ability of these methods is not good enough . Because these works require exact solutions or parameters and the source terms of PDEs , the learned solution operator is only an approximation of Green ’ s function , and the solution on the set outside the area covered by the training set will become worse . Last , for some problems such as electromagnetic wave propagation , solving PDEs in an unbounded domain is critical . However , existing methods also use a neural network to approximate the solution operators or Green ’ s function directly , which severely suffer from the difficulty of sampling in unbounded domains . To address these issues and overcome the three difficulties in computing Green ’ s function mentioned above , in this paper , we design a novel neural network based formulation to compute Green ’ s function directly . Firstly , we use the fundamental solution to remove the singularity of the Green ’ s function , such that the equation for Green ’ s function is reformulated into a general smooth high-dimensional equation . Then , neural network based methods are designed to solve this highdimensional problem which is difficult to solve by traditional methods . In particular , we introduce two neural network formulation for this problem : derivative based method and boundary integral equation based ( BIE-based ) method . The idea of derivative based method is similar to PINN ( Raissi et al. , 2019 ) , DGM ( Sirignano & Spiliopoulos , 2018 ) and some other articles ( Berg & Nyström , 2018 ) , which use the residuals of equations and boundary conditions as the loss function , and directly approximate the objective function with a neural network , and calculate each order derivative of the network by automatic differentiation . BIE-based is based on BINet ( Lin et al. , 2021 ) , which is a method for solving parameterized PDEs . BIE-based method transform the PDEs in the whole domain into the boundary integral equation such that the PDE is automatically satisfied and no extra differentiation with respect to network input is needed . It not only performs better than the derivative based method , but also can compute the Green ’ s function in an unbounded domain , which is also the solution operator for PDEs in unbounded domain . In conclusion , the main advantages of our work are summarized as follows : First , we propose a formulation to compute Green ’ s function directly , which is a difficult problem for traditional methods to deal with . In addition , we also apply the Green ’ s function computed by our formulation as the solution operator to solve a class of PDEs with high accuracy . Second , comparing with other methods of learning solution operators of PDEs , exact solutions are not required as the data set in our formulation . It not only reduces the complexity of preparation , but also improves the generalization ability of the calculated Green ’ s function as the solution operator . Third , although the Green ’ s function is very complex , we make full use of the properties of the Green ’ s function itself , so we do not need to design a complex hidden layer network for approximation . Last , our formulation can not only compute the Green ’ s function in any bounded domain , but also of the interface problems . In particular , by utilizing the boundary integral equation , the BIE-based formulation can also solve the Green ’ s function in unbounded domains . 2 RELATED WORK . The study of Green ’ s function has always been a very important issue , and there has been a lot of work . These works such as Duffy ( 2015 ) , Greenberg ( 2015 ) introduce a lot of applications of Green ’ s function . Because of the importance of the Green ’ s function , computing Green ’ s function is also a critical problem . In the notes ( Hancock , 2006 ) , authors gave the analytical expression of Green ’ s function of the Poisson equation of some simple cases , and the paper ( Kukla et al. , 2012 ) discussed Green ’ s function of the Helmholtz equation in or out of the unit disc . However , there is no good method to calculate the Green ’ s function in the general domain . An important application of Green ’ s function is that it can solve a class of PDE , so it can be seen as a solution operator . In recent years , with the development of the deep learning , there has also been a lot of works on learning solution operators of PDEs ( Hsieh et al. , 2019 ; Li et al. , 2020 ; Lu et al. , 2019 ; Gin et al. , 2020 ) . Because of the strong representation ability of neural networks , many works did not want to get only the mapping of spatial coordinates to solutions . They use the neural networks to find the mapping from parameter space to the solution space . The idea of the works ( Li et al. , 2020 ; Gin et al. , 2020 ) is closely related to Green ’ s function . In addition , there is also workZhang et al . ( 2021 ) using the idea of the Green ’ s function to calculate the solution operator . However , in the training process , these methods are difficult to avoid the dependence on the exact solution data and the information of the PDEs such as source term and boundary conditions . 3 FORMULATION . 3.1 PROBLEM DESCRIPTION . In this paper , for simplicity , we focus on the Dirichlet Green ’ s function , but the other type Green ’ s function can be easily handled in GreenNet with a small modification . We consider both problems defined on a single domain , which can be further divided into interior problem and exterior problem , and the interface problem defined on two domains with different PDE parameters separated by a interface , as will be elaborated below . 3.1.1 GREEN ’ S FUNCTION ON A SINGLE DOMAIN . Assuming Ω ⊂ Rd is a bounded domain , and Ωc = Rd\Ω . The interior problem and the exterior problem are formulated as • Interior problem , { Lu ( x ) = f ( x ) in Ω , u ( x ) = g ( x ) on ∂Ω , ( 1 ) • Exterior problem , { Lu ( x ) = f ( x ) in Ωc , u ( x ) = g ( x ) on ∂Ω + some boundary conditions at infinity , ( 2 ) where L is a differential operator . For brevity , we represent the equations in ( 1 ) and ( 2 ) as Lu ( x ) = f ( x ) in Ω∗ , ( 3 ) where Ω∗ = Ω or Ωc for the interior and exterior problems respectively . In this paper , we focus on the Poisson equations and Helmholtz equations , i.e . L = −∆ or−∆−k2 , where k is the wave number . Helmholtz equations are the expansion of wave equation in frequency domain . It is used to describe wave propagation and is widely used in electromagnetics and acoustics . Because it involves wave propagation , Helmholtz equation often appears in the problem in unbounded domain , and because of the instability of Helmholtz equation , its numerical solution has always been an important problem . For the equation ( 3 ) , we can use the corresponding Green ’ s function G ( x ) to find the analytical solution u ( x ) = ∫ Ω∗ G ( x , y ) f ( y ) dy + ∫ ∂Ω ∂G ( x , y ) ∂ny g ( y ) dsy , where G ( x , y ) is a 2d-dimensional function satisfying { LyG ( x , y ) = δ ( x− y ) , ∀ x , y ∈ Ω∗ , G ( x , y ) = 0 , ∀ x ∈ Ω∗ , y ∈ ∂Ω . ( 4 )

Summary: This paper explores a deep learning scheme to solve partial differential equations on regular domains (mostly 2D). This paper focuses on the method of Greens function for solving a class of linear PDE’s. More specifically: the Greens function corresponding to a PDE is a function that depends on (1.) the differential operator associated to the PDE (2.) the boundary and geometry of the domain on which the PDE is desired to be solved. Given this greens function, the solution to the PDE on this domain for *any* future input source function can be obtained quite simply by a convolution with the greens function (and taking care of boundary conditions). Therefore the authors motivate to “learn” this greens function using neural networks.   Furthermore, the authors propose an unsupervised formulation of this approach, by decomposing the greens function into a formulaic fundamental solution (G0) and a learnable residual ( H(.) ).  By way of simple algebraic computations, this function H(.) satisfies simpler constraints and is therefore proposed to be modelled with a deep network and optimised using two different schemes: (a) PINN - a loss function corresponding to the L2 error from constraints (b) BIE based method.