A neural network framework for learning Green's function
Authors: Guochang Lin
Fukai Chen
Pipi Hu
Xiang Chen
Junqing Chen
Jun Wang
Zuoqiang Shi
Venue: ICLR 2022 Submitted (2022)
Keywords: Green's function
partial differential equation
boundary integral
neural network

Abstract: Green's function plays a significant role in both theoretical analysis and numerical computing of partial differential equations (PDEs). However, in most cases, Green's function is difficult to compute. The troubles arise in the following three folds. Firstly, compared with the original PDE, the dimension of Green's function is doubled, making it impossible to be handled by traditional mesh-based methods. Secondly, Green's function usually contains singularities which increase the difficulty to get a good approximation. Lastly, the computational domain may be very complex or even unbounded. To override these problems, we leverage the fundamental solution, boundary integral method and neural networks to develop a new method for computing Green's function with high accuracy in this paper.  We focus on Green's function of Poisson and Helmholtz equations in bounded domains, unbounded domains and domains with interfaces. Extensive experiments illustrate the efficiency and the accuracy of our method for solving Green's function.  In addition, we also use the Green's function calculated by our method to solve a class of PDE, and also obtain high-precision solutions, which shows the good generalization ability of our method on solving PDEs.

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A NEURAL NETWORK FRAMEWORK FOR LEARNING GREEN'S S FUNCTION
Anonymous authors Paper under double-blind review
ABSTRACT
Green's function plays a significant role in both theoretical analysis and numeri- cal computing of partial differential equations (PDEs). However, most cases, Green's 's function is difficult to compute. The troubles arise in the following three folds. Firstly, compared with the original PDE, the dimension of Green's function is doubled, making it impossible to be handled by traditional mesh-base meth- ods. Secondly, Green's 's function usually contains singularities which increase the difficulty to get good approximation. Lastly, the computational domain may be very complex or even unbounded. To override these problems, we leverage the fundamental solution, boundary integral method and neural networks develop a new method for computing Green's 's function with high accuracy in this paper. We focus on Green's 's function of Poisson and Helmholtz equations in bounded do- mains, unbounded domains and domains with interfaces. Extensive experiments illustrate the efficiency and the accuracy of our method for solving Green's func- tion. In addition, we also use the Green's 's function calculated by our method to solve class of PDE, and also obtain high-precision solutions, which shows the good generalization ability of our method on solving PDEs.
INTRODUCTION
Green's '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 equa- tion, Maxwell equation, and so on. For one thing, Green's 's function can help solve PDE problems. When Green's function is known, the solutions of class of PDE problems can be written explic- itly in an integral form where Green's function serves as the integral kerne.. Therefore, Green's '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 's function a powerful tool to help study the ana- lytical 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 's functions. Firstly, solving Green's functions is nigh-dimensional problem. Compared with the original PDE, its dimension is doubled, which limits the application of traditional methods to solve Green's '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 's function is a Dirac delta function. Therefore, singularity exists in Green's 's functions, making it difficult obtain 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 recent years, ma- chine learning has made important progress in many fields such as image recognition, natural lan- guage 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 's function, which can also be seen as a solution op- erator, such FNO (Li et al. 2020). DeepONet (Lu et al. 2019), Deep Green (Gin al. 2020). In

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addition, there are also works, like MOD-net Zhang et al. 2021). using neural network to compute Green's 's function directly.
However, due to the complexity of Green's 's function and the learning of solution operators, there three main issues in most of the existing work to be solved. First, most of these works of learning PDE solution operators Green's 's function are based on supervised learning, which requires large amount of solution data as the supervisory signal. The data needed is often obtained by solving 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 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 's function men- tioned above, in this paper. design 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 's function, such that the equation for Green's ' function is reformulated into a general smooth high- nigh-dimensional equation. Then, neural network based methods are designed to solve this high- dimensional 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 (Sirigmano & Spiliopoulos] 2018) and some other articles (Berg & Nyström 2018), which use the residuals of equations and boundary conditions as the loss function, and di- rectly approximate the objective function with neural network, and calculate each order derivative of the network by automatic differentiation. BIE-based is based on BINet (Lin et al. 202I), which is method for solving parameterized PDEs. BlE-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 '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 's function directly, which is a difficult problem for traditional meth- ods to deal with. In addition, we also apply the Green's 's function computed by our formulation as the solution operator to solve 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 generaliza- tion ability of the calculated Green's 's function as the solution operator. Third, although the Green's 's function is very complex, we make full use of the properties of the Green's 's function itself, so we do not need to design complex hidden layer network for approximation. Last, our formulation can not only compute the Green's 's function in any bounded domain, but also of the interface problems. In particular, by utilizing the boundary integral equation, the BlE-based formulation can also solve the Green's function in unbounded domains.
2 RELATED WORK
The study of Green's 's function has always been very important issue, and there has been a lot of work. These works such as Duffy (0015). Greenberg (2015) introduce a lot of applications of Green's 's function. Because of the importance of the Green's 's function, computing Green's 's function is also a critical problem. In the notes (Hancock, 2006), authors gave the analytical expression of Green's 's function of the Poisson equation of some simple cases. and the paper (Kukla et al., 2012) discussed Green's 's function of the Helmholtz equation in or out of the unit disc. However, there is no good method to calculate the Green's 's function in the general domain.
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An important application of Green's function is that it solve class of PDE, so it can be as a solution operator. In recent years, with the development of the deep learning, there has also been lot of works learning solution operators of PDEs (Hsieh et al. 2019 Li al. 2020: Lu et al. 2019 Gin et al. 2020). Because of the strong representation ability of neural networks, many works did not want get only the mapping of spatial coordinates solutions. They networks to find the mapping from parameter space to the solution space. The idea of the works (Li al. 2020 Gin al. 2020) is closely related Green's 's function. In addition, there is also workiZhang 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 avoid the dependence on the exact solution data and the information of the PDEs such as source term and boundary conditions.
the neural
FORMULATION
3.1 PROBLEM DESCRIPTION
In this paper, for simplicity, focus 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 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 interface, will be elaborated below.
3.1.1 GREEN'S FUNCTION ON SINGLE DOMAIN
Assuming Ω C Rdd a bounded domain, and 92 R. The interior problem and the exterior problem are formulated as
Interior problem.
[u(()) f(x) Ω u(xx) = g(x) on an,
Exterior problem,
Lu(x) = (xx) 2
(2)
u((x) g(x) on 0n + some boundary conditions at infinity,
where L is differential operator. For brevity, we represent the equations in 1 and
Lu(x) f(x in f2",
where f2* Ω or 2 for the interior and exterior problems respectively.
In this paper, we focus on the Poisson equations and Helmholtz equations, i.c. L= -A - where l 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 acous- tics. 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 important problem.
For the equation (3) we can use the corresponding Green's function G(z)) to find the analytical aG(a.y) solution u(x) Ja- (())d) Jon 8n, g(y)dsy, where G(x, y) is 2d-dimensional function satisfying
{ (,,G(,,)) 8(x y), Vx,yER, G(I,y) = O, VzEM" y 82.
(4)
3.1.2 GREEN'S FUNCTION OF THE INTERFACE PROBLEM
Suppose Ω Rd, which can either be bounded domain just Rdd, An interface I C dd- divides into two regions, i.c., inside (Ω ) and outside (I2+) of the interface, such that = Ω U UT. The interface problem is then formulated as:
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Lu = f,
in S2
1 Ou
(5)
[2]
91, I dm
92. on r.
where is an operator with different parameters inside and outside the interface r, is the outward normal vector the interface r. The bracket denotes the jump discontinuity of the quantity approaching from Ω minus the one from Similar to the single domain case, we focus on
Poisson equations: Lu - ( Vu), where μ is a piecewise constant parameter such that J [41 in Q and - in s
Helmholtz equations: Lu - V (1 Vu) - ek where j and e are also piecewise constant parameters such that fr,
in l and 12, = in 1
Moreover, some boundary condition on A for bounded I (or at infinity for = R should be considered together to make the interface problem well-posed, which will be specified in detail in the experiments.
Utilizing the corresponding Green's 's function, the solution to the interface problem be give by
u(x)= / G( ((()d) + J. (- aG v) (1) G(æ, v)92(v)) dsy
Ony
(6)
+ item corresponding to the boundary condition on an,
and G(r, y) satisfies
L,0(r,v) 5(x,v), VI,y E
[G(x,))] [ 8G( 0, A3 veT,
(7)
Ony
along with some boundary conditions for y E ON, which will be specified in the experiments.
3.2 REMOVE THE SINGULARITY
As mentioned above, the Green's function singular function. However, we can change equation (3) into smooth equation by using the fundamental solutions Go of the original PDEs satisfiying (o, y) = 8(x - 3) for all æ, E Rd, which also can be seen as the Green's 's function of the whole space d For many important PDEs, fundamental solutions can be written explicitly. For the Poisson equation - = (x: in R22 the fundamental solution is Go (2, yl. while the fundamental solution for the Helmholtz equation -Au(x) k²u(()) [(x) in R2
In x
Go(a, y) = HH (k|x= y/) where H. is the Hankel function.
For the Green's 's function on a single domain, we set H v) = G(x,,) v) - Go v). then H is smooth function and satisfies the following equation
(x, = 0, Vz,e
(8)
H (cr - v y an.
For the Green's function of the interface problem, the fundamental solution of the Poisson equa- tion is Gu(z, y) m(y) In |r yl. while the fundamental solution of the Helmholtz equation is Go (a, v) {1 (((y))()) vl). Set (1xx)) G(z,, v) Go v). then H satisfies the following equation
{ (x,,)) 0, (xxx)))] (G v)],
Yx,yEf. Vx Ω veT,
(H(N)) Bny
(Go On,
along with some boundary conditions for y AΩ. After the singularity of the original equation successfully eliminated, we will introduce how to use neural network to solve this problem by combining boundary integral method and neural network.
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3.3 NEURAL NETWORK APPROXIMATION
After removing the singularity, we transform the problem (4) into problem (8) and (9) Although the singularities have been removed, problem (8) and (9) are still of high dimension such that traditional methods are difficult to handle. Therefore, based on the Boundary Integral Network (Lin et al. 2021 and the Physics Informed Neutral Network (Raissi et al. 2019). we introduce two neural network-base based methods. For simplicity, we take problem (8) as an example. The specific method for the interface problem can be obtained similarly.
3.3.1 1
DERIVATIVE
For the interior problem in bounded domain, can also use the PINN type method solve the equaiton (8). We use a neural network (xr, y; 0) to approximate the function (ar, y)- Then can calculate the derivative C, y; )) by automatic differentiation. The loss function can be designed as the following form
L= Σ (2)
Σ (z Go(z).u)))
(10)
3.3.2 BOUNDARY INTEGRAL EQUATION-BASER МETHON
However, the derivative based method is not stable, and the accuracy is poor. More importantly, it can not deal with the problems in unbounded domains. In the literature (Lin al. 2021). au- thors introduced the BINet method, boundary integral equation (BIE) based method to solve the parametric equation in the form of equation (8). In particular,
choose the double layer potential operator expression in our formulation. The neural network structure can be written as the kernel integral form D(h] (x, y) == - -Ja aGo(p.=) (xx, 2)ds,, where Go is the fundamental solution and the function h(x,y) y) E 0 x an is approximated by hidden layer network, which can be selected as MLP or ResNet. The output of this network will satisfy the equation (8) automatically, and we only need to fit the boundary condition, which the loss function is based This is also the reason that this method can handle the exterior PDE problem in unbounded domain. The loss function is
8p.
N
L =
Dih) v1) F (1))=2h(a))) h(2,,v)) v.) + Go(22,11)1.
(11)
NUMERICAL RESULTS
In the implementation, we choose ResNet structure neural network introduced in Deep Ritz method (Weinan & Yu 2018). In the experiments, we choose the Adam optimizer to minimize the loss function, and we training the neural network on a single GPU of Tesla V100.
4.1 THE GREEN'S S FUNCTION OF PDES IN BOUNDED DOMAINS
The Poisson equation in the unit disc. In this experiment, consider the Green's 's function of the Poisson equation in the unit disc. From the notes (anccock) 2006), we can know the Green's 's function of the unit disc of the Poisson equation has the explicit expression, G(x,y) w)
1 In ppp where
2
E S and y' = T and r = ææ - yl. ,' = |z Vl, ly]. Then we can compare the exact Green's 's function and the numerical Green's 's function calculated by GreenNet.
In the training process, for the BlE-based method, we choose ResNet with blocks with 100 neu- rons per layer and ReLU activate function. For every 500 epochs, 80 new are randomly generated. After after 50000 epochs training with learning rate 0.00001, we randomly generate 100 æ and 800 y and compute the G(z,, Mj) of these 80000 points (I;, Mj) to obtain an average relative L2 error of the Green's 's function G(x, y) of 4.66e-3. For the derivative- based method, we used the neural net- work of same size. For every 500 epochs, 40000 new (x,, v) E Ω x Ω 10000 new (zy. E Ω x On are randomly generated for the PDE loss and boundary condition loss respectively. Similarly, after 50000 epochs, the relative T.2 error of the Green's 's function G(x, y) is 5.54e-2.
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Table 1: Relative I,2 error of the Green's function section 4.1 Helmholtz equation
structure
Poisson equation
BIE-based Derivative-base based
8.56c-3 7.44c-3
5.54e-2
8.75e-1
201
Derivativ 0427..
Dedvative lusel
BIE-based
(b) 3+0002923.0.0674)
Figure 1: The difference between the exact Green's function G(z, y) of the unit disc and Green's function calculated by derivative-based and BIE-based methods with fixed æ = (0.8427, 0.4386) and (0.2923, 0.0674).
The figure show the difference of y with fixed x between the exact Green's 's function G( y) of the unit disc and Green's function calculated by derivative-based and BIE-based methods. We can see the relative 7.2 error BIE based method still small when is close to the boundary. However, the error of derivative based method increases significantly.
The Helmholtz equation in the square. In this experiment, we consider the Green's function of the following equation
-u((x) Au(x)= = [() in S2 u u(x) 0 a
(12)
where Ω = 1-, The network structure and training details of the two methods are consistent with the previous experiment. Because the Green's function of Helmholtz equation is complex number, we add output of the network to represent the imaginary part. In this example, for some fixed Fy, we solve equation (8) about y as the ground truth. should noted that equation has to be solved again for each x, so the computation cost is very expensive. The table [ also show the relative L error between ground truth and the numerical solution of Green's 's function. We can find derivative-based method failed. This is because the oscillation of Helmholtz solution will increase the difficulty for derivative based method. The superiority of BIE based method over derivative based method is shown here.
Next, we show the ability on solving the Helmholtz equation in by Green's 's function computed by BIE based method. We consider using the Green's 's function solve the equations with different source terms /, where f belongs to the following set [[(c] 2
Tz J2 (C2742)) e2 = (#1, (x)) E Ω and (a), (a (sin(a), cos(a + m/2)]. It can be seen that has 100 combinations. The histgram of the figure 2 show the distribution of the relative L error between the exact solutions and the solutions calculated by our Green's function with different f.
1.... .5),, where
The relative error of the solution of the equation is between 0.01 and 0.045, which shows the stability of solving the equation with Green's function.
4.2 THE GREEN'S FUNCTION OF PDES UNBOUNDED DOMAINS
The Helmholtz equation out of the bow-tie antenna. In this experiment, we consider the Green's function of Helmholtz equation in unbounded domains. The neural network of this experiement is same as the previous and training process also similar. The exterior PDE problems are common in scattering problems, which is closely related to Helmholtz equation. We consider the following

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nurtar oifennt
05
0.08
0.00
rOEENe L2 enor
distribution
(b) exact solution
(c) numerical solution
(d) error
Figure 2: (a) the distribution of relative L of the solution of the PDE with 100 different
source terms; (b) and (c) is the real part of the exact solution and the numerical solution with the source term f (72n² 4) cos(6.T2) + x x/2) sin(66r222; (d) is the absolute error between exact solution and numercal solution.
(a) exa,real
(b) num,real
exa,imag
(d) num,imag
(e) error
Figure 3: For fixed x=(-0.4430,0009933 (a) and (b) the real part of the exact G(x, y) and nu- merical G(r, y) respectively. (c) and (d) are corresponding imaginary part. (c) is the absolute error between exact and numerical G(x,
PDE problem.
-u((z) 16u(x) = in in
lim 0 4i)ux)) = 11/2)-1/2)
(13)
x(x) = 0 on 0n,
000 Or
where the wave number in equation 13) and the limit of |x| is the Sommerfeld condition. Let us consider more practical scenario, a receiving antenna electromagnetic simulation problem. Assume we have bow-tie antenna in length and width, which is type broad-bandwidth antenna. Its structure is 2 dimensional, implemented on printed circuit board. Therefore, for simplicity, we consider simulating the field within the 2-D space. The shape of the bow-tic antenna can be seen in the figure 3
We can also use our formulation to compute the Green's function in the domain outside the bow- tie antenna. Because the domain is unbounded, we can only used the BlE-based method. After training, the relative L error the Green's 's function is 4.98e-2. When the wave number increases and the domain of the PDE problem becomes more complex, high-precision Green's function approximation is still obtained.
4.3 THE GREEN'S S FUNCTION OF THE INTERFACE PROBLEM
The Poisson equation in a square with flower-shaped interface Let Ω be the square {(x, y) |x] < 1, |y| < 1}, I' is parameterized by (0.5 cost 0.1 cos 5t cos t, 00. - 0.1 cos 5t sin with t [0, 2x). 800 points are sampled on both I' and 0Ω for boundary integral. Take μr 0.5, p2 in the Poisson equation. Dirichlet boundary condition is imposed on an such that the Green's function this problem satisfies along with G(r, y) 0, Va E 2, aQ.
In the training process, we choose ResNet with 6 blocks with 100 neurons per layer. For every 500 epochs, 100 new randomly generated. After after 100000 epochs of training with learning rate 0.00004, we fix 100 newly generated and obtain average relative L2 error of the Green's function G(æ, 3) of 4.05%. Further more, for two fixed æ, the exact Green's 's function (obtained by traditional boundary integral method), numerical solution obtained by neural network and absolute emror shown in Fig. A The results show that the proposed method can solve the Green's 's function for the interface problem accurately.

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41
41
48
(a) exact
(b) numerical
cc) error
0.02
42
42
2005
(d) exact
(e) numerical
(1) error
x (-0.1804, -0-001804.0004114) E Q with relative L error 1.94%, while figure (f)-(f) are for x (0.7899, 0.44177 Ω with relative I,2 error 2.67%. First and second column: the exact solution and the numerical solution; Third column: absolute error between exact solution and numerical solution.
Figure 4: Figures (a)-(c) are for =
The Helmholtz equation in Re with square interface Let I be the square {(x, y) |a|| |y 1}. on which 800 points are sampled for the boundary integral. Take μl 2, #2 = 1,
1, k in the Helmholtz equation. The Sommerfeld condition is required at infinity, i... lim ( or ik /2k/222)
o[2 1/2), implying that gi & ik /E2P2)
o[[x] 1/3). which automatically satisfied if H (z. y) is written in the boundary integral form.
set the same those in the Poisson equation example. The average relative L2 сиror of the Green's 's function G(z, y) is 4.63%. For two fixed x, the exact Green's function (obtained by traditional boundary integral method), numerical solution obtained by neural network and error shown in Fig. 5 implying the effectiveness of the proposed method in computing Green's function.
The network and training hyperparameters
(a) exact, real
(b) num, real
(c) exact, imag
(d) num, imag
(c) error
ce
tl
095
(f) exact, real
(g) num, real
(h) exact, imag
num, imag
(i) error
Figure 5: Figures (a)-(e) are for = (-0.8559, -..7664 E Ω with relative 2 error 3.98%, while figure (f-()) are for x = (1.6069, -0.5065) E Ω with relative L error 2.73%. First and second columm: the real part of the exact solution and the numerical solution; Third and Forth column: imaginary part of the exact solution and the numerical solution; Fifth column: absolute emoor between exact solution and numerical solution.
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15
45
43
6:31
800
6.M
(a) distribution
(b) exact, real
(c) numerical, real
45
45
(d) exact, imag
(e) numerical, imag
(f) error
Figure 6: (a): histogram of the relative T,2 error of the 100 randomly generated equations. Figures (b)-(f) are the exact solution and numerical solution of one specific equation generated, whose pa- rameters given in the Appendix. (b) and (c): real part of the exact solution and the numerical solution; (d) and (e): imaginary part of the exact solution and the numerical solution; (f): absolute error between exact solution and numerical solution.
Furthermore, after the Green's 's function is learnt by the neural network, to show the generalization ability of the proposed method in solving PDEs, we consider the homogenuous case of problem (5). i.e., f O, while the two jump conditions 91 and 92 are generated by the superposition of class of parameterized function, which will given in the Appendix. We randomly generate 100 sets of parameters, and solve the interface problem using the learnt Green's function. The histogram of the relative 1.2 errors of the numerical solutions solution and corresponding eπгоr for one of parameters all the relative L errors are below 4%. Therefore, not only can the proposed method accurately solve class of PDEs accurately, but also this method has natural generalization ability the PDE information.
100 equations are shown Fig. (a), while the
given in Fig. 6 It can be that
5 CONCLUSION
In this paper, a novel neural network based method for learning Green's 's function proposed. By utilizing the fundamental solution to remove the singularity in Green's function, the PDEs required for Green's function reformulated into smooth high-dimensional problem. Two neural network based methods propsed solve this high-dimensional problem. The first one, based the idea of PINN. uses the neural network to directly approximate Green's 's function and take the residual of the differential equation and the boundary conditions as the loss. The second one is based on the recently proposed BINet, in which the solution is written in an boundary integral form such that the PDE automatically satisfied and only boundary terms need to be fitted.
Extensive experiments are conducted and three conclusions can be drawn from the results. First, the proposed method can effectively learn the Green's 's function Poisson and Helmholtz equations in bounded domains, unbounded domains and domains with interfaces with high accuracy. Second, BINet-based method outperforms the PINN-based method in the accuracy of Green's function and the capability to handle problems in unbounded domains. Third, the Green's 's function obtained can be utilized to solve class of PDEs accurately, and shows good generalization ability over the PDE data, including the source term and boundary conditions.
Although the proposed method exhibits great performance in computing Green's 's function, the de- pendence on the fundamental solution hinders the application of this method in varying coefficient problems or equations without explicit fundamental solution. This will be further investigated the future.

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REFERENCES
Jens Berg and Kaj Nyström. A unified deep artificial neural network approach to partial differential equations in complex geometries. Neurocomputing, 317:28 41, 2018.
Dean G Duffy. Green's 's functions with applications. Chapman and Hall/CRC, 2015.
Craig R Gin, Daniel E Shea, Steven L Brunton, and J Nathan Kutz. Deepgreen: Deep learning
of green's 's functions for nonlinear boundary value problems. arXiv preprint arXiv:2101.07206, 2020.
lan Goodfellow, Yoshua Bengio, Aaron Courville, and Yoshua Bengio. Deep learning, volume 1.
MIT Press, 2016.
Michael D Greenberg. Applications of Green's 's functions in science and engineering. Courier Dover Publications, 2015.
Matthew J Hancock. Method of green's ' functions. Lecture notes, 2006.
Jun-Ting Hsieh, Shengjia Zhao, Stephan Eismann, Lucia Mirabella, and Stefano Ermon. Learning 0900001200, 2019.
neural pde solvers with convergence guarantees. arXiv preprint
Stanislaw Kukla, Urszula Siedlecka, and Izabela Zamorska. Green's functions for interior and terior helmholtz problems. Scientific Research of the Institute of Mathematics and Computer Science, 11(1::33-2,, 2012.
Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, An- drew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895, 2020.
Guochang Lin, Pipi Hu, Fukai Chen, Xiang Chen, Junqing Chen, Jun Wang, and Zuoqiang Shi. Binet: Learning to solve partial differential equations with boundary integral networks, 2021.
Lu Lu, Pengzhan Jin, and George Em Karniadakis. Deeponet: Learning nonlinear operators for iden- tifying differential equations based the universal approximation theorem of operators. arXiv preprint
2019.
Maziar Raissi, Paris Perdikaris, and George Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686-707, 2019.
Justin Sirignano and Konstantinos Spiliopoulos. Dgm: A deep learning algorithm for solving partial differential equations. Journal of computational physics, 13311-36966,, 2018.
E Weinan and Bing Yu. The deep ritz method: deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics, 6(1):1-12, 2018.
Lulu Zhang, Tao Luo, Zhang, Zhi-Qin John Xu, and Zheng Ma. Mod-net: machine learning approach via model-operator-datta network for solving pdes. arXiv preprint arXiv:2107.03673. 2021.
APPENDIX
A EXPERIMENTAL DETAILS
A.1 HELMHOLTZ PROBLEM WITH INTERFACE
The exact solution to the interface problem is designed as
Ci (()))el (x,1)
u(x, v/ =
(14)
Σ cz(1) (1) /1 (a 2o))2 (1)))) (e.))
10

---

Under review as a conference paper at ICLR 2022
where fci, 02 K1, R2, 2o. yo is set of randomly generated parameters satisfying kj + k3 k? and (zo(t), (o()) E St, vi. Specifically, C1, C2 ~ UO,, k1 = cos 0, f/ k sin with 0 [0, 2 Zo. Yo U|00.8, 0.8]. 91 and 92 be directly computed using the exact solution, and the solution to the PDE can be given by
u(x) = (= dz g1(v) G(z, w)92(1) dsy.
Ony
take / = 3. The parameters of the exact solution corresponding to Fig. (b)-(() are given in the following table.
Table 2: Parameter setting for Fig. 6 (b)-(1)
Index
i=
=
2
= 3
ci 0.5550
0.9934
0.2986
k1 1.9959
1.6667
-0,9091
k2 -0.1282
-1,1056
1.7814
02
0.3963
0.3051
0.3642
30
0.2850
0.1625
0.2190
0.5724
-0,1154
0.6809


Reviews:
[{"review_id": "lSzCEr910qw", "reviewer": "Reviewer_dXB8", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Pros\n\n- I find the formulation of the learning problem to be elegant, specifically: by way of using information about the PDE operator and inputing the fundamental solution and learning the residual ( H(.) ) \n- The proposed solution is unsupervised and does not require precomputed accurate solutions to the PDE’s\n\nCons\n\n- I find the empirical evaluation of this paper to be significantly lacking. I see no comparisons with previous approaches like PINN, DeepGreen, DeepRitz etc. which are also deep learning solutions to solve similar PDEs. To me, even a starting comparison (error values and graphs similar to Figures 3-6) with standard solvers like Finite Difference/Finite Element based approaches would help in understanding the empirical contributions of this paper. As of now, even if I am inclined to accept that the proposed method works yielding low errors to challenging PDEs with different test-source terms, I have no idea if it works competitively to prior work. \n- The necessity of inputing the fundamental solution is a big drawback.\n- No results are shown for PDEs in higher dimensions, where the problem of estimating the greens functions is very significant, and hence the need for a “learned” solver like it is proposed here.\n\nQuestions:\n\n- Is there any particular reason for choosing that specific source set for ‘test’ (Page 6)? What happens for source functions outside this set? does the method generalise? Discuss. \n- What is the significance of the comparison between PINN and BINet? i.e. Why does the method outlined in this paper work well in conjunction with BINet and not PINN? \n- It would be convincing to see a demonstration of learning a “high-dimensional” greens function using the method proposed in this paper. ", "summary_of_the_paper": "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. \n\nFurthermore, 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.", "summary_of_the_review": "I find the idea proposed in this paper to be interesting, and the fact that the proposed method is unsupervised by carefully parametrizing a part of the PDE solution with neural networks is also a plus point. However, I am inclined to weigh heavily towards penalising this paper for a lack of an extensive empirical analysis, since I am not convinced on the efficacy of the proposed solution in comparison to existing approaches (learning or otherwise). ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "gP-zBc1sADg", "reviewer": "Reviewer_Rxs8", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Green's functions of PDEs play important roles in numerous physical problems and their numerical computation is certainly an important problem and is worth to investigate. This paper made an interesting attempt for solving Green's functions using neural networks. However, from my perspective, the contribution of this paper is incremental and may not be sufficiently novel for ICLR. Here is a list of strength and weaknesses.\n\nStrength\n\n- The authors provide extensive numerical experiments for computing 2D Green functions of Poisson and Helmholtz problems.\n\nWeaknesses\n\n- The first weakness I have is that this work is that it does not discuss why neural networks is superior than the classical numerical methods such as  finite element method or finite difference method. I do not see any numerical comparisons between neural network methods and classical methods. In fact, since the authors mainly focus on Green's function on 2D, I do not see any advantage of using neural networks compared to these classical methods. My understanding is that neural networks can be very powerful for approximation high dimensional functions, but due to the lack of robustness in the training process, the neural network approximation can have low accuracy even in low dimensions, while in this setting FEM and FD are usually more robust and accurate. The authors need to justify the precise advantages of neural networks. \n\n- The proposed method seems relying on that the singularity of the Green's function is known a priori so that one can separate that from approximation. However, what if such a priori information is not available? For example, can the authors comment on how the proposed method could be used to find the Green's function of Helmholtz equation in a non-homogeneous medium? \n\n- Here the authors presented numerical results in 2D. I would recommend the authors provide at least one example for solving the Green's function of a high dimensional PDE (e.g. dim larger than 10). This way, one can possibly see the potential benefits of  neural networks.\n\nSome typos and further comments\n\n- In the second line below section 3.3.1,  \"equaiton\" should be \"equation\". \n- Three lines below equation (12), \" is complex number\" should be \" is complex\". In the same line, \"add a output of the network\" should be \"add an output of the network\" \n- Similar typos appear here and there in the paper. The authors should run a spell check before submitting the paper.", "summary_of_the_paper": "This paper proposes a neural network based method for computing Green's functions of Poisson and Helmholtz equations defined on various domains. The idea is to first subtract the target Green's function with some background green's function which is assumed to be known and easy to evaluation. Then the next step is approximate by neural networks the difference function that satisfies the same PDE problem with nonhomogeneous boundary conditions. The authors compared two methods for solving the PDE problems, based on the PINN framework and the boundary integral equation framework.", "summary_of_the_review": "To summarize, I regret that I am inclined to reject the submission since  the contribution of this paper is incremental and is not be sufficiently novel for ICLR.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "1: The main claims of the paper are incorrect or not at all supported by theory or empirical results.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "H8If-7HFrft", "reviewer": "Reviewer_mJPD", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "I enjoyed reading the paper. Unfortunately, I do not think the approach is novel and the results are significant. The concept of using neural networks to approximate Green's function is not new. In addition, this method heavily depends on the existence of a fundamental solution hence is restricted in solving linear PDEs. \n\nFurthermore, the error of the solution seems to be high as for both cases (Poisson and Helmholtz) the maximum error is about 4%. This is not satisfying especially considering the linearity of the problem. The author should compare their results to the classical boundary element method and I suspect that BEM can bring the error below 0.1% easily. \n\n", "summary_of_the_paper": "The paper proposed a neural network based method for solving linear PDEs. The approach is based on approximating the solution using a combination of PINN and neural network based Green's function method. The architecture is then verified against the classical Poisson and Helmholtz equation. ", "summary_of_the_review": "In summary, I do not think the paper in its current form meets the standard of ICLR hence recommend a strong rejection. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "DYPOHcrkza", "reviewer": "Reviewer_YLib", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This is the wrong venue for this paper.  There is already a lot of expertise in solving PDEs.\nGreen's functions are a particular method for solving PDEs.  If this method is really better than state of the art methods for solving PDEs, the paper should be sent to a numerical analysis (PDE solving) journal, for a review by experts.  \nStrengths: the paper studies linear PDEs and compares a couple methods.\nWeaknesses: the paper does not properly motivate the challenges solving PDEs, the relative advantages of using Green's functions, versus the many other methods out there.\nMain weakness: how do we know that the methods solved here satisfy the basic requirements of any PDE solver: convergence.  Dnns are wildly unpredictable, so, in principle, anything can happen. ", "summary_of_the_paper": "This paper proposes to use neural networks to learn Green's functions.\nGreen's functions are a classical technique for solving linear PDEs.", "summary_of_the_review": "This is a very narrow paper, which focuses on a particular method for solving linear PDEs.\nHowever, there is very little motivation.  There are already decades worth of methods for solving PDEs. Using neural networks probably does worse than most off the shelf methods.  It would take a lot more evidence to be convincing that this method is better than the best standard PDE method for each problem. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: All reviewers vote for rejecting this paper. The main points of criticism shared by the reviewers are missing novelty and missing/unclear significance of the contribution. There was no rebuttal, so this is a clear reject.
Decision: Reject