Original: M ESH -F REE U NSUPERVISED L EARNING -B ASED PDE S OLVER OF F ORWARD AND I NVERSE PROBLEMS
Revised: M ESH -F REE U NSUPERVISED L EARNING -B ASED PDE S OLVER OF F ORWARD AND I NVERSE PROBLEMS

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: We introduce a novel neural network-based partial differential equations solver for forward and inverse problems.
Revised: We introduce a novel neural network-based partial differential equations solver for forward and inverse problems.

Original: The solver is grid free, mesh free and shape free, and the solution is approximated by a neural network.
Revised: The solver is grid free, mesh free and shape free, and the solution is approximated by a neural network.

Original: We employ an unsupervised approach such that the input to the network is a points set in an arbitrary domain, and the output is the set of the corresponding function values.
Revised: We employ an unsupervised approach such that the input to the network is a points set in an arbitrary domain, and the output is the set of the corresponding function values.

Original: The network is trained to minimize deviations of the learned function from the PDE solution and satisfy the boundary conditions.
Revised: The network is trained to minimize deviations of the learned function from the PDE solution and satisfy the boundary conditions.

Original: The resulting solution in turn is an explicit smooth differentiable function with a known analytical form.
Revised: The resulting solution in turn is an explicit smooth differentiable function with a known analytical form.

Original: Unlike other numerical methods such as ﬁnite differences and ﬁnite elements, the derivatives of the desired function can be analytically calculated to any order.
Revised: Unlike other numerical methods such as ﬁnite differences and ﬁnite elements, the derivatives of the desired function can be analytically calculated to any order.

Original: This framework therefore, enables the solution of high order non-linear PDEs.
Revised: This framework therefore, enables the solution of high order non-linear PDEs.

Original: The proposed algorithm is a uniﬁed formulation of both forward and inverse problems where the optimized loss function consists of few elements: ﬁdelity terms of Land L ∞ norms, boundary conditions constraints and additional regularizers.
Revised: The proposed algorithm is a uniﬁed formulation of both forward and inverse problems where the optimized loss function consists of few elements: ﬁdelity terms of Land L ∞ norms, boundary conditions constraints and additional regularizers.

Original: This setting is ﬂexible in the sense that regularizers can be tailored to speciﬁc problems.
Revised: This setting is ﬂexible in the sense that regularizers can be tailored to speciﬁc problems.

Original: We demonstrate our method on a free shape 2D second order elliptical system with application to Electrical Impedance Tomography (EIT).
Revised: We demonstrate our method on a free shape 2D second order elliptical system with application to Electrical Impedance Tomography (EIT).

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Partial differential equations are fundamental in science and mathematics with wide applications in medical imaging, signal processing, computer vision, remote sensing, electromagnetism and more.
Revised: Partial differential equations are fundamental in science and mathematics with wide applications in medical imaging, signal processing, computer vision, remote sensing, electromagnetism and more.

Original: Classical methods such as ﬁnite differences, ﬁnite volume and ﬁnite elements are numerical discretization-based methods where the domain is divided into a uniform grid or polygon mesh.
Revised: Classical methods such as ﬁnite differences, ﬁnite volume and ﬁnite elements are numerical discretization-based methods where the domain is divided into a uniform grid or polygon mesh.

Original: The differential equation is then reduced to a system of algebraic equations.
Revised: The differential equation is then reduced to a system of algebraic equations.

Original: These methods may have some limitations: the solution is numeric and may suffer from high condition number, highly dependent on the discretization and even the second derivative is sensitive to noise.
Revised: These methods may have some limitations: the solution is numeric and may suffer from high condition number, highly dependent on the discretization and even the second derivative is sensitive to noise.

Original: In the last few years, deep learning and neural network-based algorithms are extensively used in pattern recognition, image processing, computer vision and more.
Revised: In the last few years, deep learning and neural network-based algorithms are extensively used in pattern recognition, image processing, computer vision and more.

Original: Recently, the deep learning approach had been adopted to the ﬁeld of PDEs as well by converting the problem into a machine learning one.
Revised: Recently, the deep learning approach had been adopted to the ﬁeld of PDEs as well by converting the problem into a machine learning one.

Original: In Supervised learning , the network maps an input to an output based on example input-output pairs.
Revised: In Supervised learning , the network maps an input to an output based on example input-output pairs.

Original: This strategy is used in inverse problems, where the input to the network is a set of observations/measurements (e.g. x-ray tomography, ultrasound) and the output is the set of parameters of interest (tissue density etc.)
Revised: This strategy is used in inverse problems, where the input to the network is a set of observations/measurements (e.g. x-ray tomography, ultrasound) and the output is the set of parameters of interest (tissue density etc.)

Original: Feigin et al.
Revised: Feigin et al.

Original: ; Lucas et al.
Revised: ; Lucas et al.

Original: ; McCann et al.
Revised: ; McCann et al.

Original: Seo et al.
Revised: Seo et al.

Original: Unsupervised learning on the other hand is a self-learning mechanism where the natural structure presents within a set of data points is inferred.
Revised: Unsupervised learning on the other hand is a self-learning mechanism where the natural structure presents within a set of data points is inferred.

Original: Algorithms for forward and inverse problems in partial differential equations via unsupervised learning were recently introduced.
Revised: Algorithms for forward and inverse problems in partial differential equations via unsupervised learning were recently introduced.

Original: The indirect approach utilizes a neural network as a component in the solution.
Revised: The indirect approach utilizes a neural network as a component in the solution.

Original: Li et al.
Revised: Li et al.

Original: (2018) for example, proposed the NETT (Network Tikhonov) approach to inverse problems.
Revised: (2018) for example, proposed the NETT (Network Tikhonov) approach to inverse problems.

Original: NETT considers regularized solutions having a small value of a regularizer deﬁned by a trained neural network.
Revised: NETT considers regularized solutions having a small value of a regularizer deﬁned by a trained neural network.

Original: Khoo & Ying (2018) introduced a novel neural network architecture, SwitchNet, for solving the wave equation based inverse scattering problems via providing maps between the scatterers and the scattered ﬁeld.
Revised: Khoo & Ying (2018) introduced a novel neural network architecture, SwitchNet, for solving the wave equation based inverse scattering problems via providing maps between the scatterers and the scattered ﬁeld.

Original: Han et al.
Revised: Han et al.

Original: (2018) developed a deep learning-based approach that can handle general high-dimensional parabolic PDEs.
Revised: (2018) developed a deep learning-based approach that can handle general high-dimensional parabolic PDEs.

Original: To this end, the PDEs are reformulated using backward stochastic differential equations.
Revised: To this end, the PDEs are reformulated using backward stochastic differential equations.

Original: The latter is solved by a temporal discretization and the gradient of the unknown solution at each time step is approximated by neural network.
Revised: The latter is solved by a temporal discretization and the gradient of the unknown solution at each time step is approximated by neural network.

Original: Li et al.
Revised: Li et al.

Original: (2019) approximate the solution map of linear and nonlinear problems by a deep network.
Revised: (2019) approximate the solution map of linear and nonlinear problems by a deep network.

Original: Direct algorithms solve the forward problem PDEs by directly approximating the solution with a deep neural network.
Revised: Direct algorithms solve the forward problem PDEs by directly approximating the solution with a deep neural network.

Original: The network parameters are determined by the optimization of a cost function such that the optimal solution satisﬁes the PDE, boundary conditions and initial conditions.
Revised: The network parameters are determined by the optimization of a cost function such that the optimal solution satisﬁes the PDE, boundary conditions and initial conditions.

Original: Chiaramonte & Kiener (2017) addressed the forward problem by constructing a one layer network which satisﬁes the PDE within the domain.
Revised: Chiaramonte & Kiener (2017) addressed the forward problem by constructing a one layer network which satisﬁes the PDE within the domain.

Original: The boundary conditions were analytically integrated in the cost function.
Revised: The boundary conditions were analytically integrated in the cost function.

Original: They demonstrated their algorithm on the Laplace and hyperbolic conservation law
Revised: They demonstrated their algorithm on the Laplace and hyperbolic conservation law

Original: PDEs.
Revised: PDEs.

Original: Sirignano & Spiliopoulos (2017) proposed a deep learning forward problem solver for high dimensional PDEs.
Revised: Sirignano & Spiliopoulos (2017) proposed a deep learning forward problem solver for high dimensional PDEs.

Original: Their algorithm was demonstrated on the American option free-boundary equation.
Revised: Their algorithm was demonstrated on the American option free-boundary equation.

Original: Raissi et al.
Revised: Raissi et al.

Original: (2017) focused on continuous time models and solved the Burgers and Shr¨odinger equations.
Revised: (2017) focused on continuous time models and solved the Burgers and Shr¨odinger equations.

Original: In this work we address the forward and inverse PDE problems via a direct unsupervised method.
Revised: In this work we address the forward and inverse PDE problems via a direct unsupervised method.

Original: Our key contributions are four fold: (1) inverse problems can be solved in the same framework as the forward problems.
Revised: Our key contributions are four fold: (1) inverse problems can be solved in the same framework as the forward problems.

Original: In the forward part we extend the standard L 2 -based ﬁdelity term in the cost function by adding L ∞ -like norm.
Revised: In the forward part we extend the standard L 2 -based ﬁdelity term in the cost function by adding L ∞ -like norm.

Original: Moreover, (3) some regularization terms which impose a-priori knowledge on the solution can be easily incorporated.
Revised: Moreover, (3) some regularization terms which impose a-priori knowledge on the solution can be easily incorporated.

Original: (4) An important feature of our construction is the ability to handle free-form domain in a mesh free manner.
Revised: (4) An important feature of our construction is the ability to handle free-form domain in a mesh free manner.

Original: We demonstrate our algorithm by a second order elliptic equation, in particular the Electrical Impedance Tomography (EIT) application on circular and three other arbitrary domains.
Revised: We demonstrate our algorithm by a second order elliptic equation, in particular the Electrical Impedance Tomography (EIT) application on circular and three other arbitrary domains.

Original:  M ATHEMATICAL F ORMULATION
Revised:  M ATHEMATICAL F ORMULATION

Original: Let Ω be a bounded open and connected subset of R d , and
Revised: Let Ω be a bounded open and connected subset of R d , and

Original: A =
Revised: A =

Original: A ( x ) =
Revised: A ( x ) =

Original: ( a ij ( x )) be any given d × d symmetric positive deﬁnite matrix of functions for 1 ≤ i, j ≤ d .
Revised: ( a ij ( x )) be any given d × d symmetric positive deﬁnite matrix of functions for 1 ≤ i, j ≤ d .

Original: Let b
Revised: Let b

Original: = b ( x ) =
Revised: = b ( x ) =

Original: ( b j ( x )) be any given n-tuple of functions and let c
Revised: ( b j ( x )) be any given n-tuple of functions and let c

Original: = c ( x ) be any given function.
Revised: = c ( x ) be any given function.

Original: A second order operator L is said to be in divergence form, if L acting on some u has the form
Revised: A second order operator L is said to be in divergence form, if L acting on some u has the form

Original: [Equation]
Revised: [Equation]

Original: where we use the Einstein summation convention.
Revised: where we use the Einstein summation convention.

Original: Consider the partial differential problem with
Revised: Consider the partial differential problem with

Original: Dirichlet boundary conditions
Revised: Dirichlet boundary conditions

Original: [Equation]
Revised: [Equation]

Original: The forward problem solves u given the coefﬁcients θ := { a ij ( x ) , b j ( x ) , c ( x ) } while the inverse problem determines the coefﬁcients set θ given u .
Revised: The forward problem solves u given the coefﬁcients θ := { a ij ( x ) , b j ( x ) , c ( x ) } while the inverse problem determines the coefﬁcients set θ given u .

Original: The proposed algorithm approximates the solutions in both problems by neural networks u ( x ; w u ) , a ij ( x ; w ij ) , b j ( x ; w b ) , c ( x ; w c ) such that the networks are parameterized by w u , w ij , w b , w c , and the input to the network is x ∈ R d .
Revised: The proposed algorithm approximates the solutions in both problems by neural networks u ( x ; w u ) , a ij ( x ; w ij ) , b j ( x ; w b ) , c ( x ; w c ) such that the networks are parameterized by w u , w ij , w b , w c , and the input to the network is x ∈ R d .

Original: Figure 1 depicts a network architecture of u in R 2 .
Revised: Figure 1 depicts a network architecture of u in R 2 .

Original: The network consists of few fully connected layers with tanh activation and linear sum in the last layer.
Revised: The network consists of few fully connected layers with tanh activation and linear sum in the last layer.

Original: The network is trained to satisfy the PDE with boundary conditions by minimizing a cost function.
Revised: The network is trained to satisfy the PDE with boundary conditions by minimizing a cost function.

Original: In the forward problem
Revised: In the forward problem

Original: [Equation]
Revised: [Equation]

Original: and in the inverse problem
Revised: and in the inverse problem

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: The ﬁrst two terms enforce the solution to satisfy the equation.
Revised: The ﬁrst two terms enforce the solution to satisfy the equation.

Original: The ﬁrst term minimizes the error in
Revised: The ﬁrst term minimizes the error in

Original: L 2 sense while the second term minimizes the maximal error.
Revised: L 2 sense while the second term minimizes the maximal error.

Original: The second term is important since the L 2 term only forces the equation up to a set of zero measure.
Revised: The second term is important since the L 2 term only forces the equation up to a set of zero measure.

Original: The L ∞ term takes care of possible outliers.
Revised: The L ∞ term takes care of possible outliers.

Original: The third term imposes boundary conditions and the last term is a regularizer which can be tailored to the application.
Revised: The third term imposes boundary conditions and the last term is a regularizer which can be tailored to the application.

Original: There are few advantages of this setting.
Revised: There are few advantages of this setting.

Original: First, the solutions are smooth analytic functions and are therefore analytically differentiable .
Revised: First, the solutions are smooth analytic functions and are therefore analytically differentiable .

Original: In addition, this framework enables setting a prior knowledge on the solution by designing the regularizers R F and R I .
Revised: In addition, this framework enables setting a prior knowledge on the solution by designing the regularizers R F and R I .

Original: Moreover, the training procedure is mesh free.
Revised: Moreover, the training procedure is mesh free.

Original: Unlike ﬁnite differences or ﬁnite elements methods, we use random points in the domain and its boundary in the course of the optimization of equation 3 and equation 4, see Figure 2.
Revised: Unlike ﬁnite differences or ﬁnite elements methods, we use random points in the domain and its boundary in the course of the optimization of equation 3 and equation 4, see Figure 2.

Original: This means that the solution does not depend upon a coordinate mesh and we can also deﬁne an arbitrary regular domain Ω .
Revised: This means that the solution does not depend upon a coordinate mesh and we can also deﬁne an arbitrary regular domain Ω .

Revised: Suppose u ( x ) is Lipchitz with constant ν , then the approximation of the network Θ is given by

Revised: [Equation]

Revised: where p ∈ [0 , ∞ ) , N number of neurons in a layer and L number of layers Shen et al.

Original:  A PPLICATION TO E LECTRICAL I MPEDANCE T OMOGRAPHY
Revised:  A PPLICATION TO E LECTRICAL I MPEDANCE T OMOGRAPHY

Original: Let us address a special case of equation 1,
Revised: Let us address a special case of equation 1,

Original: [Equation]
Revised: [Equation]

Original: We assume that 0 < σ ( x ) ∈ C 1 (Ω) , which guarantees existence and uniqueness of a solution u ∈ C 2 (Ω) Evans (2010).
Revised: We assume that 0 < σ ( x ) ∈ C 1 (Ω) , which guarantees existence and uniqueness of a solution u ∈ C 2 (Ω) Evans (2010).

Original: The elliptical system equation 5 was addressed by Siltanen et al.
Revised: The elliptical system equation 5 was addressed by Siltanen et al.

Original: (2000) in the context of Electrical
Revised: (2000) in the context of Electrical

Original: Impedance Tomography (EIT) which is a reconstruction method for the inverse conductivity problem.
Revised: Impedance Tomography (EIT) which is a reconstruction method for the inverse conductivity problem.

Original: The function σ indicates the electrical conductivity density, and u is the electrical potential.
Revised: The function σ indicates the electrical conductivity density, and u is the electrical potential.

Original: An electrical current
Revised: An electrical current

Original: [Equation]
Revised: [Equation]

Original: is applied on electrodes on the surface ∂ Ω , where κ is the angle in polar coordinate system along the domain boundary, ϕ is the phase and ν is the normal unit.
Revised: is applied on electrodes on the surface ∂ Ω , where κ is the angle in polar coordinate system along the domain boundary, ϕ is the phase and ν is the normal unit.

Original: The resulting voltage u | ∂ Ω = uis measured through the electrodes.
Revised: The resulting voltage u | ∂ Ω = uis measured through the electrodes.

Original: The conductivity σ is determined from the knowledge of the
Revised: The conductivity σ is determined from the knowledge of the

Original: Dirichlet-to-Neumann map or voltage-to-current map
Revised: Dirichlet-to-Neumann map or voltage-to-current map

Original: [Equation]
Revised: [Equation]

Original: Mueller & Siltanen (2012); Alsaker & Mueller (2018); Bera et al.
Revised: Mueller & Siltanen (2012); Alsaker & Mueller (2018); Bera et al.

Original: ; Fan & Ying (2019).
Revised: ; Fan & Ying (2019).

Original: We demonstrate our framework by solving the forward and inverse problem of equation 5 which is a ﬁrst step towards a full tomography.
Revised: We demonstrate our framework by solving the forward and inverse problem of equation 5 which is a ﬁrst step towards a full tomography.

Original: Following Mueller & Siltanen (2012), we simulate the voltage measurement u | ∂ Ω by the Finite Elements Method (FEM) given three variants of a conductivity phantom σ ( x ) depicted in Figure 3.
Revised: Following Mueller & Siltanen (2012), we simulate the voltage measurement u | ∂ Ω by the Finite Elements Method (FEM) given three variants of a conductivity phantom σ ( x ) depicted in Figure 3.

Original: We calculate the FEM solution with different triangle mesh densities and select as ground truth the one such that ﬁner meshes do not improve the numerical solution.
Revised: We calculate the FEM solution with different triangle mesh densities and select as ground truth the one such that ﬁner meshes do not improve the numerical solution.

Original: With our suggested method, the forward problem determines the electrical potential u in the whole domain Ω given σ , while the inverse problem uses the approximated u and calculates the conductivity σ given that σ | ∂ Ω = σ 0 .[Table]
Revised: With our suggested method, the forward problem determines the electrical potential u in the whole domain Ω given σ , while the inverse problem uses the approximated u and calculates the conductivity σ given that σ | ∂ Ω = σ 0 .

Original:  F ORWARD P ROBLEM
Revised:  F ORWARD P ROBLEM

Original: In the forward problem the conductivity σ ( x i ) and boundary conditions u 0 ( x b ) are given for random points set { x i } ∈ Ω ⊂ R 2 , { x b } ∈ ∂ Ω ⊂ R 2 with sets size of N s and N b respectively.
Revised: In the forward problem the conductivity σ ( x i ) and boundary conditions u 0 ( x b ) are given for random points set { x i } ∈ Ω ⊂ R 2 , { x b } ∈ ∂ Ω ⊂ R 2 with sets size of N s and N b respectively.

Original: A neural network having the architecture shown in Figure 1 approximates u ( x ) .
Revised: A neural network having the architecture shown in Figure 1 approximates u ( x ) .

Original: Let
Revised: Let

Original: [Equation]
Revised: [Equation]

Original: The cost function equation 3 is then rewritten as
Revised: The cost function equation 3 is then rewritten as

Original: [Equation]
Revised: [Equation]

Original: The ﬁrst term is the L 2 norm of the differential operator, the second term is a relaxed version of the inﬁnity norm where we take the mean value of the top-K values of |L i | .
Revised: The ﬁrst term is the L 2 norm of the differential operator, the second term is a relaxed version of the inﬁnity norm where we take the mean value of the top-K values of |L i | .

Original: The third term imposes the boundary conditions and the last term serves as a regularizer of the network parameters.
Revised: The third term imposes the boundary conditions and the last term serves as a regularizer of the network parameters.

Original: The ﬁrst phantom is shown in Figure 3 left.
Revised: The ﬁrst phantom is shown in Figure 3 left.

Original: The impedance values associated with the background ellipses and circle were 1 , 5 and 2 respectively.
Revised: The impedance values associated with the background ellipses and circle were 1 , 5 and 2 respectively.

Original: The original piecewise constant function σ was slightly smoothed by a Gaussian kernel.
Revised: The original piecewise constant function σ was slightly smoothed by a Gaussian kernel.

Original: Figure 4 shows the forward problem results for current ψ with n = 1 and ϕ = π/ 8 .
Revised: Figure 4 shows the forward problem results for current ψ with n = 1 and ϕ = π/ 8 .

Original: The left column is the FEM solution which is referred to as ground truth, where the top row indicates the solution u and the bottom row the derivative of u with respect to x calculated as the ﬁnite difference approximation of the FEM result.
Revised: The left column is the FEM solution which is referred to as ground truth, where the top row indicates the solution u and the bottom row the derivative of u with respect to x calculated as the ﬁnite difference approximation of the FEM result.

Original: The middle column depicts the outcome of the proposed method where ∂u/∂x is an analytical derivative of our result.
Revised: The middle column depicts the outcome of the proposed method where ∂u/∂x is an analytical derivative of our result.

Original: The right column shows the outcome of the
Revised: The right column shows the outcome of the

Original: DGM method Sirignano & Spiliopoulos (2017) which is a special case of equation 3 with λ = 1 and µ = 0 .
Revised: DGM method Sirignano & Spiliopoulos (2017) which is a special case of equation 3 with λ = 1 and µ = 0 .

Original: Quantitative results of the Mean square error (MSE) and PSNR are summarized in Table 1.
Revised: Quantitative results of the Mean square error (MSE) and PSNR are summarized in Table 1.

Original: Clearly, the proposed method outperforms the DGM method since the weighting parameters and the
Revised: Clearly, the proposed method outperforms the DGM method since the weighting parameters and the

Original: L ∞ norm play a signiﬁcant role in the loss function.
Revised: L ∞ norm play a signiﬁcant role in the loss function.

Original: The forward problem was repeated using phantom 2 where the background and circle conductivities were 1 and 0 .
Revised: The forward problem was repeated using phantom 2 where the background and circle conductivities were 1 and 0 .

Original: 2 respectively (Figure 3, middle).
Revised: 2 respectively (Figure 3, middle).

Original: Four different current combinations were applied.
Revised: Four different current combinations were applied.

Original: Quantitative results are summarized in lines 1 - 4 of Table 2.
Revised: Quantitative results are summarized in lines 1 - 4 of Table 2.

Original: Figure 5 shows the results for both u and ∂u/∂x 
Revised: Figure 5 shows the results for both u and ∂u/∂x for n = 2 and ϕ = π/ 2 .

Revised: The right column presents the relative error deﬁned as e ( x, y )

Revised: = ( u fem ( x, y ) − u ( x, y )) / max( u fem ) .

Original: [Figure]
Revised: [Figure]

Original: MSE and PSNR are noted in line 2 of Table
Revised: MSE and PSNR are noted in line 2 of Table

Original: [Figure]
Revised: [Figure]

Revised: [Table]

Original: [Table]
Revised: [Table]

Revised: 1.30e45.

Revised: 8.63e37.

Revised: 3 , Ω π/6.42e47.

Original: I NVERSE P ROBLEM
Revised: 5.64e39. 3 , Ω π/1.08e34. 2.61e41. 3 , Ω π/1.08e44.

Revised: 4.51e40.

Revised: 3 , Ω π/5.93e36.

Revised: 2.74e41.

Revised: 3 , Ω π/1.22e44.

Revised: 9.17e37.

Original:  In the inverse problem, the electrical potential u ( x ) is known while σ ( x ) is unknown.
Revised: 3 , Ω π/1.74e31. 2.24e41.  I NVERSE P ROBLEM In the inverse problem, the electrical potential u ( x ) is known while σ ( x ) is unknown.

Original: Since we have a network which approximates u ( x ) , we can evaluate it at any point x .
Revised: Since we have a network which approximates u ( x ) , we can evaluate it at any point x .

Original: The objective function equation 4 then takes the form
Revised: The objective function equation 4 then takes the form

Original: [Equation]
Revised: [Equation]

Original: As in the forward problem, the ﬁrst two terms enforce σ to satisfy the PDE, where L i is deﬁned in equation 6.
Revised: As in the forward problem, the ﬁrst two terms enforce σ to satisfy the PDE, where L i is deﬁned in equation 6.

Original: The third term imposes the boundary conditions, and the fourth regularizes the network parameters.
Revised: The third term imposes the boundary conditions, and the fourth regularizes the network parameters.

Original: The last term is the total variation regularization ( p = 1 ) which promotes the solution towards a piecewise constant function.
Revised: The last term is the total variation regularization ( p = 1 ) which promotes the solution towards a piecewise constant function.

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: MSE and PSNR are summed up in lines 1 - 2 of Table 3.

Original: Additional inverse problem generalization may exploit multiple u approximations for several currents ψ j .
Revised: Additional inverse problem generalization may exploit multiple u approximations for several currents ψ j .

Original: The σ calculation thus, simultaneously relies on all { u j } , resulting a more stable solution,
Revised: The σ calculation thus, simultaneously relies on all { u j } , resulting a more stable solution,

Original: [Equation]
Revised: [Equation]

Original: Then equation 8 is generalized to
Revised: Then equation 8 is generalized to

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Revised: I

Original: [Equation]
Revised: [Equation]

Original:  Reconstruction results by the generalized inverse problem are shown in Figure 6 and lines 1 - 2 in
Revised: σ [Equation] Reconstruction results by the generalized inverse problem are shown in Figure 6 and lines 1 - 2 in

Original: Table 3.
Revised: Table 3.

Original: In both phantoms we used four current combinations ψ n,ϕ : (1 , 0) , (1 , π/ 4) , (2 , 0) , (2 , π/ 4) , and σ 0 = 1 .
Revised: In both phantoms we used four current combinations ψ n,ϕ : (1 , 0) , (1 , π/ 4) , (2 , 0) , (2 , π/ 4) , and σ 0 = 1 .

Original:  F REE S HAPE G EOMETRY
Revised:  F REE S HAPE G EOMETRY

Original: We applied the proposed method to arbitrary domains Ω 1 , Ω 2 and Ω 3 , see Figure 7.
Revised: We applied the proposed method to arbitrary domains Ω 1 , Ω 2 and Ω 3 , see Figure 7.

Original: The random sample points within the domain and along its boundary can be easily obtained as can be seen in
Revised: The random sample points within the domain and along its boundary can be easily obtained as can be seen in

Original: Figure 7 right problem results applied on phantom 3 are presented in lines 5 - 10 of
Revised: Figure 7 right. The forward problem results applied on phantom 3 are presented in lines 5 - 10 of

Original: Table 2.
Revised: Table 2.

Original: Figure 8 shows the results for n = 2 and ϕ =
Revised: Figure 8 shows the results for n = 2 and ϕ =

Original: π/ 4 for the three domains.
Revised: π/ 4 for the three domains.

Original: The outcome of the generalized inverse problem equation 11 is shown in Figure 9 and lines 3 - 5 of Table 3.  
Revised: The outcome of the generalized inverse problem equation 11 is shown in Figure 12 and lines 3 - 5 of Table 3.  A DDITIONAL I NVERSE P ROBLEM E XAMPLES 7. D IFFUSION E QUATION [Equation] Result is shown in Figure 9.

Revised: [Figure]

Original: I MPLEMENTATION D ETAILS
Revised: 7. W AVE E QUATION

Revised: [Equation] Result is shown in Figure 10.  I MPLEMENTATION D ETAILS

Original: The network architecture has 4 layers having 26 , 26 , 26 and 10 neurons.
Revised: The network architecture has 4 layers having 26 , 26 , 26 and 10 neurons.

Original: The algorithm was implemented by TensorFlow Ten (2015) using the ADAM optimizer which is a variant of the SGD algorithm.
Revised: The algorithm was implemented by TensorFlow Ten (2015) using the ADAM optimizer which is a variant of the SGD algorithm.

Original: We used the same hyper parameters set in all our experiments.
Revised: We used the same hyper parameters set in all our experiments.

Original: Batch size= 1000 , decaying learning rate starting at 5 e − 3 .
Revised: Batch size= 1000 , decaying learning rate starting at 5 e − 3 .

Original: The learning rate was factored by 0 .
Revised: The learning rate was factored by 0 .

Original: 8 every 200 epochs,
Revised: 8 every 200 epochs,

Original: Ns = 80000 , N b = 1200 , λ = 0 .
Revised: Ns = 80000 , N b = 1200 , λ = 0 .

Original: 01 , α = 1 e − 8 , K = 40 , µ = 0 .
Revised: 01 , α = 1 e − 8 , K = 40 , µ = 0 .

Original: 01 , and β = 0 .
Revised: 01 , and β = 0 .

Original: Running time on Intel i7-8650u CPU was about 15 minutes for the forward problem and 13 minutes for the generalized inverse problem. 
Revised: Running time on Intel i7-8650u CPU was about 15 minutes for the forward problem and 13 minutes for the generalized inverse problem. [Table]

Revised: Figure 11 presents the sensitivity of the algorithm to the number of sampled points.

Revised: As expected, with large number of points we obtain a plateau while with small and intermediate number (up to 50000)

Revised: the error is not monotonic due to local minima

Revised: [Table]

Original:  D ISCUSSION
Revised:  D ISCUSSION

Original: Deep networks by their nature use compositions of simple functions such as matrix multiplication and non-linear activations like sigmoid or tanh.
Revised: Deep networks by their nature use compositions of simple functions such as matrix multiplication and non-linear activations like sigmoid or tanh.

Original: This structure (i) enables the approximation of an arbitrary function Hornik (1989) and (ii) is inherently differentiable.
Revised: This structure (i) enables the approximation of an arbitrary function Hornik (1989) and (ii) is inherently differentiable.

Original: The network architecture dictates the number of degrees of freedom which in turn enables the expressibility of complex functions.
Revised: The network architecture dictates the number of degrees of freedom which in turn enables the expressibility of complex functions.

Original: In this work we present a uniﬁed framework for the solution of forward and inverse problems in partial differential equations in an arbitrary domain.
Revised: In this work we present a uniﬁed framework for the solution of forward and inverse problems in partial differential equations in an arbitrary domain.

Original: The algorithm relies on direct approximation of the unknown function by a neural network which yields an analytical smooth solution.
Revised: The algorithm relies on direct approximation of the unknown function by a neural network which yields an analytical smooth solution.

Original: The network is trained to satisfy the PDE and boundary conditions in an unsupervised fashion by the minimization of a cost function.
Revised: The network is trained to satisfy the PDE and boundary conditions in an unsupervised fashion by the minimization of a cost function.

Original: The optimization procedure depends on random points set within the domain and its boundary.
Revised: The optimization procedure depends on random points set within the domain and its boundary.

Original: The problem is therefore mesh free with free-form domain.
Revised: The problem is therefore mesh free with free-form domain.

Original: We introduce a cost function which is composed of both L 2 and L ∞ ﬁdelity terms, boundary conditions constraints and additional regularizers.
Revised: We introduce a cost function which is composed of both L 2 and L ∞ ﬁdelity terms, boundary conditions constraints and additional regularizers.

Original: The mathematical rationale of the L ∞ norm which controls possible outliers was veriﬁed in our numerical experiments.
Revised: The mathematical rationale of the L ∞ norm which controls possible outliers was veriﬁed in our numerical experiments.

Original: We also stress the robustness of our approach exempliﬁed by having the same hyper parameters set for all our experiments.
Revised: We also stress the robustness of our approach exempliﬁed by having the same hyper parameters set for all our experiments.

Original: This framework alleviates several problems of the ﬁnite differences and ﬁnite elements methods.
Revised: This framework alleviates several problems of the ﬁnite differences and ﬁnite elements methods.

Original: In particular meshing, discretization and derivatives approximation are solved in a simple and natural way.
Revised: In particular meshing, discretization and derivatives approximation are solved in a simple and natural way.

Original: Numerical solutions of PDEs in an arbitrary domain are of extreme importance, in particular in medical imaging applications.
Revised: Numerical solutions of PDEs in an arbitrary domain are of extreme importance, in particular in medical imaging applications.

Original: The framework is demonstrated by an elliptic system in R 2 applied to
Revised: The framework is demonstrated by an elliptic system in R 2 applied to

Original: Electrical Impedance Tomography for both forward and inverse problems.
Revised: Electrical Impedance Tomography for both forward and inverse problems.

Original: Promising results were achieved for complex and non monotonic functions.
Revised: Promising results were achieved for complex and non monotonic functions.

Original: Rigorous analysis of the approximation error and its relation to the network architecture and design are under current study and will be published elsewhere.
Revised: Rigorous analysis of the approximation error and its relation to the network architecture and design are under current study and will be published elsewhere.

Original: Future research includes also full tomography solution, higher dimensional problems and other classes of PDEs such as dynamic non-linear equations (Burgers, Navier-Stokes etc.)
Revised: Future research includes also full tomography solution, higher dimensional problems and other classes of PDEs such as dynamic non-linear equations (Burgers, Navier-Stokes etc.)