We introduce a novel neural network-based partial differential equations solver for forward and inverse problems. The solver is grid free, mesh free and shape free, and the solution is approximated by a neural network. 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. The network is trained to minimize deviations of the learned function from the strong PDE solution and satisfy the boundary conditions. The resulting solution in turn is an explicit smooth differentiable function with a known analytical form. Unlike other numerical methods such as finite differences and finite elements, the derivatives of the desired function can be analytically calculated to any order. This framework therefore, enables the solution of high order non-linear PDEs. 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 L 2 and L ∞ norms that unlike previous methods promote a strong solution. Robust boundary conditions constraints and additional regularizers are included as well. This setting is flexible in the sense that regularizers can be tailored to specific problems. We demonstrate our method on several free shape 2D second order systems with application to Electrical Impedance Tomography (EIT), diffusion and wave equations.
In this work, the authors propose an approach to solve forward and inverse problems in partial differential equations (PDEs) using neural networks. In particular, they consider a specific type of second-order differential operator combined with Dirichlet boundary conditions, and suggest how neural networks with suitable training objectives can be used to solve both types of problems. For a specific elliptic system relevant for Electrical Impedance Tomography (EIT), they then present numerical results obtained with their method and compare those to high-resolution finite element solutions and previous work.
The paper addresses a very general and important problem with wide ranging applications. People have solved PDEs using spectral, finite difference, finite volume or finite element methods for decades and there is a huge body of literate on this subject. The neural network based approach proposed in this paper seems general and simple with encouraging experimental results. However, there are several important points missing in the current paper based on which I would recommend rejecting it. However, I would be willing to increase the score if these points were addressed in sufficient detail.
Major comments:
1) Choice of problems
There are many important problems in the literature that belong to the general type of PDE considered in this work (e.g. in electrostatics). To appreciate and understand the merits and limitations of the proposed approach better, it would be highly necessary to apply it to a wider range of problems, including ones with known analytical solution. 
2) Convergence tests
Convergence tests are an important part that is missing in the current paper. In particular, how does the error in the neural-network solution decay with respect to the number of random points used to train the network? Plots comparing a suitable error norm versus the training dataset size would be very informative. Furthermore, what are the conditions, if any, to guarantee convergence to the exact solution? The authors mention in the last paragraph of the Discussion that a more rigorous analysis will be published elsewhere. However, as error analysis is such an integral part of the study, I think it needs to be addressed, at least to some extent, already in the current paper.
3) Related work
I don’t think the current work is put in sufficient contrast with existing work. Several related papers are mentioned in the introduction and the experimental section includes one other method (GDM method of Sirignano & Spiliopoulos) for comparison. However, a more thorough discussion on how the current approach complements existing literature on neural network based PDE solvers would be in order. 
Minor comments:
i) There are quite a few typos and grammar mistakes in the paper that need to be fixed. To give a few examples:
‘the natural structure presents’ -> ‘the natural structure present’
‘Shrödinger’ -> ‘Schrödinger’
‘approximated by neural network’ -> ‘approximated by a neural network’
‘on circular and’  -> ‘on a circular and’
‘forces the equation’ -> ‘enforces the equation’
etc...
ii) Eq. (7): There is a parenthesis missing on the LHS.
iii) The authors show many plots in Figs 3-9 but don’t comment much on the results in the main text. A lot of the results presented could therefore go into an appendix. I would find it better to present fewer results/plots and discuss those in more detail.
iv) It would be good to define the Nabla operator in Eq. 5 for completeness.
v) Please put references to equations and papers in parentheses or, at least, separate them otherwise from the sentence, to avoid confusion. For example:
- the reference to ‘Evans (2010)’ below Eq. 5, or
- the list of dangling references before the last paragraph on page 3,
- the reference to Sirignano & Spiliopoulos in the caption of Table 1,
etc...
vi) Please expand the captions in Fig. 3 to make the figure meaningful as a stand-alone, without having to read the entire text to understand what 'phantom' means. The same applies to all other captions.
vii) Perhaps it would be clearer to use ‘row’ instead of ‘line’ when referring to the results in the table.
viii) In Sec. 7 ‘s’ in ‘Ns’ should be a subscript.
ix) Explain what the acronym PSNR means.
x) Please elaborate on why you take the top k values in Eq. (7). What happens if you take more/less?
xi) First paragraph in Discussion: I am not sure the robustness w.r.t. the hyperparameters is really so surprising given that you apply the solver to very similar problems. If you could show that the same hyperparameters worked on a completely different problem, that would be much more interesting.
xii) Please add references for the first paragraph of the introduction.
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I increased my rating based on the revisions. 
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