Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: This paper proposes to incorporate energy-based priors to improve the training of physics-informed neural networks (PINN), for solving partial differential equation (PDE) based inverse problems. The method is mainly validated in the application of electrical impedance tomography (EIT).
strength_and_weaknesses: Strength:
- The paper is well written and organized.
- The studied problem and the proposed idea is interesting intersecting two areas of PINN and energy-based model, particularly denoising score matching (DSM)
- The results of simulation experiments and ablative study support the improvement of incorporating energy-based priors in the application of EIT

Weakness:
- How to train and obtain a EBM prior is an important and interesting part in the proposed method. For example, DSM always may require a large-scale dataset for training, but the paper “generates 6512 such smoothed phantoms for EBM prior training and 1628 for testing.” How can the author decide this is a sufficient dataset to train the EBM prior. Moreover, the distribution captured by EBM prior is always dependent on the training data. How could the trained EBM prior generalize to other cases or how easy is it to train an adaptive EBM prior for different application cases?
- The paper mentions that “ training of PINN is extremely sensitive to interplay between different loss terms and hyper-parameters, including the learning rate.” But for the proposed method, as shown in Eq. (11) as well as the training of EBM prior, there are still many hyper-parameters that need to be decided. Although Sec. 4.4 shows that with EBM prior, the model seems to have a higher training success rate, the training of EBM prior also depends on the different hyper-parameters choices and may also raise other problems during the training. How practical it could be to choose a reasonable set of hyper-parameters when generalized to different application cases?
- The paper conducts all the experiments on the simulation data. Is there any real dataset that could be used to validate the effectiveness of the proposed method in real applications? Especially, for the real data, is it still possible to get a decent scale of dataset to train the EBM prior and how efficient is it?
-For comparison among different baseline methods, beyond the quantitative results shown in Table 2, can the paper also provide the qualitative results comparison? It would be interesting how these methods may distinguish in the real application?
- The idea of combining PINN and DSM prior for solving inverse problems is interesting. The author introduces this mainly from the perspective of PINN and  solving the EIT inverse problem as in Sec. 5. There are also a series of researches recently for using DSM models for inverse problem solving such as medical image reconstruction by incorporating physics priors. How could this paper be relevant to those line of research also for inverse problem solving?
- In Table 2, the paper citations of titles are all missing?


clarity,_quality,_novelty_and_reproducibility: The paper is well written and organized. It proposes an interesting idea incorporating PINN and DSM. No code is provided in the paper but there are some training and model details described in the appendix. 
summary_of_the_review: The paper studies an interesting problem and proposes an interesting idea to incorporate energy-based priors to improve the training of physics-informed neural networks (PINN). But there may still be some concerns as aforementioned.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: This paper introduces the use of Bayesian approach as prior for stabilizing and helping the training of Physics Inspired Neural Network (PINN) for solving the semi-inverse Electrical Impedance Tomography. 
strength_and_weaknesses: Strengths:
1.	The proposed approach was demonstrated to be outperforming the strong solution proposed by Bar and Sochen, with faster convergence and more robustness toward the learning rate.
2.	The authors demonstrated the added value of EBM priors for different EIT approach, and also highlight the added value of added hinge loss.
3.	
Weaknesses:

1.	The authors have only focused on the semi invers problem, whereas in real EIT experiments, one will only have access to current measurement on the body torso. Could the authors explain why they restricted themselves to the semi-inverse problem? 
2.	Why did the authors not implement the classical FEM approach for solving the EIT problem, in order to compare their approach to a standard technique? 
3.	Also, it would be interesting for the authors to report computational times and also memory usage. As this would help for comparison purpose.
4.	Could the authors also discuss how their proposed approach could extended to the 3D case?

clarity,_quality,_novelty_and_reproducibility: The paper is well written> the use of EB: priors is novel for solving EIT semi-inverse problem, and the proposed technique seems to outperform the state-of-the-art. It seems that the authors are not intending to share their source code, which would hinder reproducibility.
summary_of_the_review: The authors present an interesting approach for stabilizing the training of PINN for EIT problems. The paper is well presented, and the results are convincing. It is a pity that the authors have only focused on the EIT semi inverse problem, and not the full inverse problem.
review_title: null

final_score: rating: 10: strong accept, should be highlighted at the conference
review_content: summary_of_the_paper: The manuscript is using PINN to solve the problem of electrical inverse tomography. It provides examples where PINNs circumvent the instability of the EIT problem, it creates a data-driven energy-based model to improve the accuracy of the EIT reconstruction. The problem is not only mathematically interesting but has a significant practical use in multiple domains, not least in medicine.

Careful analysis is done on the different regularisation methods. These  show that viable solutions are found with regularisation. The traditional solutions are computed for comparison with different degrees of noise. 
strength_and_weaknesses: This is an excellent paper. Introduction of the energy based criteria that quantifies the non-physical domain for the solutions is innovative and effective. 
clarity,_quality,_novelty_and_reproducibility: High quality paper trying to stuff a lot of information to small space that somewhat reduces the clarity. For example, one could bring the Full EIT Joint Training image to the actual text and move some of the text to the appendix. Just to get an earlier visual about the architecture in question. Novel solutions that solve essential stability problems.
summary_of_the_review: Excellent and informative paper.
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: The paper develops an energy based model based regularizer in conjunction with a PINN to solve the ill-posed inverse problem of electrical inverse tomography. The method is shown to be faster than a classical PDE solution and more robust.
justification_for_why_not_higher_score: The method is evaluated in the limited context of EIT. Its utility beyond this domain is yet unknown.
justification_for_why_not_lower_score: The method is clearly novel and well evaluated for the problem of EIT.
Decision: Accept: poster