Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: IFN provides an implicit neural network layer, which can be used to predict unlabeled edge flows (f) from nodal supply and demand (u), which can be formulated as a nonlinear inverse problem. It also takes advantage of the physics of the flow network, namely conservation of mass at nodal points and constitutive relationship (Eqns 1 and 2) to train h_\theta. Analysis is given for the convergence and existence of the IFN. To demonstrate, examples in the context of AC power networks are given.

strengths_and_weaknesses: Strengths:
The method to enforce physical laws of eqns 1 and 2 into the training is elegant. It comes with guarantees on the existence of the solution (contraction of T, which also makes the inverse problem a well-posed one), algorithm with clear mathematical foundation (with a finite number of iteration for convergence k*), and convergence analysis on FN (forward pass). 
The authors further compute gradients required for training in manner specific to IFN.
The distinction with optimization-based approaches is clear.

Weaknesses:
The implicit flow network represented by a neural network is not new; but addition of physical laws (conservation and constitutive) is novel. Authors also develop a nice method to achieve this. But although they try to generalize this for a verity of physical systems in section 3.1, I don't think it is clear how can one extend the framework for more complicated networks such as traffic flow or collective system of rigid bodies. Furthermore, seems like the method can only be applied to steady-state systems which is a very serious limitations. I think such limitations should reflect in the title of the paper as the current one may sound too generic.  
questions: Minor suggestion: maybe theorem 2.2 can be expressed as a corollary right after theorem 2.3. Of course. authors can still mention the well-posedness of IFN before section 2.1, whose proof will be provided later.  

Can we have an estimation of k* a-priori? Since it is a function of h_{\theta}, it seems like it can be only found after the training. But it might be nice to have an approximation of the number of iterations.

Any comments for the time-dependent flow networks? Please note that for various physics-informed networks, a major limitations is capturing the time-dependencies, so it's a non-trivial task (but feasible). 

The authors clearly show how nonlinearities in equation 2 are captured, but seems like equation 1 needs to be linear. This may be a limiting factor. Any comments for a nonlinear version of equation 1 (for example conservation of momentum). 

Although IFN does not include measurements; can you consider it an advantage? Let's say constitutive law is not completely understood. So at the time of training, h_\theta and f are evaluated based on the current model but such model may bifurcate or change in action. Since IFN does not take advantage of measurements directly, it may be prone to error in such circumstance. Any comments on robustness of IFN?


limitations: Yes.
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: The submission proposes a solution to solve a conservation law and the existence of a constitutive relationship via implicit function. They shows that this model can estimate flows in AC networks. 
strengths_and_weaknesses: Strengths:

- It improved the previous general models predicting network flows not only by considering conservation law but also by considering a constitutive relationship.
- This work clearly and accurately incorporates the constitutive relationship in the implicit model.
- This work shows detailed descriptions for the forward and backward pass, where a contraction algorithm is used for evaluation and backward pass.

Weaknesses:

- This work is limited to an undirected graph.
questions: Is this work still working when the number of nodes of the graph increases or decreases?
limitations: This work is limited to the undirected graph.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The paper introduces an implicit neural network to compute (nonlinear) network flows, extending previous works that predict flows based on a divergence minimization principle.
strengths_and_weaknesses: Strengths:
- well written paper
- clear theoretical characterization of results

Weaknesses
- somewhat limited numerical evaluation (only power flow datasets)
- related literature could be improved
questions: Why do the authors only consider primarily power flow problems in their evaluation? 

It appears that one of the strengths of the proposed framework would be that it can be adjusted to different constitutive laws, covering other scenarios as well. It would be nice to see an evaluation of this. Several further datasets (traffic, etc.) are available from Ref [3] here:
https://github.com/000Justin000/ssl_edge/tree/master/data

I think a somewhat broader discussion of related literature would be of interest to the reader.

In the context of solving power flows with (graph) neural networks, see. e.g.:
Böttcher, Luis, et al. "Solving AC Power Flow with Graph Neural Networks under Realistic Constraints." arXiv preprint arXiv:2204.07000 (2022).
There are also several neural networks for edge flow predictions:

Roddenberry, T. Mitchell, Nicholas Glaze, and Santiago Segarra. "Principled simplicial neural networks for trajectory prediction." International Conference on Machine Learning. PMLR, 2021.
Ebli, Stefania, Michaël Defferrard, and Gard Spreemann. "Simplicial Neural Networks." NeurIPS 2020 Workshop on Topological Data Analysis and Beyond. 2020.

Finally, related works can also be found in the graph signal processing literature, e.g.,
Yang, Maosheng, et al. "Simplicial Convolutional Filters." arXiv preprint arXiv:2201.11720 (2022).


limitations: n/a -- limitations are not discussed
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: This paper incorporates two physical laws, namely conservation of mass and the existence of a constitutive relationship between edge flows and nodal states, into an implicit neural network layer to solve the estimation problem of unlabeled edge flows.

Rating raised from 4 to 5 after revision.
strengths_and_weaknesses: Theory: 
1. Applying these two physical laws to flow prediction seems natural and reasonable. I’m not sure if others have made similar attempts. 
2. The mathematics in the paper is cool, although it might be a little hard to follow.

Presentation: 
1. How the physical laws are incorporated is not highlighted in the paper and is hard to find. Section 3.1 is too far away from Equations 1 and 2.  After the revision, this point has been demonstrated more clearly.
2. The physical laws incorporated are not elaborated.
3. Figures sometimes are too far away from the corresponding text. Solved after revision.

Experiment: Although the paper lists a few potential applications of flow prediction, the experiment focuses only on AC power networks. Thus, only one example of the second physical law, which is Ohm’s law, is examined. More experiments are included after the revision.

questions: N/A
limitations: Yes. The authors address the limitations of the implicit neural network layer that needs to be improved in the future.
review_title: null

Meta-review: metareview: This paper proposes Implicit Flow Network to estimate flow over network edges under certain physical constraints (conversation law and constitutive relationship such as Ohm’s law). Most reviewers agreed that the proposed idea of augmenting IFNs with physical constraints is novel and interesting, and experimental validation is sufficiently convincing.  
confidence: Less certain
Decision: Accept