Physics-Informed Implicit Representations of Equilibrium Network Flows
Abstract
Flow networks are ubiquitous in natural and engineered systems, and in order to understand and manage these networks, one must quantify the ﬂow of commodities across their edges.
This paper considers the estimation problem of predicting unlabeled edge ﬂows from nodal supply and demand.
We propose an implicit neural network layer that incorporates two fundamental physical laws: conservation of mass, and the existence of a constitutive relationship between edge ﬂows and nodal states (e.g., Ohm’s law).
Computing the edge ﬂows from these two laws is a nonlinear inverse problem, which our layer solves efﬁciently with a specialized contraction mapping.
Using implicit differentiation to compute the solution’s gradients, our model is able to learn the constitutive relationship within a semisupervised framework.
We demonstrate that our approach can accurately predict edge ﬂows in AC power networks and water distribution systems.
Introduction
Network ﬂows are a fundamental aspect of modern society, from trafﬁc and communication networks to power and water distribution systems.
Many critical infrastructures are well-modeled as graphs, with edges that transport vital commodities [1].
Beyond infrastructure, network ﬂows are also central to models in epidemiology, ecology, medicine and chemical networks.
Their dynamics have been well-studied in compartmental systems theory [2].
Given the prevalence of ﬂow networks in natural and engineered systems, predicting ﬂows in these networks is an important learning task that may facilitate monitoring, control, optimization, and protection of these networks.
While domain-speciﬁc tools to predict network ﬂows have been around for a while, the machine learning community has only recently taken an interest in general-purpose models for network ﬂows.
[3] predicts edge ﬂows from partial measurements by making a smoothing assumption i e., by minimizing nodal ﬂow divergence.
[4] improves on this approach by adding a trainable regularizer that can incorporate side information.
Both of these approaches are centered on a notion of approximate conservation, i e., that the net inﬂow to each node should be near zero.
Since conservation of mass is a universal constraint on network ﬂows, imposing this conservation law is an important step toward embedding physics into the model.
But the conservation law alone is not enough to uniquely determine ﬂow, which is why both [3] and [4] rely on heuristic regularizers to select the “best” conservation-respecting ﬂow.
In fact, physical networks are often governed by a pair of physical laws: the conservation law, and a constitutive relationship , which speciﬁes the magnitude and direction of each edge ﬂow based on “effort” variables at each incident node (e.g., pressure or voltage).
For example, in DC circuits, currents are conserved according to Kirchoff’s current law, and Ohm’s law is the constitutive relationship that relates current ﬂows to nodal potentials.
The conservation law and the constitutive relationship together deﬁne the unique edge ﬂows (and nodal efforts).
1.1 Contributions
This paper proposes a model for network ﬂows that embeds both the conservation law and existence of a constitutive relationship.
Our model, which we call an Implicit Flow Network (IFN), predicts each edge ﬂow using a trainable nonlinear function of latent nodal variables.
These latent variables are constrained to a manifold wherein the conservation law is satisﬁed.
In addition to introducing
IFN, we offer the following contributions: (i) a contraction algorithm that is able to both evaluate the IFN layer and backpropagate gradients through it, (ii) an explicit upper bound on the number of iterations required by this algorithm, (iii) a rigorous theoretical comparison between IFN and the state-of-the-art ﬂow estimation methods in [3, 4], and (iv) numerical experiments from several
AC power networks and water distribution systems that indicate IFN can signiﬁcantly outperform these baselines on the ﬂow estimation task.
Additionally, because IFN requires a nonlinearity with a constrained slope, we provide (v) a novel “derivative-constrained perceptron”, which is essentially a trainable activation function with upper and lower bounds on its slope.
1.2 Related Work
Network Flow Estimation Flows on graphs are a classical topic in computer science [5], and ﬂow forecasting has long been studied in speciﬁc domains like trafﬁc [6], but interest in the ﬂow estimation task from a machine learning perspective appears to be relatively recent.
Deep learning algorithms have been used to predict trafﬁc ﬂows [7, 8] and power ﬂows [9], but [3] and [4] appear to be the ﬁrst papers to propose methods for domain-agnostic ﬂow prediction, based on the notion of divergence minimization.
Implicit Neural Networks IFN belongs to a growing class of models called implicit neural networks, which do not explicitly state the output of the model; rather, they describe a desired relationship between the model’s inputs and outputs.
In the prevailing implicit framework, the output is deﬁned as a ﬁxed point of a trainable perceptron.
This approach was introduced in [10] as a “deep equilibrium network”.
Subsequent work has developed new frameworks for ensuring the existence of the ﬁxed point and computing it [11, 12, 13, 14, 15].
Other types of implicit neural networks include neural
ODEs [16] and layers that solve convex optimization problems [17] and Nash equilibria [18].
Graph Neural Networks Graph neural networks (GNN) are a diverse family of models for networkrelated learning tasks that incorporate graph structure directly into the model.
GNNs can typically be classiﬁed into three types, in increasing order of generality [19, §5.3]: convolutional models [20, 21], attentional models [22], and message-passing models [23, 24].
Recently, [25] proposed an implicit graph convolutional network.
Analogously, IFN can be interpreted as an implicit message-passing
GNN, with ﬂows serving as messages and latent nodal variables acting as an embedding.
1.3 Preliminaries and Notation
Given a directed graph G =
( V , E ) , the signed incidence matrix B ∈ {− 1 , 0 , 1 } |V|×|E| is the matrix with entries
B i,e =   1 , i is the head of e − 1 , i is the tail of e 0 , else , ∀ i ∈ V and e ∈ E
For an undirected graph, the signed incidence matrix is obtained by assigning an aribtrary orientation to each edge.
For each i ∈ V , let N in ( i ) , N out ( i ) ⊂
V be the in-neighbors and out-neighbors of i .
Given a vector x ∈ R n , we use the notation [ x ] to denote the diagonal matrix diag( x ) ∈ R n × n .
Where such notation would be unclear (e.g., may be confused with brackets to indicate order of operations), we fall back on the diag( · ) notation.
We write x ⊥ to refer to the vector space that is orthogonal to x , i.e., the space { x (cid:48) ∈ R n : x (cid:62) x (cid:48)
= 0 } .
Given a positive deﬁnite diagonal matrix
D ∈ R n × n , we write || x
|| 2 ,D to represent the weighted
[Table]
Implicit Flow Networks
IFN is inspired by the physics of network systems.
In many physical networks, nodes “communicate” through the exchange of a commodity, like power, water, or force, which can be represented as edge ﬂows.
Flows obey a conservation law: for all i ∈ V ,
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation] [Equation]
) where u ∈ R |V| are nodal inﬂows from outside the network, and f ∈ R |E| are the edge ﬂows.
Furthermore, the ﬂows are related to nodal variables through a constitutive relationship (CR); there is some strictly increasing function h such that, for all ( i, j ) ∈ E ,
[Equation]
(2) where a ∈ R |E| are edge weights and x ∈ R |V| are nodal “efforts” or “potentials.”
For example, in
DC power networks, the CR is Ohm’s law f
( i,j ) =
r − 1 ( i,j ) ( x i − x j ) , where r are resistances and x are voltages.
In lossless AC networks, the CR is the active power ﬂow equation f
( i,j )
= a ( i,j ) sin( x i − x j ) , where the edge weights are a function of line parameters and x are voltage angles [26, §6.4].
In water distribution systems, the CR is the Hazen-Williams formula [27, Sec. 8.15].
Table 1 lists several ﬂow networks, the physical interpretation of the effort variables x , and the ﬂow function h .
We propose IFN as a layer that predicts edge ﬂows based on these two physical laws—conservation and the existence of a CR:
Deﬁnition 2.1 (Implicit Flow Network) .
An implicit ﬂow network (IFN) is a module with the following components:
ﬁxed parameters 0 < d min ≤ d max , 2.
trainable parameters θ ∈ R r for some r , and 3.
a family of differentiable functions h θ : R → R such that d min ≤ h (cid:48) θ ( y ) ≤
d max for all y ∈ R and θ ∈ R r , which we call ﬂow functions .
The module requires each of the following inputs:
a weighted, connected, undirected graph G =
( V , E , a ) with edge weights a ∈ R |E| > 0 , and 2.
a supply / demand vector u ∈ R |V| such that (cid:80) i ∈V u i = 0 .
The module outputs the unique vector f ∈ R |E| for which there exists x ∈ R |V| such that
[Equation]
where B ∈ {− 1 , 0 , 1 } |V|×|E| is the signed incidence matrix of G , and h θ is applied element-wise.
We use the notation FN h,θ ( G, u ) to represent the solution f given inputs G and u , ﬂow functions h , and parameters θ .
[Figure]
[Figure]
We will prove that IFNs are well-posed in Theorem 2.2.
Note that (3) and (4) are just vectorized statements of the conservation law (1) and the CR (2), so these two physical laws directly deﬁne the output.
The IFN’s only trainable component is its ﬂow function, parameterized by θ .
In practice, we will only make calls to the inverse of the ﬂow function when evaluating and backpropagating through
IFN layers, so it is convenient to learn the inverse ﬂow function directly.
We emphasize that IFNs are layers that can be situated in more complex architectures, with other models upstream estimating the supply / demand vector, edge weights, or even the topology.
For example, in power systems, demand forecasting is a very well-studied problem [28, 29], and one can solve the economic dispatch problem to forecast power generation at each node [30], collectively leading to an estimate of the supply / demand vector.
2.1 Evaluating the Implicit Flow Network
Our approach to evaluating the implicit ﬂow network is adapted from [31] and is illustrated in Figure 1.
Any undirected graph G induces a direct decomposition of the edge ﬂow space R |E| : given the incidence matrix B ∈ {− 1 , 0 , 1 } |V|×|E| , the cycle space ker( B ) and cutset space Img( B (cid:62) ) are orthogonal, and R |E| =
ker( B ) ⊕ Img( B (cid:62) ) .
We refer the reader to [32, §9.4] for a primer on cycle and cutset spaces.
Accordingly, we decompose the vector f = FN
h,θ ( G, u ) as f = f cyc + f cut , where f cyc ∈ ker( B ) and f cut
∈ Img( B (cid:62) ) .
The cutset component is readily determined from (3), since Bf = Bf cut = u implies that f cut
= B † u .
Then we must analyze (4) to solve for f cyc .
Deﬁne a cycle projection matrix P ∈ R m × m as the oblique projection onto ker( B ) parallel to Img([ a ] B (cid:62) ) :
[Equation]
Based on this projection, we deﬁne a map T : ker( B ) → ker( B ) for all f cyc ∈ ker( B ) by
[Equation]
We can show that f cyc is the unique ﬁxed point of T , and that T is a contraction mapping, leading to a simple algorithm to compute this ﬁxed point.
Theorem 2.2 (Properties of T ) .
Consider an implicit ﬂow network with parameters d min , d max , and θ , with ﬂow functions h θ .
Suppose that the inputs G =
( V , E , a ) and u ∈ 1 ⊥ |V| are given, and let
B ∈ {− 1 , 0 , 1 } |V|×|E| be the signed incidence matrix of G .
The following are true:
T is a contraction mapping with respect to || · || 2 , [ a ] − 1 , with Lipschitz constant
[Equation]
the sequence of iterates f ( k +1) cyc
= T ( f ( k ) cyc ) starting from any initial condition f (0)cyc ∈ ker( B ) converges to a unique ﬁxed point f cyc , 3.
the output of the implicit ﬂow network is unique and given by
[Equation]
Consequently, IFN is well-posed.
Theorem 2.2 provides a simple algorithm for computing the IFN output f : pick any f (0)cyc ∈ ker( B ) , repeatedly apply the map T until approximate convergence, then add B † u .
Some care is required when implementing this map.
Since P is a dense matrix with |E| 2 entries, it is undesirable to explicitly construct the cycle space projection matrix for large networks.
Instead, in order to project a vector v ∈ R |E| , we can use the fact that
[Equation]
so the projection is evaluated as Pv = v − [ a ] B (cid:62) w .
Using this method of projection to implement
T , the ﬁxed point iteration to compute FN h,θ ( G, u ) is stated in Algorithm 1.
Algorithm
1: B ← signed incidence matrix of G 2: f cut ← argmin f cut ∈ R m {||
Bf cut − u || 2 } 3: f cyc ← 0 m 4: ∆ f cyc ← ∞ 1 m 5: while || ∆ f cyc || 2 , [ a ] − 1 > (cid:15) do 6: v ← d
[Equation]
a
[Equation]
h
[Equation]
[ a
[Equation]
f
[Equation]
[ a ]
[Equation]
f
[Equation]
(cid:1) 7: w ← argmin
[Equation]
8: ∆ f cyc ← v − [ a ] B (cid:62) w 9: f cyc ← f cyc − ∆ f cyc 10: end while 11: f ← f cyc + f cut 12: return f
Theorem 2.3 (Implicit Flow Networks, Forward Pass) .
Consider an implicit ﬂow network with parameters d min , d max , and θ , with ﬂow functions h θ .
Suppose that the inputs G =
( V , E , a ) and u ∈ 1 ⊥|V| are given, and let B ∈ {− 1 , 0 , 1 } |V|×|E| be the signed incidence matrix of G .
The following are true of Algorithm 1, with a tolerance of (cid:15) > 0 :
for each iteration k = 1 , 2 , . . .
of the loop, let f ( k ) cyc represent the new value of f cyc deﬁned on line 9; and let f (0)cyc = 0 m .
Then
[Equation]
the algorithm converges with at most k ∗ iterations of the while loop, where
[Equation]
and ρ = || [ a ] 12 h − 1 θ ([ a ] − 1 B † u ) || 2 ; and 3.
the algorithm returns f ∈ R |E| , where
[Equation]
If evaluating h − 1 θ is sufﬁciently simple, then the most expensive step in the iteration is solving the ordinary least squares problem on line 7.
Using a general-purpose solver, the complexity of this operation is roughly O ( |V| 3 ) .
But B [ a ] B (cid:62) is a sparse Laplacian matrix, so we can use a specialized
Laplacian solver that reduces the complexity to O ( |E| log k |E| ) for some constant k [33].
The bound on the number of iterations k ∗ can be computed before any forward pass, since evaluating h − 1 θ does not require solving the IFN equations.
But we can further simplify the bound by approximating h − 1 θ (0) = 0 , which is often justiﬁed because physical ﬂow functions generally have a root at the origin.
Using the fact that ( h − 1 θ ) (cid:48) ( y ) ≤ d − 1min y , we can then eliminate the dependence on h − 1 θ :
[Equation]
2.2 Computing the Gradients
In order to train the ﬂow function and any upstream models, it is necessary to backpropagate gradients through the IFN layer.
We can perform this backward pass using implicit differentiation, and it turns out that the gradients of FN h,θ ( G, u ) with respect to the parameters θ , a , and u can also be computed using Algorithm 1, i.e., by writing the gradient as the output of an auxiliary implicit ﬂow network.
Theorem 2.4 (Gradients) .
Consider an implicit ﬂow network with parameters d min , d max , and θ , with ﬂow functions h θ .
Suppose that the inputs G =
( V , E , a ) and u ∈ 1 ⊥ |V| are given, and let
B ∈ {− 1 , 0 , 1 } |V|×|E| be the signed incidence matrix of G .
Let f = FN
h,θ ( G, u ) , and let w be a scalar entry of θ , a , or u .
We can compute the derivatives dfdw as follows.
Deﬁne a vector of ﬂow functions g : R |E| → R |E| by
[Equation]
[Equation]
where D ∈ R |E|×|E| is the diagonal matrix with entries
[Equation] [Equation]
and v = [ a ] h − 1 θ ([ a ] − 1 f cyc + [ a ] − 1 B † u ) .
Then
[Equation]
12) (We use the notation · in place of θ , since g has no trainable parameters.)
Furthermore, the derivative constraint parameters d min , d max from the original implicit ﬂow network are valid for the new implicit ﬂow network.
In other words, to compute the gradient with respect to a parameter, we perform a single evaluation of the implicit ﬂow network.
In order to compute the derivatives with respect to some parameter or input w , we ﬁrst evaluate the partial derivatives ∂v∂w and the total derivatives dudw .
Then we construct the ﬂow functions g according to (10), and solve an implicit ﬂow network to ﬁnd dfdw according to (12).
It is easy to evaluate dudw , but for convenience, we provide the values of ∂v∂w below:
[Equation]
Comparison with Optimization Models
Both of the state-of-the-art methods for ﬂow estimation, from [3] and [4], use an optimization problem to predict ﬂows.
After a suitable transformation to incorporate external ﬂow injections u , we can state this optimization problem as
[Equation]
where λ > 0 , and q > 0 m is some vector of edge weights.
In [3], q = 1 m , while [4] allows q to be the output of a neural network.
IFN is not explicitly an optimization problem, but it can be cast as one that is similar to (13):
Theorem 3.1 (Optimization Form of IFN) .
Consider an IFN with ﬂow function h θ .
Suppose that the inputs G =
( V , E , a ) and u ∈ 1 ⊥|V| are given, and let B ∈ {− 1 , 0 , 1 } |V|×|E| be the signed incidence matrix of G .
Then the IFN output can be stated as the solution of a convex optimization problem:
[Equation]
Theorem 3.1 can be interpreted as a nonlinear generalization of the Thomson principle from electrical circuits theory [34].
Interestingly, the theorem sets up a direct comparison between IFN and the models in [3] and [4].
If the ﬂow function h θ is the identity map, then (14) can be simpliﬁed as
[Equation]
Ignoring the constraints from labeled ﬂows, we can interpret (13) as using a penalty method to approximate the output of an IFN with a linear ﬂow function.
Thus, we have three distinct differences between IFN and the optimization-based approaches.
First, IFN allows for a nonlinear ﬂow function, while [3] and [4] implicitly assume a linear CR.
Second, IFN imposes ﬂow conservation as a hard constraint rather than an approximate constraint (which is a limitation if u is uncertain).
Finally, IFN does not incorporate ﬂow measurements directly; rather, the model exploits these measurements during training to learn the proper ﬂow function (and train any upstream models for the IFN inputs), making it less sensitive to noise in the labeled ﬂows.
Models for Flow Functions
In order to implement an IFN, it is necessary to parameterize its inverse ﬂow function h − 1 θ .
Since the ﬂow function is essentially a trainable activation function, i.e., a scalar nonlinearity, simple models are likely to be sufﬁcient.
The main difﬁculty with selecting a ﬂow function is that its slope must be bounded by d min ≤ h (cid:48) θ ( y ) ≤
d max for all y ∈ R .
This section proposes a simple scalar nonlinearity that is guaranteed to respect arbitrary upper and lower bounds on its slope.
Deﬁnition 4.1 (Derivative-Constrained Perceptron) .
Let k ∈ Z > 0 be a hidden layer size, let a, b, c ∈
R k be freely trainable parameters (encoded within the parameter vector θ ), and let σ be a nonexpansive activation.
Let p, q ≥ 1 such that p − 1 + q − 1 = 1 , and let ¯ d min ≤ ¯ d max ∈ R .
Then the derivative-constrained perceptron N ( x, θ ) is the 3-layer neural network deﬁned by
[Equation]
Intuitively, (L1) re-scales c so that the perceptron in (L2) is guaranteed to be non-expansive in x .
Then (L3) re-centers and re-scales the derivatives of the perceptron from the range [ − 1 , 1] to [ ¯ d min , ¯ d max ] .
Theorem 4.2 (Derivative-Constrained Perceptron) .
Let N( x, θ ) be a derivative-constrained perceptron with ¯ d min ≤ ¯ d max ∈ R .
Then for all parameter values θ , ¯
[Equation]
[Equation]
Note that the values ¯ d min , ¯ d max in Deﬁnition 4.1 and Theorem 4.2 are distinct from the IFN parameters d min , d max .
Since we parameterize the inverse ﬂow function h − 1 θ in IFN, one shoudl set ¯ d min =
d − 1max and ¯ d max = d − 1min to implement h − 1 θ with a derivative-constrained perceptron.
Numerical Experiments
We studied the transductive task of predicting unlabeled ﬂows, given that some labeled ﬂows in the same network are known.
If the edges E are partitioned into a labeled set E l and an unlabeled set E u , the task is to predict the missing ﬂows { f e : e ∈ E u } given the labeled ﬂows { f e : e ∈ E l } .
For each network, we randomly selected a fraction of the edges to be labeled edges, and we trained IFN and baselines on the labeled edges.
Then we evaluated the RMSE of the ﬂows predicted for the unlabeled edges E u to compute the testing error.
See Appendix B in the supplementary material for full details.
Code is available at https://github.com/KevinDalySmith/implicit-flow-networks .
5.1 Datasets
AC Power We selected 6 standard power network test cases.
The ﬁrst 4 test cases (IEEE-57, IEEE118, IEEE-145, and IEEE-300) are synthetic transmission system test cases, while the remaining cases ACTIVSg200 and ACTIVSg500 are similar to the Illinois and South Carolina power grids, respectively [35].
Each test case contains the topology and electrical parameters of the power network, as well as baseline demands and power injections at each node.
While branch resistances are typically small, we set them to zero to ensure lossless transmission lines.
We used the MATPOWER toolbox [36] to solve the power ﬂow equations, then recorded the active power ﬂows on each branch ( f ), computed the net active power injections at each node ( u ), and selected relevant electrical parameters as edge attributes (series reactance, tap ratio, and voltage magnitude at the two incident nodes).
Water Distribution We selected 3 sample water distribution networks from the ASCE Task Committee on Research Databases for Water Distribution Systems database [37], representing municipal water distribution systems in Fairﬁeld, CA, Bellingham, WA, and Harrisburg, PA.
Each network contains the topology of the distribution system, as well as the characteristics of pipes and other network elements and nodal demands.
We used the WNTR package [38] to compute the ﬂow rates through each pipe ( f ), net inﬂow rate at each node ( u ), and edge weights associated with each pipe.
5.2 Models and Experiment Details
IFN Architecture In order to use the IFN layer to predict power ﬂows, we created a two-layer model.
The ﬁrst layer estimates positive edge weights a ∈ R |E| according to a e = exp ( L ( z e )) for all e ∈ E , where L is a linear module, and z e is the log-transformed vector of edge attributes.
The second layer is an IFN.
To predict water ﬂows, we used an IFN layer alone, supplying the edge weights from the dataset as input (rather than learning them from other edge attributes).
For both water and power, the IFN layer uses a derivative-constrained perceptron as the inverse ﬂow function ( k = 128 , p = q = 12 ) with a ReLU activation function.
For power, we set d min = 0 .
4 and d max = 2 ; and for water, d min = 0 .
2 and d max = 20 .
Baselines We compared the IFN model against four baselines.
The minimum divergence method ( Div ) from [3] minimizes the nodal divergence ||
Bf || 22 and a regularization term
λ ||
f ||
The bilevel optimization methods from [4] replace the uniform regularizer with a weighted regularizer
||
f || 22 , [ q ] , where q is a vector of weights.
In Bil-MLP and Bil-GCN , q is the output of either a 2-layer MLP or GCN model with edge attributes as inputs (we use 64 nodes in each hidden layer with ReLU activations).
In Bil-True , we specify q as the reciprocal of the coefﬁcient in the linearized CR for
AC power networks, so that Bil-True approximates (15) with a as the ground-truth edge weight.
For water experiments, Bil-True uses the same edge weights as the IFN model.
All of the baselines assume that nodal divergence Bf should be approximately zero, but nodes in power networks inject and withdraw power according to the supply / demand vector u , resulting in nonzero divergence.
Thus, when we evaluate the baselines, we transform the power network into a divergence-free network by introducing a “source node”, adding an edge from the source node to all nodes in V , and treating the entries of u as the ﬂows along each corresponding virtual edge.
5.3 Results
Figure 2 reports the results for the AC power networks, and Figure 3 reports the results for water distribution systems.
In both types of networks, the IFN model signiﬁcantly outperforms the baselines on all of the networks when a small fraction of edges are labeled (less than 80% in power and less than 60% in water).
While the other baselines tend to improve as more labeled edges are made available for training, IFN achieves near-optimal performance with as few as 10% of the edges labeled.
[Figure] [Figure]
Conclusion
In this paper
[Figure]
we have introduced an implicit model for network ﬂows that incorporates physics through a conservation law and through the existence of a constitutive relationship between ﬂows and nodal variables.
We have demonstrated that a simple architecture using this model can learn to accurately predict active power ﬂows in AC networks and water distribution systems.
Future work may investigate more elaborate architectures using IFN as a layer, wherein the supply / demand vector, edge weights
[Figure]
or even the graph itself could be predicted from upstream models
[Figure]
and the ﬂows themselves used for downstream tasks.
Another interesting extension may be to extend our method to networks with higher-order interactions, i
[Figure]
e., hypergraphs [39] and simplicial complexes [40, 41].
IFN has some limitations that should also be addressed in future work.
IFN assumes that the graph is undirected, which does not adequately model networks with unidirectional ﬂows (e.g., trafﬁc) or lossy ﬂows (e.g., resistive power grids).
IFN also assumes a CR that depends on the difference between nodal variables.
This form appears frequently in physical systems, but in other network ﬂow models (like Daganzo trafﬁc models [42]), the CR has a more general dependence on the nodal variables.
These limitations may be addressed with extensions of IFN’s contraction algorithm.
Acknowledgments and Disclosure of Funding
This work was supported by AFOSR under grant FA9550-22-1-0059, US Army ERDC under grant
W912HZ-22-2-0010, and DTRA under grant HDTRA1-19-1-0017.