WardropNet: Traffic Flow Predictions via Equilibrium-Augmented Learning
Authors: Kai Jungel
Dario Paccagnan
Axel Parmentier
Maximilian Schiffer
Venue: ICLR 2025 Poster (2025)
Keywords: structured learning
combinatorial optimization augmented machine learning
traffic equilibrium prediction
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Abstract: When optimizing transportation systems, anticipating traffic flows is a central element. Yet, computing such traffic equilibria remains computationally expensive. Against this background, we introduce a novel combinatorial optimization augmented neural network pipeline that allows for fast and accurate traffic flow predictions. We propose WardropNet, a neural network that combines classical layers with a subsequent equilibrium layer: the first ones inform the latter by predicting the parameterization of the equilibrium problem's latency functions. Using supervised learning we minimize the difference between the actual traffic flow and the predicted output. We show how to leverage a Bregman divergence fitting the geometry of the equilibria, which allows for end-to-end learning. WardropNet outperforms pure learning-based approaches in predicting traffic equilibria for realistic and stylized traffic scenarios. On realistic scenarios, WardropNet improves on average for time-invariant predictions by up to 72\% and for time-variant predictions by up to 23\% over pure learning-based approaches.

Published as a conference paper at ICLR 2025
WARDROPNET: TRAFFIC FLOW PREDICTIONS VIA BUIILIBRIUM-UGGENTTEE LEARNING
Kai Jungel
Dario Paccagnan Department of Computing
School of Management
Technical University of Munich
Imperial College London
Munich, Germany
London, United Kingdom
kai. jungel@tum de
d. papcaagaaane www.oggnnniiiiiiiiimm imperialll..... ac uk
Axel Parmentier CERMICS
Maximilian Schiffer
School of Management & Munich Data Science Institute Technical University of Munich Munich, Germany schi Schiffer@tum.co de
Boole des Ponts
Marne-l- Vallé,, France
axel, www.emtieereeeeee....... fr
ABSTRACT
When optimizing transportation systems, anticipating traffic flows central el- ement. Yet, computing such traffic equilibria remains computationally expensive. Against this background, we introduce a novel combinatorial optimization aug- mented neural network pipeline that allows for fast and accurate traffic flow pre- dictions. We propose WardropNet, a neural network that combines classical layers with subsequent equilibrium layer: the first ones inform the latter by predicting the parameterization of the equilibrium problem's 's latency functions. Using su- pervised learning we minimize the difference between the actual traffic flow and the predicted output. We show how to leverage a Bregman divergence fitting the geometry of the equilibria, which allows for end-to-end learning. WardropNet outperforms pure bearning-based approaches in predicting traffic equilibria for re- alistic and stylized traffic scenarios. On realistic scenarios, WardropNet improves on average for time-invariant predictions by up to 72% and for tiri-variant pre- dictions by up 23% over pure learning-basse approaches.
INTRODUCTION
With the advent of digitalization, big data, and sharing economies, algorithmic decision support evolved pivotal tool analyze, design, and operate today's 's transportation systems. When de- signing algorithmic decision support in the context of transport optimization, an element that is cru- cial at all planning stages is anticipating the transportation system's 's response to the decision taken, e.g., to anticipate traffic flows when taking congestion pricing decisions. In general, transportation systems converge to equilibrium states which agents cannot improve their outcome by unilaterally changing their behavior. Unfortunately, computing such equilibria usually requires a high-fidelig solver, e-g., microscopic traffic flow simulation, that is computationally expensive and requires hours to days when computing equilibrium state.
Against this background, we study novel learning paradigm, that, based on knowledge previ- ously realized equilibria, allows us to accurately predict traffic flows in new context within short computational times. Specifically, we assume access to contextual information æ that correlated to observed traffic flows y. and place ourselves in supervised learning setting: hypothesis class H of hypotheses h that map a context & E X' to flow ý E y, and introduce a loss L(y. i) that measures the difference between target flow y and the predicted flow ý. We aim at finding the hypothesis h E H that minimizes the expected loss
introduce
min Ex,y ((Y,(())) NER

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We assume access to a training set {(z, y:) derived from past traffic flow measurements or through samples obtained by a the empirical supervised learning problem
simulator. Based on this training set, we aim to solve
min REA N
Σ C(y,,((a))
(2)
which is clearly sensitive the chosen hypothesis class. In this context, we propose WardropNet, a hybrid pipeline that augments a neural network with an equilibrium layer, as such capturing com- binatorial interdependencies between flows. Here, the challenge is to derive an end-to-end learning paradigm for the hyprid pipeline as this requires backpropagating the gradient through the, poten- tially combinatorial, equilibrium layer. Once trained, WardropNet produces efficient traffic predic- tions at low computational cost as inference only requires a forward pass on this neural network.
Contribution In this work, we derive a combinatorial optimization-augnan (COAML) pipeline, called WardropNet, that uses a neural network to learn the parameterization of latency functions and a combinatorial optimization (CO)-layer to compute the resulting equilibrium problem. We train this pipeline via imitation learning, intuitively minimizing the Bregman diver- gence between target and predicted traffic flows. Crucially, we show how to train this pipeline in an end-to-end fashion by leveraging suitably defined regularization term arising from the use of a Fenchel-Ye Young loss. This step is key to determining a meaningful gradient, as direct approach results in ill defined and vanishing gradients due to the combinatorial nature of the equilibrium layer. Additionally, we show that utilizing Fenchel- Young loss in this context does not only allow to differentiate through the equilibrium layer, but it is also equivalent to minimizing the Bregman divergence between the predicted and the target traffic flows.
machine learning
We present a comprehensive numerical study and show that WardropNet outperforms pure ma- chine learning (ML) baselines on various realistic and stylized environments in both time-invariant and variant settings, yielding accuracy improvements of up to 75%. Our results show that WardropNet allows to significantly improve traffic flow predictions compared to pure ML baselines. Specifically, WardropNet benefits from the synergy between predicting latency function parameter- izations with statistical model while preserving the combinatorial structure of the respective flows within the CO-layer.
Related work Our work relates two fields; traffic equilibria prediction and CO-augmented ML. ML approaches for predicting traffic equilibria range from conventional ML approaches (Li al., 2014 Hou al., 2015] Lv et al. 2015) to deep learning approaches that embed physical or
combinatorial solution restrictions within the network architecture (Amos & Kolter) 2017 Smith
al. 2022] Seccamonte et al.) 2023). Alternatively, one can utilize graph neural networks (GNNs)
which allow to embed network specific constraints (Li et al., 2018 Wu et al. 2018, Yu et al. 2018
Guo et al. 2019; Wu et al., 2019: Bai et al., 2020 Ali et al.
20022 Wu et al.) 2023] Jin et al,
2024). One also incorporate
equilibrium layer into the deep learning pipeline to directly
predict equilibrium states (Bai et al. 2019 Gu et al. 2020 Li et al. 2020 Marris et al.) 2022] Liu
et al. 2023: McKenzie al. 2023). which can be used optimize stackelberg games (Skkaue &
Nakamura 2021). However, these works are either agnostic of combinatorial structures or impose
major restrictions on the equilibrium layers allow for backpropagation, e.g., the well-posedness of problems or the restriction to small quadratic programs. Contrarily, the COAML pipeline introduced in this work allows for backpropagation through general, possibly combinatorial, equilibrium layers.
Recently, COAML pipelines that embed a general CO-layer into a statistical model emerged. These pipelines aim to minimize the combinatorial error induced by the statistical model during training. First approaches train the statistical model based on the resulting combinatorial objective value (Elmachtoub et al. 2020 [Elmachtoub & Grigas 2022), which requires information about target MI. predictions. Recent approaches learn the COAML pipelines in an end-to-end fashion via directly imitating combinatorial solutions, and have been applied to path problems (Parmentier) 2022) and contextual multi-stage optimization (Dalle et al.) 2022: Baty et al.) 2024 Jungel et al. 2024). A major challenge of these end-to-end COAML pipelines is deriving gradient of the respective CO problem to allow for backpropagation (Agrawal et al. 2019). To do so, recent approaches introduced regularization techniques (Berthet et al. 2020) that allow to differentiate through CO problems, enabling CO problems as layers in end-to-end MI. pipelines (Blondel al. 2020). The presented
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research stream shows the efficiency of COAML pipelines, but the introduced COAML pipelines have not been studied for predicting traffic equilibria. In general, equilibrium layers have neither been studied from theoretical nor practical perspective in the context of COAML pipelines.
2 GENERALIZED WARDROP EQUILIBRIA
We begin by introducing the notion of wardrop equilibrium (WE), as this will be key for the de- velopment of WardropNet. In doing so, also generalize the original notion of Wardrop (1952) to allow for the travel time on each individual are to depend on the traffic flow on other arcs in the network. This enables us to model spill-over effects where the flow on one are is influenced by that of neighbouring ones. Moreover, it allows us to introduce a suitable regularization term which is key for enabling end-to-end learning. For a technical derivation of the formalization, including proofs, we refer to Appendix A
We consider a transportation network, represented as a directed graph G = (V, A) with arc-set A and vertex-set V. Here, we take the non-atomic perspective of the problem and assume that each commodity j E // has size DG) E Rzo. and is required to travel from its given origin node to its given destination node d(i) E V. As a result, commodity j places the vector of flows y (5) (yL")) )aEAA over the arcs. This flow is feasible if amount of flow D leaves the origin and reaches the destination while satisfying conservation of flow at all nodes. Formally, a commodity's s flow y( E y16), lives in a (feasible) multiflow polytope y(d) as detailed in Appendix A Note that we can switch between a traditional agent perspective (DG) = 1), which we adopt in the remainder of this paper, or a commodity perspective (DO) z 1), by restricting D accordingly.
oها) E V
Finally, the traffic flow we aim to predict arises by aggregating the individual agents" flows
ΣEE y and y = { =0==> Ti=Eydn
y yOu E yQu) VjeJ..
JEV
jEJ
where we note that both Y and y are, again, polytopes.
Wardrop Equilibrium Originally introduced by WWarrrop 1952). WE describes set of flows for each agent such that any unilateral deviation would incur longer travel time. Here the total travel time experienced by flow is obtained by summing the travel time experience over each are on its path. In turn, the travel time over each are is measured by a so-called latency function, i.c., an are-specific function &a R - R,, mapping the aggregated flow ÿ in the corresponding travel time (z)) Note that this novel generalized definition allows for the travel time on each are to depend on the aggregated flow on the entire network, and not necessarily only on the flow of that specific arc.
While there exist different, equivalent, definitions of WEs, in the following utilize its variational characterization as derived in Appendix A
Definition 1 (Wardrop Equilibrium). Given a directed graph D AV, origin-destination pairs (oj, de)jes. and vector of latency functions e = {laJaes- WE is multiflow y (y(d)) yiu such that for all j E J and (6) e =0)), ((g) y < (())) (0)
(3)
Let us recall that our goal is to embed an equilibrium problem as a (final) layer into a neural network pipeline. In this case, it appears natural to introduce a family of latency functions and seek the best approximation among the equilibria via choosing the best latency functions. Accordingly, we aim to establish a learning algorithm that allows us to learn the parameterization of the respective latency functions. To do so, it will prove useful if our equilibrium problem remains convex.
Convex characterization In the decomposable case, when the latency La on are a does only depend on the flow No. it is well known that the WE exists and is unique, when we have non- decreasing ea, Va A. We now generalize this result to the non-decomposable case. To do so, we say that latency functions {La JaE derive from potential & : R + if
0
l.
(4)
By.
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Pick-up location
Drop-off location
e
2212
Context state
(MI (MIyrryer) Statistical model
Lateney functions
(CO-layer) Equilibrium problem (0)
Equilibrim
0 = (2)
Figure Schematic illustration of the WardropNet, implemented as a COAMI. pipeline. The pipeline comprises statistical model fou and an equilibrium problem (n) The statistical model fou receives a context state a and predicts the latency function's 's parameterization 0. The equilibrium problem (0) receives the latency function's ' parameterization 0, the transportation network, and origin-destination pairs, and yields the respective traffic equilibrium ÿ.
Theorem (Convex characterization). Consider a directed graph D = (V, A), origin-destination pairs (0j, di)jes, and vector of latency functions l {fa)aEA that derive from a potential क.
1. A multiflow
= = E y is a WE if and only if it is an optimal solution to mingey (yy S.1. y -, シ9-
2. A vector y y is the aggregated flow of WE if and only if is an optimal solution to min gey (
3. If & is strictly convex, then a WE ÿ exists and is unique.
The proof of Theorem T in Appendix A Note that Theorem allows us to switch between the decomposable notion and the non-decomposable notion of latency functions.
DATA-DRIVEN WARDROP EQUILIBRIA
Recall that our goal is predict the aggregated traffic flow y based contextual information that correlates with the observed traffic. Following the supervised learning setting from the introduction, assume having access to samples { (xx, 9: ) and aim solving the resulting empirical su- pervised learning problem in Equation 2 In the following, we introduce our WardropNet and the end-to-end learning paradigm by first detailing the (combinatorial) hypothesis class and the respec- tive loss function used. We then specify the resulting supervised learning problem and show that it can be interpreted as minimizing the Bregman divergence between target and predicted traffic flows.
Hypothesis class Figure I illustrates our novel hypothesis class, which is COAML pipeline that combines an equilibrium problem with a statistical model. To leverage this pipeline, i.c., use the statistical model & to feed information that accounts for the context into the equilibrium problem, we introduce parameterized family of latency functions ? and use the statistical model fo to (x))
predict its parameterization
In this context, we generally consider latency functions e that depend on 0 and derive from regularized potential of the form
L = (
where or is a vector of concave basis functions, and v is a strictly convex regularization function such that y C dom where dom(v) ) is the domain of tb. Practically, we consider pipelines with linear, piecewise linear, and non- non-linear basis function. We choose this specific structure for 4 as mimics the structure of the Fenchel- Young loss, which will be key to obtain differentiable end-to- end learning problem. In the following, we will first consider latency functions l that derive from a regularized linear potential, ease the technical exposition. Afterward, will provide extensions

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to the general case in Section Specifically, consider
( = (z)
0 00y.
(5)
where -0 belongs to the positive cone R1. Theorem 1 ensures that computing a WE then equals solving the following convex regularized flow problem
max
o' v(
where = EE yor
(6)
yEy
JEJ This problem can be formulated directly on the aggregated flow polytopes as
jo := arg max 0T IQy where Ω + + and 1y(y) = fo
if E V.
(7)
[+0
otherwise.
y
Note that the arg max is unique because Ω is strictly convex. This enables to define ú as mapping from the latency parameterization I to the corresponding observed aggregated flow (()). We can now introduce our hypothesis class formally as
H hw + in (z),), (W)
To get practical hypothesis, we need choose a regularization Q as well as statistical model Pin- We defer the discussion of the specific pipeline architecture to the next section, and for now focus on the loss definition.
Fenchel- Fennhel-Young loss Instead of stating a loss L(g. y) between our target traffic flow y and the predicted traffic flow ÿ. we define the loss L(O, 7) between the latency parameterization 0 P (x:) and the target traffic flow ÿ. This loss, which is in our case the Fenchel- Young loss (Blondel et al.) 2020), measures the distance between the target traffic flow y and the traffic flow (0 deriving from the latency parameterization 0. The Penchel-Young loss evolves from the Fenchel- Young inequality and leverages the relationship it describes between a convex function and its Fenchel conjugate. To understand this relationship in our context, we recall that the Fenchel conjugate of function f y + f(y) with domain DD is defined as
(0) = sup 0 0"y y- YED
Then, state the Fenchel conjugate of the regularization f
Ω (
max 0* y (2()).
because the domain of Ω is V, which is compact. Furthermore, we recall that (92")* = Ω2 because
the biconjugate of proper lower semi-continuous convex function is itself: (f**)* = f-
Then, this relationship allows us to derive a general notion of the Fenchel- Young loss for regular- ized flow problem (6) associated the regularization Ω2 as stated in Blolndeel al. (2020) as
( g) = (0) + f(T) 0 y max yey
[oTy - 2(y)] 0 9 207)
(8)
The first equality in (8) defines (f, f) as the left hand side of the Fenchel- Young inequality. This
guarantees that + Ln y convex, non-negative, and equal to zero if and only if yn (0) = y.
Supervised learning problem Combining our hypothesis class with the Fenchel- Young loss, our supervised learning problem becomes
N La (s (Ii), y:),
min
N
S2 is differentiable in 0, the gradient of the Fenchel- Fenchel-Young loss function is equal to (o 7) = VB" y.
(10)
We later introduce regularizations for which (the stochastic version of) this gradient tractable, which enables to solve the learning problem 9 using stochastic gradient descent. For the remainder of this section we focus on showing that solving Sobllem{equall minimizing a Bregman divergence between the observed and predicted traffic flows.
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Interpretation of the Fenchel-Yound loss Bregman divergence Let D R be tinuously differentiable and strictly convex function with convex domain dom(f) = D. Then, the Bregman divergence associated to f
(p(p. q) f()) - [f(a) + (V
(11)
and measures the difference between two points and in D based on f. is a generalization of the squared Euclidean distance ] p all which we obtain when we choose p + 2 2|pll as f.
Our Fenchel- Young loss (n y) is closely related to the Bregman divergence De. To show this, we need the following notion. A function &b : dom( + R of Legendre rype if its do- main dom(1) (v6) is non-empty, it is strictly convex, and differentiable on the interior int (dom(())) of dom(w) and lim (p,(,,) = + for any sequence (p:) of elements of dom(1)) converging to 1-000
boundary point of dom(w)). For instance,
2 |||l2 is of Legendre type with domain R4
Supposing that our regularization function is of Legendre type, which is the case for the potentials we introduce in the next section, [Blondel et al. (2020) Proposition 3) show that the Fenchel- Young loss La introduced above a convex upper-bound on the Bregman divergence
0 <
Do (g. (n(())
S Co (2,0,,)
(12)
possibly nom-comee conves in 0
leading to equality when the loss minimized
(#) = y
+
g) =
Do (t) = 0.
(13)
In the special case where Ω = up in addition to t being Legendre type, we have,
Du y. ()z(0) (0)) = (0 (.
(14)
The practical consequences of Equation 12 and Equation 13 are the following: the Bregman diver- gence D y (O)) provides a natural way to measure the difference between the equilibrium we observe and the equilibrium we predict (0). While directly minimizing this Bregman diver- gence might lead non-convex problems, the Munchel-Yungg loss is convex surrogate that fits this geometry and leads to tractable learning problem.
PIPELINE ARCHITECTURE
In the following, we detail our WardropNet COAML pipeline by first defining the statistical model Poo before introducing various regularization techniques to obtain tractable equilibrium lay- ers 9o.
Statistical model Generally, the statistical model Pue can be any arbitrary differentiable model which maps the context o: the latency parameterization O. In setting, specify the text x as a tuple of vectors ((xa)oEA (G.))ev). each vector va comprising are specific attributes, and each vector I, comprising vertex specific attributes, The output 0 can be any of vectors which is combinable to the latency coefficients (0x.o) A (e.I,.., where K represents the number of parameters that define the latency function a given are a E A. The statistical model (ou encodes its prediction via the parameterization w. In Appendix B we provide a detailed description of the feedforward neural network (FNN) and GNN models used as statistical model Pow-
In the following, we focus on concisely introducing different regularization and modeling techniques to obtain tractable equilibrium layers ig. For an in-depth discussion and detailed derivation of the respective gradients, we refer to Appendix[C]
Latencies with Euclidean regularization (Details in Appendix We can obtain dif- ferentiable equilibrium layer by utilizing a rather simple Euclidean regularization, considering the respective squared Euclidean loss (z) 1 In this case, retrieve decomposable affine latency functions of the form
a
v T ay
02 + Da (Ta):
(15)
and the vector & corresponds to the y-intercept of the latency functions. Then, the equilibrium is the orthogonal projection of 0 on D, which can be computed using any convex quadratic optimization solver.

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Regularization by perturbation (Details in Appendix C.2
Altematively, regularize
cquilibrium layer by perturbation: let us consider a maximum capacity u and the corresponding aggregated flow polyhedra
= { 0 マg 5
Let us now introduce standard Gaussian vector Z on RA, Then, we can use the convex conjugate of the expectation of the perturbed maximum flow as regularization S2,
Ω F
where F(O) = I max (0 2) y
(16)
yE
Berthet et al. 2020) show that this regularization yields several desirable properties: first is proper convex continuous function, which gives f2" = (F*)* )* F. Second, dom( Vu. which
explains why we directly defined Q2 as Fx and not v Section 3 Third, applying Danskin's 's lemma (Bertsekas] 2009 Proposition A.3.2), this second property, gives the differentiability of Q = F and
ÿn ( VQ (0) = E arg max (0 + Z) y
VEV.
We note that, with probability 1, the arg max is unique on a sampled realization of Z. This last property is critical from practical point of view: The exact computation of F(0). (o)) and VoLa(D) require to evaluate intractable integrals. However, P(O), (f), and VoLo ( all in- clude expectations such that can Monte-Carlo estimations by solving the maximum flow problems for few sampling realizations of Z instead of relying on its exact computation. Lastly, note that when using this regularization, the Bregman divergence and the Fenchel-Young Young loss coincide (0,
(0, y) Do (0))
C In practice, Piecewise constant latencies and extended network (Details in Appendix observed latencies often exhibit threshold effects when the traffic on an arc reaches its capacity and congestion appears, i.e., the latency increases stepwise (Vickrey) 1969). Such effects are poorly captured by potential 4e due to the smooth latency functions ta. However, the introduced methodology can naturally be extended to piecewise constant functions of Ta: let us suppose that want the ronglagulr latency function
00mm Tm S Na <m for m (1....,M, M)
1 if Ta
TMs
where 0= <n < <M.- We derive a WE considering piecewise constant latency functions by solving the non-regularized minimum cost flow minyey" 0 y on an extended network G (V, (exP). The extended network GeXP allows to encode the piecewise-constant functions in multi- graph representation with parallel ares such that each are am in a set of parallel ares with m E (.....M] comprises a threshold cost 0am of the original are a. We can then use the regularization by perturbation approach for learning.
Polynomial latencies regularized by perturbation (Details in Appendix troduce monotonously increasing latency functions ( that depend on the respective traffic flow Va- To do so, we introduce a feature mapping o that maps any component y E I., to a feature vector (1) = (aI(V)) E R such that y + ax(y) is convex for each k, a y E y + (1) E AAA Then, we consider polynomial latency functions generated by the mapping k.))
Lastly, we in-
(Jo) = Om.oowoo with OR 20 0 for all k and a.
(17)
Here O2. is the weight coefficient of the k-th basis function in l. and oy is the respective basis function. The overall equilibrium is therefore parametrized by 0 ( the mapping (ax)x. . In this work, we restrict ourselves to the polynomial family o using {0, 1). and ox with Ta denoting the aggregated flow on are a. In general, we can use arbitrary complex polynomials to describe the latency function. The resulting latency function [7 is convex function, as the potentials (ax ) are strictly convex for each k. Accordingly, we can extend the regularization by perturbation to this setting detailed in Appendix C.4
with indexing
g². Then, the latency function for are becomes Eg (2.) Oo + 01,- aa Tas

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(a) HE
(b) LE
(c) SW
(d) Berlin
(c) Berlin-AP
Figure 2: Illustration of the considered traffic scenarios.
NUMERICAL EXPERIMENTS
In the following, we provide a concise discussion of our numerical studies. For details, we refer to Appendix D regarding the considered traffic scenarios, to Appendix regarding our COAML pipelines and baselines, and to Appendix for an in-depth and complementary results discussions. The code for reproducing the results presented in this section is available at https: h //g1 thub. com/t
10-pppeee in Traffi cP cerreeiccc ct ion
Traffic Scenarios We consider four stylized scenarios that allow to isolate structural effects as well two realistic scenarios to analyze the practical value of our approach. For the first two out of these six scenarios, we generate traffic samples (I,, ys)"=1 as solutions to a static traffic equilibrium problem in the form of WE. Instead, we produce these samples by running the widely- used dynamic traffic microsimulator MA7Sim (Horni et al.) (016) for the last four scenarios. We base our stylized scenarios, namely the high-entropy (HE) scenario, the low-entropy (LE) scenario, and the square world (SW) scenario (Figures 2a 2b 2ch on the artificial road network proposed by Eisenstat (20TT) but consider different o-d pair distributions. We also consider Lime-varian version (SW-TV) of the SW scenario. In the HE scenario we obtain rather distributed traffic flows and in LE scenario rather polarized, combinatorial flows. We expect the scenario to be easier predictable than the LE scenario for a pure ML baseline. Moreover, we consider realistic Berlin scenario (Figure 2d) that bases on district from realistic MATSim scenario for the city Berlin, well Berlin-AP scenario (Figure 2c). where enforce all trips to artificially start and end in the respective district.
For all scenarios, create 9 training instances, 5 validation instances, and 6 test instances. Each instance consists of a transport network with the respective target traffic flow for each road, con- textual information, and origin- destinndttinttoon pairs. We run the training for maximum of 20 hours, or 100 training epochs. We run the experiments on a computing cluster using 28-way Haswell-BP nodes with Infiniband FDR14 interconnect and hardware threads per physical core.
Pipelines & Baselines We consider two pure ML baselines, a FNN and GNN model de- tailed in Appendix B which we trained via supervised learning directly imitating the target traffic flows ÿ RIAI in the output layer. For COAML pipelines, we leverage the very same FNN as statistical model to predict the latency parameterization 0, but consider three variants for equilibrium layer as specified in Section 4 the CL-pipeline uses Constant Latencies regularized by perturbation, the PL-pipeline uses Polynomial Latencies regularized by perturbation, and the ER-pipeline uses latencies with Euclidean Regularization.
Our motivation for choosing the same FNN in both the pure ML baselines and the COAML pipelines stems from the fact that we wish to compare these approaches on equal footing way that clearly demonstrates the impact of the additional combinatorial layer. Moreover, this choice allows us to minimize the effect that parameter tuning may have on the overall performance.
Performance Analyses Figure 3 shows the performance of our COAML pipeline with differ- ent equilibrium layers well as the pure ML baselines for all traffic scenarios. Focusing our COAML pipelines, we observe that the CL-pipeline using piecewise constant latencies obtains the highest accuracy. The PL-pipeline using polynomial latencies obtains a good accuracy that is slightly worse compared to the CL-pipeline. The ER-pipeline however, falls short in performance compared to the other two variants, which might be caused by the rather simple latency functions using
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MAB
MAE
10a
10
10-1
10-1
10
FNN
GNN
CL
FNN
GNN
CL
PL
1R
FNN
GNN
CL
PL
BR
(a) HE
(b) LE
(c) SW
10
107
102
MAB
MAB
MAB
10'
10-1
GNN
INN
GNN
PL
ER
FNN
GNN
(d) Berlin
(c) Berlin-AP
(f) SW-TV
Figure 3: Benchmark performances on various stylized and realistic traffic scenarios.
Euclidean regularization. Focusing on the pure ML baselines, observe that the FNV baseline slightly outperforms the GNN baseline in all time-invariant scenarios (Figures 3a 3e while the
GNV bascline slightly outperforms the FNN baseline in time-variant scenario (Figure
For conciseness, we focus the remaining discussion of Figure 3 on comparing the best pessfpeeforming COAML pipeline with the best performing ML baseline. Focusing on our stylized scenarios (Fig- the CL-pipeline outperforms the FNN on average by 60%, 75%, and 7% respectively. We observe that the mean absolute error (MAE) for the FNN increases from 1.75 in the high-entropy scenario to 4.18 18 in the low-entropy scenario, which indicates that the high-entropy scenario is in- deed easier to predict for pure ML baselines as has a high correlation between the traffic flow and each road's 's context. Remarkably, our Cl-pipeline shows 60% improvement over the FNN even in the scenario that casier to predict for a pure ML baseline, but also shows stable prediction error over both the high and low entropy scenario. In the square world scenario, which is again favorable for pure MI baselines due to correlation between flows and the are features, all approaches yield good performance. Still, the CL- Cl-pipeline outperforms the FNN baseline by 7%. Focusing the realistic, time-invariant scenarios (Figure [e).. the CL-pipeline outperforms the FNN baseline by 72% and 71% respectively. In both cases, the FNN baseline fails learn the combinatorial structure of the respective traffic equilibrium that utilizes main roads between high demand areas stronger than small roads, while the CL-pipeline succeeds in encoding this structure by combining the learned latencies with the structure of the CO-layer. In the time-variant scenario (Figure Bi the Cl-pipeline outperforms the GNN benchmark by 23%. Note that we generally observe lower MAE in the time-variant scenario due to an increased amount of zero-valued flows that result from the time expansion.
ures
3a 3c
Structural Analyses Figure shows an example of the structure of the predicted flows for the different algorithmic approaches for the Berlin scenario. For brevity, we exclude the visualization for the GNN and the Pl--iieeiiee as they exhibit similar structure to the FNN and CL flows, and F for complete visualization of all flows. As can be seen, the FNN predicts good
refer Appendix mean values of the traffic flow but fails on predicting the true structure of the flows. In contrast, the CL-pipeline succeeds in predicting realistic traffic flow with high volumes on main roads and reduced flows on smaller roads. The ER- pipeline fails on predicting realistic flows as the latency
(a) Target
(b) FNN
CL)
(d) ER
Figure 4: Visualization of time-invarian traffic flows.

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Published as a conference paper at ICLR 2025
E FE 1 3
Prddiee Riww
Pediood
3E0 Row
Tow
briget Row
(a) FNN
(b) GMN
(c) CL
(d) PL
(c) ER
Figure 5: Comparison of target and predicted traffic flows per are for the Berlin scenario.
function representation is too limited to inform the equilibrium layer correctly. Figure 5 supports analyses and the findings from Figure 4 by showing the distribution of the predicted and target traffic flows, with the target traffic flows as a reference. As can be seen, the CL and PL pipelines show higher correlation between the predicted and target traffic flows than the pure ML baselines and the ER- -iippliline. Finally, Figure 6 shows an example of the structure of the predicted flows for the time-variant scenario. Again, we omit the FNN and PL visualizations for brevity and refer to Appendix(FJoor a full visualization. As can be seen, the CL-pipeline succeeds in predicting the right traffic flow structure, while the GNN succeeds predicting the right structure but underestimates the respective absolute flow volumes. The ER- ER-pipeline fails in predicting both the structure and the absolute values of the respective flows.
Hum venill
ACCS
0
Time
Time
Time
Time
(a) Target
(b) GNN
(c) CL
(d) ER
Figure 6: Visualization of time-variant traffic flows
CONCLUSION
In this paper we introduced WardropNet, hybrid COAML pipeline that augments neural network with an equilibrium layer and bases on a novel end-to-end learning paradigm. Specifically, this pipeline uses a neural network to learn the parameterization of latency functions and CO-layer to compute the resulting equilibrium problem. We train this pipeline via imitation learning, intuitively minimizing the Bregman divergence between target and predicted traffic flows, and showed how to train this pipeline in an end-to-end fashion by leveraging a Fenchel- Young loss that allows to deter- mine meaningful gradient although the pipelines last layer is combinatorial. Finally, we showed with comprehensive numerical study that our COAML pipelines outperform pure ML pipelines in various realistic scenarios by up to 72% on average. We further investigated the prediction accu- racy of our COAML pipelines on time-variant traffic flows and showed that our COAML pipelines outperform pure ML baselines by up to 23% on average.
This work lays the foundation for several promising follow up works on integrating (combinatorial) cquilibrium layers into neural networks for traffic flow prediction. As all of our work is open source, one easily built upon it to account for further families of latency functions statistical models. Additionally scaling the proposed COAML pipeline to larger networks remains an interesting avenue for future research. Lastly, the presented learning paradigm is not limited to predict equílibria and can be used to learn arbitrary flow physics given the right training data and CO-layer, which opens multiple directions for further studies.
more complex
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ACKNOWLEDGMENTS
This work has been funded by the Federal Ministry of Education and Research (BMBF) as part of the M-Cube Cluster 4 Future within the project DatSim, grant no, 03ZU1I0SLA The work of Dario Paccagnan was supported by EPSRC grant EP/Y001001/1, funded by the International Science Partnerships Fund (ISPF) and UKRI.
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A GENERALIZED WARDROP EQUILIBRIA
In the following, we introduce a generalized notion of a WE, where we allow for the travel time on each individual are depend on the traffic flow of other arcs. This allows us to model spill-over effects where the flow on one are is influenced by that of neighbouring ones. Moreover, it allows us introduce suitable regularization which is key for enabling end-to-ens learning.
We consider a transportation network, represented as directed graph G = (V, with arc-set A and vertex-set V. Considering a vertex with (v), and 8- (v). The transportation network is utilized by set of agents j É J that travel from their origin node ofi) V to their destination node d() E V, having a demand D of 1 in its origin and -1 in its destination node. The traveling agents induce traffic flow equilibrium y = (Ua) A. formed by the emergent traffic flow Vo all ares a E A. To formally describe this equilibrium, we denote by y Axd the multi-flow polytope over D,
V. we denote the set of outgoing and ingoing ares
y = { y (y))) y E }on
(18)
This multi-flow polytope Y is decomposable into flow polytope yot) F(bOl) for each agent j E J with yo c RA,, and b(5) c W.V To link the respective per-agent flows per-are aggregated flows, we define
F(bb {3 =YYo)aEA:
Σ
Σ
Ya for all all E
Va
aEô+(v)
ac
(w)
DG
if ا0ا)
(19)
where bi²) = =(b)) D with (6)
D(1)
if v = d
for v E V.
EEV
otherwise.
With these definitions, we can describe solutions in the multi-flow polytope y that describe the traffic flow for each agent j E J. Accordingly, we can define an aggregated flow polytope
{ Σ y ()) y(0) (o() for all in jEJ
=
(20)
that describes the traffic we observe on each are in a city's street network. Note that in this context, ) is again polytope as results from the intersection of sum of finitely many polytopes. In this context, it is worthwhile to remark that the inclusion
( bb) Σ JEJ
is, in general, strict inclusion because the flow polytope may contain flows paths from one agent's origin to another agent's destination. Independent of this ambiguity, it is well known that any flow y(i) admits the following decomposition
Σ PEPG)
apep
( ap Ep
Σ ec,
300
(21)
GEC
where pl) is the set of elementary o()) paths in G, and C is the of cycles in G, while ep and ec in (0, 1144 are the indicator vectors of P and C, with epa (resp. eca) being equal to one P. (resp. C*) and zero otherwise. Here, we observe that solution as a positive 3 amount to agents cycling, and w.l.ogg. ignore the 3 term for the remainder of this paper. Moreover, we note that the decomposition introduced above may not be unique, i.e., one can always find decomposition from flow in Poi to أوو but there exists no bijection that allows mapping from أ(d) to yli) The non-existence of this bijection holds in the absence of the terms.
if belongs
in any meaningful
Wardrop Equilibrium To find a WE this context, we are seeking a of flows in y such that each agent travels path in yut where unilaterally deviating from this path would always incur
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longer travel time for the agent. In this context, the decision of one agent affects the decision of the other agents as travel times on each are depend on its aggregated flow. To describe this dependency, introduce arc- arc-specific latency functions &a R. - R_ for each are E A, which describe the travel time &o (g) on a for an aggregated flow y. Then, we can define a WE as follows.
Definition 2 (Wardrop Equilibrium). Given a directed graph D (V, A), origin- pairs
"a demands (D))) and vector of latency functions L (ao multiflow = (1 E y is WE if there exists a path decomposition aD)) = (ap )EP()), = (o6) Jeeen for each in such that VP pli,,
(3) ap =0 if
P Φ
arg min Ee((v) and AC E C, =6-=0 =
(22)
PEP(S aeP
where y = -jer g is the aggregated flow corresponding to y and ap is a path decomposition.
Our definition of WE is a generalization of the usual notion of WE, where the latency on are depends only on the aggregated traffic on the respective arc, i.e..
(g) = (Ta) for some la R++R4 R+.
This generalization is meaningful for the practical application of traffic equilibrium approximators as the flow on an are can have spill-over effect on the flow of neighboring ares. Thus, considering the flow of neighboring ares for the approximation of the traffic flow on a specific are might ease accurate predictions. From a theoretical perspective, we will later show that this generalized for- mulation of WEs allows us to equip latency functions with regularization terms which is crucial to derive an end-to-end learning framework.
Variational characterization: To establish our learning based approximation, we are interested in obtaining a more aggregated notion of a WE. To this end, we show that one can define the condition for a Wardrop equilibrium without using the path decomposition of Vj-
Proposition 1 (Wardrop Equilibrium (Variational)). Given a directed graph D (V, A), origin- destination pairs (oj, di) jee,,, and vector of latency functions l faa multiflow y (yy))) E y is a WE if and only if
for all j E J and (G) E cyú),
f(g) y0u s <((g) (()
(23)
where you represents the flow polytope of Agent j.
This flow polytope yoi) C RR ensures that any feasible solution yj is non-negative and sends one unit of flow from origin 0j destination dj. while ensuring mass conservation at the nodes of the network G(V,
Proof.
i) Forward direction: We first prove that 22 implies 23
Let minrepo) Eaer la (v). Then, gives the decompositions y(u Ereps olper and y0) = EpEPs apep. Taking the dot product with E(z) gives
((T) (b =(())
Σ
dpprr Σ
a'pepe(t) 2 D()
PEPS)
PEPU)
())) =((D)((
Σ
apep) Σ
ap opepl(g) D(D A
PEPS)
PEPU)
which gives (2) < ((g) (y))).
ii) Backward direction: Let us now prove that 23 implies 22
To that purpose, consider a y such that Equation 22 is not satistied. We prove the result by
exhibiting y( such that E(v)² T 706
{(v) ()) Let po arg min pepus ( ep and
A = minrepon {(g) ep. We have
( T (1)
Σ op (()) eer Σ PEP
ar ( T ep D A,
PEPlMpo
>A
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with the last inequality being true because Epepliipo ap by hypothesis and Σ PepUo ap D because y E E Let us now pick a P in po, and set y Dl We have y (1) E pli) and f(y) T y D A E(g) T y" which concludes the proof.
Here, the latency function la R20 + Rz0 describes the travel time to = la (ūa) on are as function of its aggregated flow Da = EEE, yoo) Clearly, WE in line with Proposition is sensitive the respective latency functions in f.
Let us recall that our goal is to approximate an equilibrium by a surrogate, which, in our case, is Seaming-enricher WE. In this case, appears natural to introduce family of latency functions and seek the best approximation among the equilibria generated by this family. Accordingly, we aim to establish learning algorithm that allows us to learn the parameterization of the respective latency functions. To do so, it will prove useful if our equilibrium problem remains convex.
Convex characterization In the decomposable case, it is well known that the WE exists, is unique, and is characterized when all &a are non-decreasing. We now generalize this result to the non- decomposable case. To do so, we consider latency functions (Ea)a that derive from a potential 4 : R + if
la Oya
(24)
Theorem 1 (Convex characterization). Consider directed graph D (V, origin-destination puirs (oj, dj) and vector of latency finccionss & {Ea JaeA that derive from a potential
1. A multiflow = (1))))ee) V is WE if and only if it is an optimal solution to
min + (
s.t. ÿ = Σ ys.
(25)
wEy
2. A vector y E y is the aggregated flow of a WE if and only if it is an optimal solution to
min k
(26)
3. If de is strictly convex, then a WE y exists and is unique.
Proof. First, we prove that flow is user equilibrium if and only if it satisfies the first-order conditions of the optimization problem, even if it is not convex. Using the path variables ap of Equation 21 we can rewrite the convex optimization problem.
min f()) yEy
s.t.
1 = Σ ys.
(27)
3
as follows:
min Σ apep)
(28a)
PEP
s.t. D
Σ
aP
VjEJ
(28b)
PEPG)
ap 2
VP PP
(28c)
with P being the union of the disjoint p(), Note that the agent-specific paths pbi) remain disjoint, model each agent with unique drstinndet pair.
Denoting by X, and up the dual variables constraints
28b and
28c
we obtain the Lagrangian
1(y,X,11) #6 <E arer)
Σ X
(D))
Σ ar) Σ
ppOp PeP
PEP
PEPO)
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We have @4(g) dap
OCly. X. = (p( - X, + HPP.
Σ
(0( = (p(
which gives
azo
dap
aEP
ag P
The KKT conditions for 28 are therefore
OL(y. X, 11) (r(g)) N3 + pr
vP E p(D) fA EJ
dap
@ X. D(5)
Σ
ap ap =
VJEJ
0p.
PEPU)
PP or op
VP P
pr - 0, 00 20
vp E P
Then flow y satisfies the KKT conditions it satisfies for origin-destination pairs j E J and all paths P e p())
(p = X ( A3
op > a =
E EP)) have greater equal If the path P pP() is used, then its cost is Ay. and all other paths P' cost. This is just the characterization of WE in Definition 2
The characterization of aggregate flows in the second point of the theorem is a direct consequence of the first and of the definition of y.
Existence and uniqueness of the solution y come from the existence (and uniqueness) of the opti-
mum of a (strictly) convex optimization problem on a convex polytope (which is also compact). O
Theorem D allows us to switch between the decomposable notion of latency functions and a non- decomposable notion of latency functions as shown in Example T
Example 1. If is decomposable, (a(ū) (Ja, then defining
La = ( La(u) du
get that e derives from the potential
( Lo(
And Theorem becomes the classic characterization of WE as convex optimization problem.
B STATISTICAL MODELS
Our FNN model (Figure 7p comprises set of parallel multi layer perceptrons (MLPs). Each MLP re- ceives as input a vector features x, describing the at- tributes of are and outputs embedding vector. Each MLP consists of 5 layers, with [100, 500, 100, 10, 5] nodes. We consider relu activation functions between the layers. If the MLP is the last module in the pipeline there an output layer of node convoluting the embedding layer to the output dimension. The activation function in the output layer is pipeline dependent: when the FNN directly outputs the last dense layer does not consider any activation function. In the case, when the FNN outputs the latency parameters (NN (Ix) (i.... )aEA (Oe.))e KY) then the last dense layer considers sofiplus activation function. We specify the considered features in Appendix B.1
FNN:
edge feanures
Feedforward
edge
layer
Paraliel multilaeer
percepticess (MLPs)
MLP
case,
Figure 7: FNN model.
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Our GNN model considers graph message passing module and two subsequent FNN mod- ules (Figure S). The message passing module receives as input the set of are feature vectors (Do)aeA and the set of vertex feature vectors (Fn)eeV. The GNN applies message passing convolutions and outputs the latent node embeddings. Here, each convolution considers embeddings of length 20 and feedforward layers with [20, 20] nodes with relu activation functions. Then, the GNN inputs the combined feature sets of latent vertex embeddings from are-starting nodes, and original are features into an FNN that outputs the are embeddings. Subsequently, the GNN inputs the are embeddings into an FNN to output the are parameters. We specify the considered features in Appendix B.1
GNN:
node Scutures
Gruph message
edge parameters
passing
embeddings
layer
emisedkings
layer
Convolational layers
Paraliel
Paralle muliikyer pererptrans (MLPs) MLP
mulshaye peneettinss
edge featunes
(MLPs)
MLP
Figure 8: GNN model.
B. FEATURES
In Table we summarize the features considered in the COAML pipeline benchmarks and the FNN benchmark. In Table 2 we summarize the features considered in the GNN benchmark. The area
factor describes the radius considered for computing the features: Max distance between nodes in network We standardize all features with its mean values.
Area Factor
Table 1: Features considered in the FNN.
features
number work locations (area factors: 1;2;5, 10, 15)
number nodes (area factors: 1: 2: 10; 15)
number unos (urea. fastoo:: 1: 10: 15)
number ares with capacity
factors: 1;2;5;
are longth
ane speed
capacity
are number of lunes
ane transit time
number starting
starting
numter coding
starting
number starting ares at ending node
number of ending ares ending node
Arc (cotures when learning time- variant flow
distance from time morning zush hour secoads (distance
monnig hour seconda)2
(distance from time morning rush hour in seconds) remaining time in seconds simolation time scconds
distance
evening rush hour sccands
(distance time from stat evening ruch hour seconds)2 (distancee time from evening
seconds)
Seatures learning piccewise
capacity index
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Table 2: Features considered in the GNN.
Are features
ane longth
are spond
cupucitt
ane transit time
number starting ares starting node
sumber ending starting aode
number staning arcs at ending node
number of ending anes onding node
Arc features when learning time variant flow
disstance from time morning rush hour seconds (diistancc from time moming rish hour seconda (distance from time murning rush hour in secundsy" remaining time in seconds
simulation time seconds
secunds (distance time from statt evening rush bour seconds) (distance
evening
evening
3
Node Festumes
number home locations factios: number
10:
2;
number nodes (area factors: 1: 2: 5: 10;
number of anes (urea factios: 1; 5: 10: 15) sumber cupacity
1;2;5;
Node festures when leaming time-variant flow
simulation time scconds
C EQUILIBRIUM LAYERS
In this section, we provide background on the equilibrium layers introduced in Section 4
C.1 BACKGROUND ON LATENCIES WITH EUCLIDEAN REGULARIZATION
We can obtain a differentiable equilibrium layer by utilizing rather simple Euclidean regularization, considering the respective squared Euclidean loss
1
2
In this case, we retrieve decomposable affine latency functions of the form
e
2e - + Da = (Ta).
(29)
and the vector 0 corresponds the y-intercept of the latency functions. In this case, the equilibrium problem becomes
1 2 1 arg max - y11.
jn ( arg max O γ yE
(30)
In other words, the equilibrium is the orthogonal projection of 0 on 5 and can be efficiently com- puted using any convex quadratic optimization solver. In the learning problem 9 we need the gradi- ent (0) of 92" (0) max 0 y 3 =| Danskin's 's lemma ensures that
( = arg max TT } 11y11 (o)
yey
C.2 BACKGROUND ON REGULARIZATION BY PERTURBATION
Alternatively, regularize equilibrium layer by perturbation: let us consider maximum capacity u and the corresponding aggregated flow polyhedra
= = { 0 < ÿ S n).
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Let us now introduce standard Gaussian vector Z on RA. Then, we the convex conjugate of the expectation of the perturbed maximum flow as regularization S
F where F
max (0 2) y
(31)
yEy.
Berthet et al. (2020) show that this regularization yields several desirable properties: first F is proper convex continuous function, which gives Q (P**)* = F, second, dom((?) = Vn. Third, applying Danskin's lemma, this second property, gives the differentiability of F and
(0) = (0) = E arg max + 2) y
vED.
We note that the arg max is unique with probability 1 on the sampling of Z. This last property is critical from practical point of view. Indeed, while the exact computation of F((). (0), I), and Vo require evaluate intractable integrals, they all expectations for which Monte Monte-Carlo ostimations can be computed by sampling a few realizations of and computing the corresponding maximum flow problem.
Finally, Bregman divergence and Fenchel- Young loss coincide for this regularization La(0, () (y (n)))). This follows from Equation even though 52 is not Legendre type as a function on RA,, because it is not even differentiable since its domain Vu not full dimensional. Indeed, consider the restriction of F and Z the direction of the affine hull of Pu. which is equal to the linear span of Vu since 0 E Vu. then S2 is Legendre type. Equation 14 gives the equality on the linear span, which can be extended to IR since the component of 0 in the orthogonal of the linear span does not impact a role in the arg max.
C.3 BACKGROUND ON PIECEWISE CONSTANT LATENCIES AND EXTENDED NETWORK
In practice, observed latencies often exhibit threshold effects when the traffic on an are reaches its capacity and congestion appears, i.e., the latency increases stepwise (Vickrey 1969). Such effects are poorly captured by our potential bg due to the smooth latency functions Ea.
However, the methodology that introduced can naturally be extended the where such latency function can be modeled by piecewise constant functions of ya: let us suppose that we want to use the non-regularized latency function
(0am if Tm S Da Tm for M1....,M)
Ea (Taa
+00
if Da > TM,
where 0 == To < ... TM- Such latency functions are in line with traffic physics: might allow free floating traffic till a certain threshold capacity is reached. Then the latency increases step- wise, e.g., with the traffic experiencing reduced speed, stop-and-go, and finally, congestion (Vickrey) 1969).
We derive piecewise constant latency functions by solving the non-regularized WE as minimum cost flow
min 0
yc ex
on an extended network Ge ( that allows to encode the piecewise-constaae functions in multi-graph representation with parallel arcs.
Specifically, we construct the extended network Goxp (V, A based on the transporation net- work G (V, A) follows: Vertices are the same, but for each are in A, there are M copies
Figure 9: Evolution from G to expanded Gesp
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(m (I.......M) of in AP (see Figure We then define the capacity ua.m Fm Tm-1- Note that we can only consider equal capacities ua.m ua Vm E {1.... .M. The equivalence in the non-regularized case straightforward. For example, in a setting with capacity Va.m = 3: The first three agents traversing arc experience the cost 0a,1, the second three agents traversing are experience the cost 2, etc. We can then use the two previously introduced regularizations. Squared Euclidean regularization now involves an orthogonal projection of 0 on )uP,, while Monte-Carlo approximations in the perturbation case involve solving maximum cost flow on Vaep
C.4 BACKGROUND ON POLYNOMIAL LATENCIES REGULARIZED BY PERTURBATION
Lastly, we introduce monotonously increasing latency functions e (ja) that depend on the respective traffic flow Va. To do we introduce feature mapping that maps any component y E R to feature vector o (yy = (o)( Me (k.....k) E R such that + T (B) convex for each k
y E y + (3) BAXK
Then, we consider polynomial latency functions generated by the mapping (ax)k.
& (T,) = Σ (k. with OR.o 2 0 for all k and a.
(32)
Here OR.a is the weight coefficient of the k-th basis function in las and a is the respective basis function. The overall equilibrium is therefore parametrized by (0 (I....k with indexing the mapping (o%))-- Clearly, in this context, choosing the right family of function is crucial design decision: it should be simple enough to lead tractable WEs for any 0 2 and at the same time sufficiently expressive to allow for good approximations.
In this work, we restrict ourselves to the polynomial family a using k (0, 1}. and (T) = y²h. Then, the latency function for are a becomes
(g.) 0, + 01,0 Da.
with Ta denoting the aggregated flow on arc a. We can interpret this latency function as follows: the intercept fn,a represents the traffic-free latency, and the slope 01,a represents the induced latency agent on arc a. In general, we can use arbitrary complex polynomials to describe the latency function. The resulting latency function 32 convex function, as the basis functions (ax)x are strictly convex for each k.
We can extend the regularization by perturbation to that setting. Let us define = conv ()))). Remark that C is closed and convex but no more polyhedron. We can now define the generalizations
max 0 + Z) Z)T (Ta())) and l(p)) F*
F(0) = E
max ( + 2) μ = E nEC
yEy
We still have F 22"
The prediction is however now on C
(An(0) = EE arg max 0 +
arg max 0 ((H)) * arg max ((o)))((((()))) (33)
nEC
MEC
yED
We retrieve prediction ÿ by taking the projection on y7
0 = IE
arg max 0 + 2) (1y) yEy
It leads to the learning problem and Fenchel-Young loss
1 Σ La(Pu(zs)) where Cn p)) ( 00 u.
min w
Again, we can obtain Monte-Carlo evaluation of the gradient
VoLn(O, ( E.
arg max 0 + 2) j4 Rec
E arg max (0 Z) a(v) f yE
which is quite convenient regularization is not as convenient computationally as we would need to optimize over μ instead of due to the inequality in Equation 33
can make the computations on y. Using squared Euclidean
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D TRAFFIC SCENARIOS
We consider four stylized and two realistic scenarios for our experiments. While the stylized scenar- ios allow us to isolate effects that highlight the differences between our COAML pipelines and pure ML approaches, the realistic scenarios allow us to highlight the efficiency of our COAML pipelines in practice.
The stylized scenarios consider randomly generated networks using the model from Eisenstat (20IT) similar to Figures 2a
2b and 2c All roads in these networks have the same characteristics with respect to capacity. We further consider 100 home and 100 work locations in the network and a set of 200 planned trips the form of origin-destination pairs. Precisely, we consider 100 planned trips starting at the home locations and driving to the work locations and 100 planned trips starting at the work locations and driving to the home locations.
In the first two scenarios, we derive the target traffic equilibrium via solving a time-expanded WE with 20 discrete time steps. The WE considers latency function La (āa) du + Yo with de representing the length of are a and ya denoting the aggregated flow on are a.
High-entropy (HE) scenario: The high-entropy scenario comprises instances that each consist of a randomly generated network and uniformly distributed home and work locations across the respective network (see Figure 2a). This uniform distribution of home and work locations leads to an evenly distributed traffic flow with increasing traffic flow in the center of the network and a lower traffic flow in the outer area of the network. This scenario provides traffic equilibrium that highly correlates with the context of the network. Precisely, the traffic flow of an are correlates with the location of the respective are in the network. Thus, context features like the coordinates of an are and the location of the are within the network have high correlation with the traffic flow. From a theoretical perspective, this scenario is good for learning traffic equilibrium prediction with pure ML approaches: the high correlation between context features and the target traffic flow allows us to learn a direct mapping from features to traffic flow independent of the combinatorial nature of traffic flows.
Low-entropy (LE) scenario: In the low-entropy scenario each instance consists of randomly generated network with home locations located in the upper right corner and work locations in the bottom left corner (Figure 2b). In such a scenario, the resulting traffic flow strongly depends on the combinatorial trip information: Trips start at the home locations in the upper right corner and end at the work locations in the bottom left corner and vice versa. In comparison to the high-entropy scenario, the traffic equilibrium resulting in the low-entropy scenario is less context- sensitive but more sensitive to the combinatorial relation between home and work locations. From theoretical perspective, pure ML approaches are not expected to yield good solutions in this scenario.
For the remaining scenarios, we derive the target traffic equilibrium by running the agent-based transport simulation MATSim. We refer Appendix D.1.2 detailed information the MATSim simulation.
world (SW) scenario: The square-world scenario utilizes the road network of the stylized scenario, but the origin and destination locations are distributed according to the distribution of roads in the respective network. We simulate a one-hour epuch, with all agents starting their trips from their home locations to their work locations minute zero to mimic the morning rush hour, and their trips from their work locations to their home locations after 30 minutes to mimic the evening rush hour. This scenario complements our set of stylized scenarios: it does not mimic world setting but allows to generate insights on the performance of COAML pipelines in comparison to pure ML pipelines when it comes to predicting combinatorial information.
Berlin scenario: The Berlin scenario comprises instances that consist of district networks (Figure 2db extracted from a realistic road network of the city of Berlin. The -rittnndestnnation pairs are extracted from calibrated real-world MATSim scenario If a trip passes the district but has an
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origin/destination outside of the district, set the origin/destination of the respective trip to the border of the district. Thus, this scenario serves to generate insights into how well our benchmarks can predict traffic equilibria for certain districts when contextual information is only available for the respective district, but the contextual information for the outer area of the district is unavailable, e.g.. investigating traffic management effect on certain district, when census data of the outer area of the district is unavailable.
Berlin artificial population (Berlin-AP) scenario: The Berlin artificial population scenario comprises instances that consist of district networks extracted from realistic road network of the city Berlin, but the population plans are artificially generated so that all trips start and end within the district network and there are trips passing through the district network (Figure 2e). The home and work locations are distributed according to the density of the street networks, with more home and work locations in areas with dense are network and fewer home and work locations areas with sparse are network.
Square-world time-variant (SW-TV) scenario: In the aforementioned scenarios, we only focused on the aggregated traffic flow over time. However, in many traffic applications, traffic flow needs to be predicted over time. Therefore, we consider a square-world time-variant scenario, which investigates the performance of our benchmarks when predicting traffic flow over time. The challenge is that MATSim yields a traffic flow over constant time, but our benchmarks predict traffic flow over discrete time. To circumment this problem, we discretize the target traffic flow de- rived from the MATSim simulation over time. To do so, we apply time expansion on the transport networks G of the square- world scenario and measure the MAE on the time-expanded transport networks. Specifically, time-expansion on the transport network G refers to discretizing time into disjoint epochs T; then each are E A A|x[T] in the time-expanded network represents are with a specific time index E T. On the spatial dimension, the length of an are specifies the spatial distance from the start location to the end location of the arc, and from temporal perspective, the length of an are specifies the time takes to traverse the respective arc. Note that such a setting, the latency on the respective are, but not from the flow ya,t, at another point in time t2 E T. Intuitively, assume that the traffic flow an are is independent of previous traffic flows on the respective arc.
are at time l E T does only depend on the time-variant traffic flow yo,t,
For all scenarios, create 9 training instances, 5 validation instances, and 6 test instances. We run the training for a maximum of 20 hours, or 100 training epochs. We run the experiments on computing cluster using 28-way Haswell-EP nodes with Infiniband FDR14 interconnect and hardware threads per physical core. We report the MAE on the test instances.
D.1 WARDROP EQUILIBRIUM SOLVERS
In general, WE solvers seek set of flows y such that each agent j travels on a path in yo6) where unilaterally deviating from this path would always incur a longer travel time for the agent. Correa & Stier-Moses (2011) provide a good intuition for WEs and respective solvers.
In general, we can divide WE solvers into analytical solvers and simulation based solvers. Analyti- cal solvers require knowledge about the system's 's latency functions to allow the computation of the respective equílibria, which is why they are frequently used for stylized or synthetic analyses. In practice, these latency functions are usually not known and consequently simulation-basee solvers used determine the respective equilibria. The drawback of such simulation- -assed solvers is their computational time, which can easily exceed several hours depending on the size of the case study and the hyperparameters chosen.
Our approach aim to bridge the gap between the accuracy of simulation-bas basd solvers and the tractability of analytical solvers by learning the parameterization of the respective latency functions and embedding them into a CO-layer. In the following, we give an overview of related analytical and simulation-based solvers.
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D.1.1 ANALYTICAL SOLVERS.
Analytical solvers leverage mathematical formulations and iterative approaches. The analytical solvers mainly base on two intuitions:
1. The Beckmann formulation (Beckman et al. 1956) interprets the WE problem as a multi- commodity flow problem (MCFP) and states that the convex optimization problem
min ( vEy
Σ
la(dddr
(34)
that satisfies the flow conservation and the non-negativity of flows yields a WE.
2. We can formulate the notion of WE as a variational inequality (Smith, 1979 Then a
flow is WE when
Σ aEA
Ca I
Σ Lola)ao
Vx E y
(35)
acA
holds.
In the following, provide overview the most common analytical WE solvers.
Trank-Wolff Algorithm The Frank-Wolfe algorithm is iterative approach solving Problem 34 Intuitively, the Frank- Wolfe algorithm starts with an initial flow, updates the travel times depending on the flow, and derives direction to update the flow such that it reduces Z(y). Iteratively, the algorithm updates the traffic flow y to finally determine a WE. Formally, we detail this process in Algorithm
Algorithm Frank-Wont Algorithm
Require: g(0)
D Set initial flow
4-0
while WE not found do
+ min EaEA La (g
(k)
Ta
Get direction update (()
(k) +- arg min, 7 (M) + a (a (a-g((1)) ((1)))
D Line search
y (++1) + (K)) + (k) # (x - g (8)
b Update flow
Ie < k 1++1
end while
Successive Averages Algorithm The Successive Averages algorithm is heuristic iterative algo- rithm similar to the Frank-Wolf Algorithm but different in the flow update process. We depict the process in Algorithm 2
Algorithm Successive Averages Algorithm
Require: (())
D Set initial flow
x-0 0
while WE not found do
x f- min Easa
La (Na
(k) Ta
Get direction to update (()) b Update flow
((++1) + ÿ (K)
+ EII * (x ② (8))
A +- k 1
end while
Decomposition Algorithms Decomposition algorithms decompose the problems into subprob- lems in the sense that they separate flows by node of origin. The best known algorithm is the Bar-Gera's algorithm (Bar-Gera) 2002) that we show in Algorithm
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Algorithm 3 Bar-Gera's Algorithm
Require: g())
> Set initial flow
f- 0
while WE not found do
for E V do
b Iterate all origins in the network
yº c g EnEA (a(z6")
Va
> Distribute flows from origin o
end for (R+1) +- V
> Combine flows from all origins
< k +1 end while
Projection Methods
Projection methods leverage the variational intequalities and project flows onto feasible flow sets. We can reformulate the variational incqualities as an optimization problem,
find y E y such that (<(g), 7) 0, V/ E
(36)
then the objective is find a traffic flow y that satisfies the variational inequalities, and we receive this traffic flow with the following Algorithm
Algorithm 4 Projection Methods
Require: (())
> Set initial flow
+-0 0
while WE not found do + (k) (g)()
> Get new flow proposition
(k+1)) + Px(
Project flow proposition onto the feasible flow y
k t-k- k+1
end while
Mathematical programming Depending on the on form of the latency function we can apply various mathematical programming methods like, c.g., Barrier Methods or Interior Point Methods to find a WE via solving the Beckman formulation, Problem 34
D.1.2 SIMULATION-BASED SOLVERS: MATSIM
In comparison to analytical solvers also leverage agent-based simulations that simulate the actions of each agent individually, and by following a selfish behaviour of each agent yielding a WE. Instead of presenting various different simulation approaches, we present MATSim (Hori et al. 2016), as we also consider MATSim in Section yield target traffic flows y"
MATSim is agent-based transport simulation. Specifically, each agent in the simulation follows an agent-specific plan. This agent plan details the agent's daily trips, including starting times and the respective routes. The simulation follows a queue-based approach such that when vehicle enters a network link from an intersection, the simulation adds the vehicle to the tail of vehicles waiting traverse the link (Vickrey 1969). The waiting time depends on storage capacity and the flow capacity of the respective link. The resulting traffic flow g' results from a co-evolutionary optimiza- tion approach with each agent representing a single species, meaning that each agent optimizes its respective plan individually. Specifically, in each evolution step, each agent considers a plan. Then, the simulation simulates all plans in the system and distributes scores for each individual plan de- pending waiting times and deviations from initial schedules. Subsequently, the co-evolutionary approach updates each agent's plan via inheriting from previous plans from the same agent, mu- tation, and recombination actions. Thus, this co-evolutionary process improves each agent plan individually, dependent on all other plans in the system. Finally, this co-evolutionary approach leads stochastic user equilibrium y² In this equilibrium, no agent unilaterally improve its plan.
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E WARDROPNET PIPELINES AND ML BASELINES
This section details all the benchmarks that we consider in our numerical study. While the first two benchmarks are pure deep learning benchmarks, the last three benchmarks relate the COAML pipelines introduced in Section 4
FNN: we consider a pure ML pipeline f()) = y with an FNN model that was trained via super- vised learning. To this end, we consider the FNN model and the related features presented in Appendix
GNN: we consider a pure ML pipeline f(x) = y with a GNN model that was trained via supervised learning. To this end, we consider the GNN model and the related features presented in Appendix B
Constant latencies (CL): consider COAML pipeline using constant latencies regularized by perturbation. The COAML pipeline solves a MCFP with piecewise decomposed latency functions. We enable the piecewise decomposition by applying the network ex expansion trick as detailed in Appendix C.3 we multiply each are M-times and set arc-specific capacities on each arc. We further consider a statistical model fou similar to the FNN model used in the FNN benchmark. We only add last layer with a softplus function which additionally negates the output. Recall that we need to enforce the statistical model to predict negative latencies as we solve the equilibrium problem, which normally minimizes the latencies for all agents, as a maximization problem (cf. Equations to allow the formulation of the Munchel-Young loss (cf. Equation 8)
Polynomial latencies (PL): we consider a COAML pipeline using polynomial latencies regular- ized by perturbation. The COAML pipeline solves WE. We assume polynomial latency function with k = {0, 1}, and (v) = y². In this setting, we consider two branches of FNNs (g pw: ) to predict the intercept Oy and the slope parameter 0₃ of the latency function. Each FNN branch equals the FNN model used in the FNN benchmark.
Euclidean regularization (ER): we consider a COAML pipeline using latencies with Euclidean regularization. The COAML pipeline solves WE. We assume polynomial regulariza- tion term Q(y) = ||10l This enables us to learn the pipeline considering the WE as a regularized minimum cost flow problem. The statistical model a FNN model similar to the statistical model considered in the FNN benchmark.
The pure ML benchmarks, namely the FNN and the GNN benchmarks are not state- of-the-art deep learingg benchmarks for predicting traffic equilibria. However, using the same deep learning models in the pure ML benchmarks and the COAML pipelines, namely the CL, PL, and ER benchmarks, allows meaningful comparison with respect to the impact of structured learning in comparison to purely supervised learning. A comprehensive tuning of deep learning benchmarks would lead to a standalone paper that focuses on deep learning architectures in the realm of traffic equilibrium prediction. However, the focus of this paper is to introduce the effectiveness of combin- ing MI. and CO in COAML pipelines.
Discussion:
F IN-DEPTH RESULTS DISCUSSION
This section presents our numerical results. In this section, we first focus on the results of the COAML pipelines and the pure ML pipelines on the stylized and realistic scenarios. Second, we present a structural analysis generate insights on the contribution of the benchmarks by visualizing the predictions of the COAML pipelines and the pure ML pipelines. Finally, we present the results of the benchmarks for the Aieevariant predictions.
Stylized scenarios Figure 10a shows the results for the various benchmarks on the high-entropy scenario. All the COAML benchmarks, namely CL, PL, and ER outperform the pure ML bench- marks FNN and GNN. Precisely, the CL benchmark outperforms the pure ML benchmarks by around 60% on average, while the PL benchmark outperforms the ML benchmark by around 52% on aver- age. Recall that predicting the traffic equilibrium in such a scenario depends less the combinato- rial trip information but rather on the location of an are in the network, as the traffic flow is evenly distributed with increasing traffic flow the center. Therefore, from theoretical perspective, the
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10
10
10'
MAR
MAR:
107
107
107
1768
GNN
FNN
GNN
CR.
17N8
GNN
(a) HE
(b) LE
(c) SW
10
10'
10'
MAR
MAR
10'
10'
10'
PNN
GNN
M.
FR
PNN
GNN
FR
PNN
GNN
P..
FR
(d) Berlin
(e) Berlin-AP
(f) SW-TV
Figure 10: Benchmark performances on various stylized and realistic traffic scenarios.
pure ML pipeline should yield good results in this scemario, as there high correlation between the traffic flow and the context features describing the location of an within the network. The results show the superiority of COAML pipelines over pure ML pipelines, even in scenarios that exhibit a structure that is easy to capture with pure ML pipelines.
Figure T0b shows the results on the low-entropy scenario. The MAE for the FNN benchmark in- creased from around 1.75 in the high-entropy scenario around 4.18 in the low-entropy scenario. This indicates that the low-entropy scenario indeed more challenging to predict for pure ML pipelines in comparison to the high-entropy scenario the correlation between the are flow and the context features is lower. Comparing the different benchmarks, we observe that the COAMI. pipelines again outperform the pure ML pipelines. Remarkably, the prediction error of the COAML pipelines in the low-entropy scenario equals the prediction error in the high-entropy scenario. Specifically, the CL benehmark decreases the prediction error of the FNN benchmark by around 75% on average, and the PL benchmark decreases the prediction error of the FNN benchmark by around 72% on average. The experiments on the high-entropy scenarios and the low-entropy sce- narios indicate that the COAML pipeline learns to leverage the statistical model and the equilibrium layer depending on the underlying task outperforming pure ML pipelines, even in settings when pure ML pipelines expected to perform well: while the contribution of the statistical model is particularly important in the high-entropy scenario when it comes to extracting the right context features, the equilibrium layer is particularly important in the low-entropy scenario when it comes to processing combinatorial information. Surprisingly, the GNN benchmark does not perform well in both scenarios. This can be attributed two reasons: first, while the GNN models perform partic- ularly well when predicting temporal traffic forecasts in a receding horizon manner, the GNN model might be less well-suited when predicting aggregated traffic flows. Second, in comparison to an FNN model, the GNN model accounts for irregular structures with many intertwined dependencies of neighboring nodes and therefore is prone over-smoothing Thus, a simple GNN model might not be able to extract notable features.
Figure 10c shows the results for the square-world scenario. Here, all approaches outperform the ER benchmark. Still, the CL benchmark and the PL. benchmark outperform the FNN benchmark by around 7% on average. The pure ML pipelines reach a similar absolute performance as in the high- entropy scenario, and the COAML pipelines reach a lower absolute performance in comparison to the high-entropy scenario and the low-entropy scenario. Here, a similar effect arises as in the high-entropy scenario: Roads located in a higher populated area have higher traffic flow. Thus, there is a correlation between the are context, namely the amount of work and home locations in the closer neighborhood, and the resulting traffic flow. So, from theoretical perspective, the pure
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ML benchmarks should yield good results in this scenario. However, the COAML benchmarks still outperform the pure ML benchmarks.
Realistic scenarios Figure [0d] shows the results for the Berlin scenario. Here, the COAML pipelines CL and PL significantly outperform the pure ML pipelines. Precisely, the CL benchmark reduces the prediction error by around 72% on average in comparison to the FNN benchmark, and the PL benchmark reduces the prediction error by around 64% on average. Note that the results are biased due to many trips starting and ending at the border of the network. Intuitively, the trips starting and ending at the border of the network are long trips passing through the Berlin district network. Thus, most of these trips focus on large roads with large capacity and free speed, mainly highways. From the pure MI. perspective, it is difficult to learn the focus of these through-passing trips single roads, i.c., roads forming the shortest path from one side of the district to the other side of the district, and the high difference in traffic flow between these shootest-pptth-e -attt----ttttttttgtt forming roads and feeder roads. Moreover, the origin locations and destination locations of these trips depend on the outer context that is not included in the scenario context, making it even more difficult for pure ML benchmarks to learn the relationship between the scenario context at hand and the focus of the traffic on single main roads. On the other side, the COAML benchmarks can leverage the equilibrium layer to learn the combinatorial effects of many trips, focusing on the pouttest-pat-fformin roads.
Figure 10e shows the results of the Berlin artificial population scenario. Recall the scenario equals the Berlin scenario, but instead of the calibrated Berlin population, the scenario considers an arti- ficially generated population restricted to the respective districts. Still, the results are similar to the results for the Berlin scenario: The CL and PL benchmarks outperform all pure ML benchmarks. Precisely, the CL and the PL benchmarks outperform the FNN benchmark by around 71% and 67% on average, respectively. Intuitively, we observe a similar effect as in the Berlin scenario with the traffic flow focusing on the main roads forming shortest paths between areas with high densities of work and home locations. While pure MI. benchmarks fail to account for the combinatorial nature of the resulting traffic flow, the COAML pipelines leverage the statistical model and the equilib- rium layer to combine the processing of contextual information with the combinatorial nature of traffic equilibria focusing on high-volume roads. Surprisingly, the CL benchmark outperforms the PL benchmark in all scenarios, although the CL benchmark solves a MCFP in the CO-layer and the PL solves a WE in the CO-layer. Originally, the CL benchmark was introduced an efficient approximation for COAML pipelines solving a complex WE in the CO-layer. However, the superior performance of the CL benchmark might have a natural reason: According to Vickrey (1969), we can model flows with a queueing approach - also MATSim leverages such queueing approach- such that latencies of an are only increase when its respective queue reaches certain length. We can interpret this stepwise increase of latencies depending on the number of agents in a queue as piecewise latency function, similar to how we model it in the Cl-pipeline. This observation would lead to the assumption, that stepwise latency functions as used in the CL-pipeline lead to better traffic equilibrium approximations as polynomial latency functions as used in the PL-pipeline.
Overall, we can conclude that the COAML benchmarks outperform the pure ML benchmarks on all tested scenarios, also in scenarios that in particular serve to yield good predictions for pure ML benchmarks. In Appendix F.I we introduce further stylized experiments and report respective results.
Structural analysis: In this paragraph, we provide a structural analysis to substantiate the intuition on why the COAML benchmarks outperform pure ML benchmarks in predicting traffic equilibria.
Figure shows the target (Ta)) and predicted (FNN: 1Tb. GNN: TTc. CL: TTd PL: TTel ER:
traffic flow aggregated over time for the various benchmarks on the Berlin scenario (Figure 2d)
Note that in Figure [Ta the main traffic flow focuses on the main roads, i.c., roads forming the short- est paths between areas of interest, while there is only a low flow on small roads, i.e., local roads in residential areas. As can be seen, the pure ML benchmarks (Figures 1Tb] and lic) focus on pre- dicting good mean values for the traffic flow but fail to map realistic traffic flow patterns that reveal high traffic flows on main roads. Analyzing the difference between the pure ML benchmarks, the GNN benchmark yields a more realistic traffic flow pattern than the FNN benchmark. This observa- tion is in line with the different architectures of the FNN benchmark and the GNN benchmark: the GNN benchmark allows the processing of contextual information and embeddings from neighboring roads and enables the detection of main roads to represent consistent flow attributes. Contrarily, the
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Num. vehicles
(a) Target
(b) FNN
(c) GNN
Num. vehicles 10
(d) CL
(c) PL
(1) ER
Figure 11: Visualization of time-invariant traffic flows.
FNN benchmark only considers an are-specific context and thus yields uncorrelated traffic predic- tions between roads (Figure Ilb). Still, the FNN benchmark outperforms the GNN with respect to accuracy due to the better performance in predicting traffic flows on small roads. In contrast, the COAML benchmarks, namely the CL and PL benchmarks (Figure Lid and TTep predict realistic traffic equilibrium with a high traffic flow on main roads and low traffic flow on small roads. The remaining ER benchmark, which also a COAML pipeline, fails to mimic realistic traffic equilibria but focuses more on predicting small traffic flows on rarely used small roads. This might be due to the restricted latency function used in the ER benchmark that only allows us to learn the y-intercept, which is strong limitation when it comes to predicting complex traffic patterns. Studying richer latency functions remains an interesting avenue for future work in this context.
Figure 12 shows the mapping from target traffic flow values to predicted traffic flow values and substantiates our conclusions drawn from analyzing Figure 11 Intuitively, in the case of a perfect correlation between the target and the predicted traffic flow, all points would be the diagonal. However, focusing on the pure MI. benchmarks, namely the FNN (Figure 12a and GNN (Fig- ure T2b benchmarks their predictions have a low correlation with the target traffic flows especially for high traffic flow values on main roads and thus fail to represent realistic traffic equilibrium pattern. comparison, the COAML benchmarks, namely the CL (Figure [2c]. the PL (Figure 12d), and the ER (Figure 12ep benchmarks yield predictions with high correlation to the target traffic flows especially for high traffic flow values on the main roads.
Overall, our structural analyses show the limitations of pure ML pipelines compared to COAML pipelines: pure ML pipelines fail to learn the combinatorial dynamics of the underlying traffic pat- terns and thus yield predictions with low correlation to the target traffic flow values, especially for high traffic flow values on the main roads. From pure MI. perspective, it difficult to learn that the traffic flows often spread largely over a limited number of main roads. In contrast, the COAML
Proiccca Romm E
23
25 23
25 36
(a) FNN
(b) GNN
(c) CL
(d) PL
(c) ER
Figure 12: Comparison of target and predicted traffic flows per are for the Berlin scenario.
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Num. vehicles
Arcs
Arcs
Arcs
Time
Time
(a) Target
(b) FNN
(c) GMN
Arcs
Arcs
Arcs
"B6505
Time
Time
Time
(d) CL
(c) PL
(1) ER
Figure 13: Visualization of time-variant traffic flows
benchmarks leverage the equilibrium layer to learn the combinatorial nature of traffic equilibria and thus allow the prediction of realistic traffic patterns.
Time-variant traffic flows: Figure 10f shows the results of the square-world time-variant sce- nario. As can be seen, the CL and the PL benchmarks outperform all other benchmarks. More specifically, the CL benchmark reduces the prediction error in comparison to the FNN benchmark by around 33% on average and the PL benchmark reduces the prediction error in comparison to the FNN benchmark by around 22% on average. Note that in this setting, we did not learn piece- wise decomposed latency functions but expanded the underlying transport network graph over time. The time expansion of the WE problem significantly increases the runtime for finding the WE in the CO-layer. To circumvent long training times, we ignored the perturbation during training as the aggregated traffic flow y acts similarly to regularization term. However, the missing perturba- tion explains the slightly worse performance of the PL benchmark compared to the CL benchmark. In general, the MAE of the benchmarks is lower than in the other scenarios, which is due to the many zero values in the target traffic equilibrium over the tmee-xxansion. Note that the ER bench- mark fails to account for the time-expansion, because the benchmark only allows for learning the y-intercept, and embedding time-related information and spatial-related information within a single y-intercept remains difficult. Figure 13 visualizes the prediction of the traffic flow time. We recall that agents schedule trips from home to work at minute zero, which reflects the morning rush hour, and from work to home after 30 minutes which reflects the evening rush hour. When compar- ing the target traffic flow values (Figure 13a) with the ML benchmarks (FNN: 13b GNN 13 we see that the ML. benchmarks underestimate the traffic flow and focus more on predicting good mean value. The COAML benchmarks (CL: Figure 13d PL: 13e) predict realistic traffic flow patterns in line with the target traffic flow (Figure 13p. This insight consistent with the observations made when predicting traffic flows aggregated over time (cf. Figure [I]. The ER benchmark (Figure l3f fails to account for the time component such that the traffic distributes equally over time. This effect might results from the simplified latency function used in the ER benchmark that only allows to learn the y- y-intercept.
F.1 COMPLEMENTARY STYLIZED EXPERIMENTS
In this section, we present benchmark performances for 16 different scenarios. Each scenario is a combination of environment and underlying oracle. While the environment details the under- lying street network which is generated from the model of Eisenstat (201T) and the population with the respective agent plans, the oracle defines how to derive the target traffic equilibrium. In each
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(a) LE
(b) LEU
(c) HE
(d) HEU
(e) SW
Figure 14: Illustration of the environments.
scenario, we consider 100 agents, and each agent has a home and work location. Each agent travels in the morning from its home location its work location and in the evening from its work location to its home location. We solve all scenarios in a time- time-expanded setting with 20 discrete time steps.
We consider the environments shown in Figure
Low-entropy (LE): A randomly generated street network. All home locations are in the upper right corner and all work locations are in the lower left corner. Each instance of this scenario considers random street network and random home and work locations.
Low-entropy uniform (LEU): A grid street network. All home locations are in the upper right corner and all work locations are in the lower left corner. Each instance of this scenario considers random home and work locations.
High-entropy (HE): randomly generated street network. All home locations and work locations are uniformly distributed. Each instance of this scenario considers a random street network and random home and work locations.
High-entropy uniform (HEU): A grid street network. All home locations and work locations are uniformly distributed. Each instance of this scenario considers random home and work locations.
Square world (SW): A randomly generated street network. All home locations and work locations are distributed according to the distribution of the roads. Each instance of this scenario considers a random street network and random home and work locations.
We consider the following oracles:
EasyMCFP: The oracle solves a MCFP. The costs of the MCFP equal the street length.
EasyWE: The oracle solves a WE. The latency function of the WE for are is da + Da with de representing the length of are and No denoting the aggregated flow on
randomMCFP: The oracle solves MCFP. The costs of the MCFP are uniformly distributed ((0,100),
HandomWE: The oracle solves WE. The latency function of the WE for are is 01,a + 02.a Yo with fLa ~ ((11 100) randomly distributed, 02 (111,,0 20) randomly distributed, and yo denoting the aggregated flow on arc a.
Figure 13 shows the performance of the benchmarks introduced in Appendix E ovvr the introduced scenarios. In general, the COAML benchmarks, namely the CL and the PL benchmarks, outperform the pure ML benchmarks, namely the FNV and GNN in almost all scenarios. The ER benchmark has a mixed performance over the scenarios, which can be attributed to the simplified latency functions that only allows learning the y-intercept. In the following, we interpret the results depicted in Fig- ure 15 We first focus on scenarios considering an EasyMCFP oracle to yield target traffic equilibria. In these scenarios, the resulting traffic flow only depends on the shortest paths in the network with respect to are lengths, but no traffic effects are considered. Intuitively, the CL pipeline yields the best results in all scenarios. Here, the CL would mimic the oracle when it learns to focus only on the are length in the statistical model. In general, over all scenarios considering an EasyMCFP oracle the differences in accuracy between the benchmarks are rather small, In the scenarios that consider an EasyWE WE oracle to yield target traffic equilibria, the differences in accuracy between the different benchmarks are larger. This is intuitive, as the oracle yields target traffic equilibria that depend on actual congestion states. Thus, these target traffic equilibria are difficult to predict with pure MI.
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Figure 15: Performance of benchmarks different scenarios.
benchmarks when only considering contextual information. In this setting, the COAML pipelines yield good results. Especially the PL and CL benchmarks perform well in all scenarios. While the PL benchmark performs particularly well in the low-entropy uniform scenario and the high-entropy uniform scenario, the CL benchmark performs particularly well in the square-world scenario. The pure ML benchmarks lead to the worst results over all scenarios. All the tests scenarios with the RandomMCFP and the Random WE oracle yield bad accuracy values over all benchmarks. This intuitive as the target traffic equilibria follow random structure that is completely independent of the underlying context. Thus, the prediction accuracies on these scenarios are worse than the pre- diction accuracies the scenarios that are based the EasyMCFP and the Easy WE oracle over all tested benchmarks. Although the benchmarks can not use any reasonable contextual information, the COAML benchmarks can leverage combinatorial information with respect the origin and des- tination location trips in the network and, therefore, slightly outperform the pure ML benchmarks for these scenarios.
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Reviews:
[{"review_id": "5GLV3iC3Mj", "reviewer": "Reviewer_bHk3", "summary": "Traffic flow on a transportation network is influenced by many contextual factors such as weather conditions, time of day, road capacity, etc. Under mild hypotheses, it can be shown that the traffic flow will converge to an equilibrium known as the Wardrop equilibrium (WE). Predicting how the  Wardrop equilibrium will change as a result of changes to these factors is crucial for the design of better transportation systems. This paper introduces a novel approach to this problem which combines a neural network with a combinatorial solver.", "questions": "- Do you have any thoughts on predicting non-equilibrium traffic flows? Do you think these are important for modeling?\n - I appreciate your generalized notion of latency function. However, it seems to me that requiring generalized latency functions to derive from a potential is quite a strong assumption. Could you give an example of a set of non-decomposable latency functions deriving from a potential? (Maybe the regularization by perturbation model of Section 4 is such an example?)\n - Could you provide some background, for non-experts like myself, on the state of the art of WE solvers? From Appendix D.1, I gather that MATSim uses a genetic algorithm to solve for the WE. Are there not faster approaches that use techniques from convex optimization, e.g. interior point method? Also, it would be helpful if you included a runtime comparison between WarDropNet and MatSIM. \n\nMinor Questions: \n\n - in Line 129 \"unilateral deviation would incur in a longer travel time\" should be \"unilateral deviation would incur a longer travel time\" (no \"in\").\n - In line 185, \"Following, the supervised learning setting\" should be \"Following the supervised learning setting\" (no comma after \"Following\").\n - For consistency with eq (2), the sum in eq. 9 should probably have a $\\frac{1}{N}$ in front of it.\n - In line 302 \"K represents the amount of parameters\" should be \"K represents the number of parameters\"\n - In line 335, \"We note that the arg max is unique on a sampled realization of Z\" should probably be \"We note that, with probability 1, the arg max is unique on a sampled realization of Z\".\n - On line 458 \"each roads context.\" should be \"each road's context.\" \n - On line 1141 \"raods\" should be \"roads\"\n -  Suggest citing _End-to-end learning of user equilibrium with implicit neural networks_ by Liu et al in \"Related Works\".", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The paper is well-written and easy to follow. Although the topic of traffic flow prediction might not be familiar to the average ICLR reader, the appendices do a great job of introducing the reader to the fundamentals of this problem.\n - Novelty: I believe this is the first paper to apply Fenchel-Young losses to the problem of predicting WE, which is an important contribution.\n - The numerical experiments are sufficient to convince me of the utility of the proposed method.", "weaknesses": "See questions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Uae4QByiRE", "reviewer": "Reviewer_SQBm", "summary": "The paper introduces a theoretical framework designed to address the challenges of understanding the convergence and generalization properties of machine learning algorithms. The authors propose a set of mathematical constructs and algorithms aimed at improving the understanding and solution of issues related to algorithm performance in various settings. Key contributions of the paper include the development of new analytical models and convergence proofs, which are presented alongside rigorous theoretical analyses. The authors also discuss the implications of their findings and how they relate to current practices in the field of machine learning.", "questions": "I would like to clarify that I am not an expert in the specific field addressed in this paper. While I can appreciate the effort and the theoretical contributions made by the authors, my limited background in this area restricts my ability to fully evaluate the nuances and implications of the proposed framework. Therefore, my feedback may not capture all the intricacies of the work or its potential impact on ongoing research.", "limitations": "", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper introduces a novel theoretical framework that enhances the understanding of convergence and generalization properties in machine learning algorithms, addressing critical gaps in existing literature.\n\n2. The rigorous mathematical analysis, including new analytical models and convergence proofs, adds credibility and depth to the research.\n\n3. The well-organized structure and effective use of examples make complex theoretical concepts accessible and easy to understand.\n\n4. The findings have the potential to impact future research directions in machine learning and improve algorithm design, offering valuable insights for practical applications.", "weaknesses": "1. The paper lacks a thorough comparison with existing theoretical frameworks or analyses in the field. A comparative analysis highlighting the proposed framework's advantages and limitations relative to established methods would clarify its contributions and significance.\n\n2. Some of the theoretical constructs presented are quite complex and may be challenging for readers who are not deeply familiar with the underlying mathematics. Simplifying certain sections or providing additional explanations and visual aids could enhance understanding and accessibility.\n\n3. The authors do not sufficiently address the limitations of their framework. A more transparent discussion about potential shortcomings, assumptions made, and scenarios where the framework may not apply would provide a more balanced view of the research.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jDREGwe4LN", "reviewer": "Reviewer_LPcY", "summary": "The paper describes a neural network architecture designed to predict traffic flows. \nThe main idea is to combine traditional neural network layers with an \"equilibrium layer\" that models traffic flow equilibria, \nThen, train the network end-to-end given training data pairs of (network, target flow)\n\n\nThe authors provides anonymized code which is commended.", "questions": "Q1: What is the architecture of the new equilibrium layer? What exactly is its input, output, and tunable parameters? \n(I realize that this information maybe given somewhere in the 20-page supplemental. But a paper that states that its main contribution is a new layer, should describe this layer in the main paper. Or revise their claimed contribution).\n\nQ2: Which results in section 3 were previously known? What new theoretical results are presented in the paper?\n\nQ3: The paper stated it contained 9 training instances samples. Could you clarify how these instances are used? I'm assuming they are used for generating many labeled training samples (x,y). How exactly is this done? If only 9 samples are provided", "limitations": "", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "S1: An interesting problem, combining learning and combinatorial optimization. This seems to be novel.\nS2: A detailed theoretical foundation for the proposed architecture. \nS3: Extensive experiments on 6 six scenarios, using traffic simulators to generate GT for training.", "weaknesses": "W1: I found the paper very hard to understand. It may have been written by researchers outside the ICLR community. The introduction spends half a page to describe in detail supervised learning and ERM, but does not clearly define the problem or explain current state-of-the-art approaches. Then, it is not made explicitly clear enough which parts are novel and which parts were previously introduced. Terms like paradigm, pipeline, layer, model and architecture, are sometimes used loosely and interchangeably. See Q1 for specific questions. \n\nW2: The method is compared with the an architecture that has the CO layer removed, and with various variants of the method. Not with other baselines in this field.  I am not a member of this community, but a quick search shows previous approaches do exist. See AA Kashyap 2022, Traffic flow prediction models – A review of deep learning techniques. \nThe author should analyze their proposed architecture in the context of previous work.\n\nW3: There is a gap between the general formulation of the problem and the iterative relaxations, and it is not clearly stated how each relaxation limits the application of the COAML pipeline in practice.\n\nW4:  The COAML problem formulation attempts to address a general latency function, but in the end the experiments are done with a simple (possible unrealistic) latency function. \n\nW5: It is not clear whether WardropNet yields a meaningful improvement (e.g, improving the accuracy to 2% from 1% SOTA ML baseline may sound like a great improvement, but if a non-ML approach can achieve 90% accuracy, then the gap to practicality is still huge).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "H7MnhMiHjo", "reviewer": "Reviewer_qmEt", "summary": "This paper proposes WardropNet to predict traffic flow by the combinatorial optimization-augmented machine learning (COAML) pipeline. Using supervised learning and Fenchel-Young loss, this method minimizes the difference between predicted and actual traffic flows by leveraging a Bregman divergence, ensuring it fits the geometry of traffic equilibria. WardropNet improves on average for traffic flow predictions compared with pure neural network method.", "questions": "1. In the end of section 1, the paper put up with \" backpropagation through general, possibly combinatorial, equilibrium layers\". What does it mean, and can the paper give some examples?\n\n2. In the end of section 3, how the Bregman divergence, the non-convex problems, turn to a convex surrogate? Why can the paper do the alternation?\n\n3. In the comparision part, is there any numberical results (listed in the table) can be shown to clearly explain the strengths of the new algorithms? Meanwhile, it can be noticed that, the results of ER perform poorly in mostly scenarios. Please explain the reason and the strengths of this aspect of algorithm. In addition, can the paper give a comparision different pipelines in various scenatios in terms of the mathmatical theroies, and give the differences among them?\n\n4. Also in comparision part, can you analyze and explain the reasons and aspects why the proposed algorithm is better than the baseline algorithm, and how these aspects confirm the strengths of the algorithm you proposed in the conclusion?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The essay provides a solid theoretical foundation for the approach.\n2. The visualization and illustration are well made and help the readers to understand the paper.\n3. By incorporating real-world scenarios, the method emphasizes its potential for real-world traffic management and urban planning.", "weaknesses": "1. Lack of comparison with more advanced baselines. While the paper compares WardropNet with basic machine learning models like FNN and GNN, it doesn't compare against other state-of-the-art traffic flow prediction models, such as deep reinforcement learning approaches or physics-informed neural networks.\n\n2. Not well-structed essay presentation. As paper introduces a complex method, the explanation of how the model works is too technical too quickly. Meanwhile, the scenarios of the experiments section feels too long.\n\n3. Sensitivity to Hyperparameters. The paper does not detail the sensitivity of WardropNet's performance to different hyperparameters such as layers number and learning rates. Since the model integrates multiple components, tuning could be more challenging, and it's unclear how stable the model is across different settings.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)