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WardropNet: Traffic Flow Predictions via Equilibrium-Augmented Learning

Kai Jungel, Dario Paccagnan, Axel Parmentier, Maximilian Schiffer

TL;DR

This work proposes 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, and shows how to leverage a Bregman divergence fitting the geometry of the equilibria.

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 architecture 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.

WardropNet: Traffic Flow Predictions via Equilibrium-Augmented Learning

TL;DR

This work proposes 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, and shows how to leverage a Bregman divergence fitting the geometry of the equilibria.

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 architecture 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.

Paper Structure

This paper contains 50 sections, 3 theorems, 62 equations, 15 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related work
  4. Generalized Wardrop Equilibria
  5. Wardrop Equilibrium
  6. Convex characterization
  7. WardropNet---Learning data-driven Wardrop equilibria
  8. Hypothesis class
  9. Fenchel-Young loss
  10. Supervised learning problem
  11. Interpretation of the Fenchel-Young loss as Bregman divergence
  12. Pipeline architecture
  13. Statistical model
  14. Latencies with Euclidean regularization
  15. Regularization by perturbation
  16. ...and 35 more sections

Key Result

Theorem 1

Consider a directed graph $D = (V, A)$, origin-destination pairs $(o_j,d_j)_{j\in J}$, and a vector of latency functions $\boldsymbol{\ell} = \{\ell_a\}_{a\in A}$ that derive from a potential $\Phi$.

Figures (15)

  • Figure 1: Schematic illustration of the WardropNet paradigm, implemented as a pipeline that comprises a statistical model $\varphi_{{\boldsymbol{w}}}$ that predicts the latency function's parameterization ${\boldsymbol{\theta}}$ and an equilibrium problem yielding the respective prediction $\hat{\bar{\boldsymbol{y}}}_\Omega({\boldsymbol{\theta}})$.
  • Figure 2: Illustration of the considered traffic scenarios.
  • Figure 3: Benchmark performances on various stylized and realistic traffic scenarios.
  • Figure 4: Visualization of time-invariant traffic flows.
  • Figure 5: Comparison of target and predicted traffic flows per arc for the Berlin scenario.
  • ...and 10 more figures

Theorems & Definitions (8)

  • Definition 1: Wardrop Equilibrium
  • Theorem 1: Convex characterization
  • Definition 2: Wardrop Equilibrium
  • Proposition 1: Wardrop Equilibrium (Variational)
  • proof
  • Theorem 1: Convex characterization
  • proof
  • Example 1