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arxiv: 2604.11380 · v3 · submitted 2026-04-13 · 📡 eess.SY · cs.SY

End-to-end differentiable network traffic simulation with dynamic route choice

Toru Seo This is my paper

Pith reviewed 2026-05-10 15:55 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords differentiable simulationtraffic flow modelautomatic differentiationdynamic user optimumcongestion tolllink transmission modelgradient optimization
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The pith

An end-to-end differentiable traffic simulator using automatic differentiation enables efficient optimization of dynamic congestion tolls on large urban networks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a network traffic simulator that is fully differentiable, allowing automatic computation of gradients with respect to parameters like tolls. It achieves this by combining the Link Transmission Model, whose min and max operations have subgradients, with a Dynamic User Optimum route choice model that produces continuous diverge ratios. Sympathetic readers would care because this removes the barrier of deriving complex gradients by hand or using slow numerical methods, opening the door to optimizing real-world traffic systems with thousands of variables.

Core claim

The paper introduces an end-to-end differentiable simulator based on the Link Transmission Model and Dynamic User Optimum route choice, where piecewise-linear operations admit subgradients and diverge ratios are continuous, thus supporting automatic differentiation for solving large optimization problems such as dynamic toll setting on the Chicago-Sketch network with 2500 links and 15000 variables in 40 minutes.

What carries the argument

Automatic differentiation applied to the Link Transmission Model's piecewise-linear min/max operations and the continuous diverge ratios derived from the Dynamic User Optimum model.

Load-bearing premise

Subgradients of the piecewise-linear min/max operations and the continuous diverge ratios from the DUO model provide sufficiently accurate gradients for stable convergence in gradient-based optimization.

What would settle it

A failure of the proposed simulator to produce a high-quality toll optimization solution on the Chicago-Sketch dataset within 3000 iterations would falsify the claim of its practical effectiveness.

Figures

Figures reproduced from arXiv: 2604.11380 by Toru Seo.

Figure 1
Figure 1. Figure 1: The simulation framework combining LTM traffic flow model and DUO route choice. 3.1 Parameters and state variables, and what the differentiable simulator means A road network is represented as a directed graph G = (N ,L), where N denotes a set of nodes and L denotes a set of directed links. Each link l ∈ L connects an upstream node to a downstream node. For each node ν ∈ N , we define L in ν and L out ν as… view at source ↗
Figure 2
Figure 2. Figure 2: Triangular FD. The LTM uses the cumulative vehicle counts (i.e., the number of vehicles that passed the location from a certain reference time to the current time) at the upstream and downstream ends of each link as its state variables. They are denoted as Nl,U (t) for upstream and Nl,D(t) for downstream, respectively, of link l at time t. The LTM is derived from Newell’s simplified kinematic wave (KW) the… view at source ↗
Figure 3
Figure 3. Figure 3: Mechanism of LTM. Top: time–space diagram on link l. Bottom: its cumulative curve plot. Link l determines its demand Dl and supply Sl considering its traffic state. Then, the node model determines inflow f in l and outflow f out l considering demand and supply of connected links. where tp denotes the exit time from the p-th link. The total travel time is tP − t0. The path P itself can be determined by a ti… view at source ↗
Figure 4
Figure 4. Figure 4: Vehicle trajectories and cumulative counts (adapted from Seo (2023)). (e.g., Dl(t) is denoted as Dl) [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Node with multiple incoming and outgoing links. Let Bν = [blo] denote the turning fraction matrix, where blo is the fraction of outflow from inlink l directed to outlink o, and let αl denote the merge priority of inlink l. The value of Bν is obtained from the route choice model. The INM initializes inflow allocations qˆl = 0 for all l and outflow allocations qˆo = 0 for all o, then iterates the following s… view at source ↗
Figure 6
Figure 6. Figure 6: Conceptual illustration of logit-DUO. In summary, the DUO and logit-DUO models preserve end-to-end differentiability: the gradient flows from the objective through the per-destination cumulative counts, through the diverge ratio computation, through the node models, and back to the FD and demand parameters. Furthermore, the logit-DUO typically has a non-zero gradient with respect to link or path cost. The … view at source ↗
Figure 7
Figure 7. Figure 7: Merge network (a) t = 300 s (b) t = 800 s [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Network speed (normalized by free-flow speed). The width represents the density at the segment. First, we define the total travel time (TTT) as an objective function: TTT = X l∈L T XS−1 t=0 max{Nl,U (t) − Nl,D(t), 0} · ∆t + X ν∈Norig T XS−1 t=0 rν(t) · ∆t, (29) where Norig denotes the set of origin nodes. The first term accounts for vehicles on links, and the second term accounts for vehicles in vertical q… view at source ↗
Figure 9
Figure 9. Figure 9: Time-space diagrams of density (normalized by jam density). Note that these values are obtained by numerically differentiating N for visualization purposes, and thus some numerical noises exist, especially near the link borders. With respect to the FD parameters, the following values are obtained: ∂TTT ∂u1 = −1278.282, ∂TTT ∂u2 = −616.873, ∂TTT ∂u3 = −2024.685. (31) All values are negative, meaning that in… view at source ↗
Figure 10
Figure 10. Figure 10: Two-routes network (a) Fast route. The bottleneck is at 1000 m location. (b) Slow route [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Time–space diagrams of normalized density in the two-route network. We compute partial derivatives with respect to the bottleneck capacity q ∗ BN. The results are summarized in [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Simulated average link delay in the Chicago-Sketch data scenario without pricing. “Delay” is the ratio of the excess of the average link travel time over the free-flow travel time. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: shows the average network state in the best pricing case. By comparing with the no-toll case ( [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Convergence of the objective function, TTT, and gradient [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Average link toll in network. 4.2.4 Comparison with SPSA In order to quantify the advantage of the proposed AD-based gradient computation, we solved the same congestion pricing problem using SPSA (Spall, 1998), a conventional derivative-free optimization method in the simulation and DTA literature (Balakrishna et al., 2007; Lu et al., 2015). SPSA estimates the gradient by evaluating the objective function… view at source ↗
Figure 16
Figure 16. Figure 16: Time-series of toll and traffic states. where the step size ak = a/(A + k) α and perturbation magnitude ck = c/kγ follow the standard decay schedule recommended by Spall (1998) with A = 100, α = 0.602, γ = 0.101, and the initial parameters c = 30 and a = 0.0001 are calibrated to achieve the best performance as much as possible. The objective function for SPSA is the same as that for AD, Eq. (36). In order… view at source ↗
Figure 17
Figure 17. Figure 17: Macroscopic Fundamental Diagram. set to 17 000, which took 8532 sec [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Comparison between AD and SPSA. software on GitHub (https://github.com/toruseo/UNsim) and PyPI (pip install unsim). All code that reproduces the presented results is also published in the same GitHub repository. For future research, the following directions are worth considering. First, there is room for improvement in the model and algorithms to facilitate numerical applications. During this study’s nume… view at source ↗
read the original abstract

Optimization using network traffic models requires computing gradients of objective functions with respect to model parameters. However, derivation of such gradients has often been considered difficult or impractical due to their complexity and size. Conventional approaches rely on numerical differentiation or derivative-free methods that do not scale well with the parameter dimension, or on adjoint methods that require manual derivation for each specific model. This study proposes a novel end-to-end differentiable network traffic flow simulator based on automatic differentiation (AD), employing the Link Transmission Model (LTM) and a Dynamic User Optimum (DUO) route choice model. The LTM operates on continuous aggregate state variables through piecewise-linear min/max operations, which admit subgradients almost everywhere and thus require no smooth relaxation for AD. The DUO is also suitable for AD: although the shortest path search is itself discrete, the resulting diverge ratios at each node are continuous functions of per-destination vehicle counts and are thus differentiable. In order to demonstrate the capability of the proposed model, we solved a dynamic congestion toll optimization problem on the Chicago-Sketch dataset with approximately 2500 links, 1 million vehicles, a 3-hour duration, and 15000 decision variables. The proposed model successfully derived a high-quality solution in 3000 iterations, taking about 40 minutes. The simulator, implemented in Python and JAX, is released as open-source software named UNsim (https://github.com/toruseo/UNsim).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes an end-to-end differentiable traffic simulator combining the Link Transmission Model (LTM) with piecewise-linear min/max operations and a Dynamic User Optimum (DUO) route-choice model whose diverge ratios are continuous in vehicle counts. Implemented in JAX, the simulator is used to solve a dynamic congestion-toll optimization problem on the Chicago-Sketch network (~2500 links, 1 million vehicles, 3-hour horizon, 15 000 decision variables), reporting a high-quality solution after 3000 iterations in roughly 40 minutes. The code is released as open-source UNsim.

Significance. If the automatically differentiated subgradients prove accurate and stable, the work would enable gradient-based optimization of traffic models at scales that previously required derivative-free or manually derived adjoint methods, representing a practical advance for large-scale transportation network design. The open-source release strengthens reproducibility and potential follow-on use.

major comments (2)
  1. [Abstract] Abstract: the central claim that LTM min/max operations 'admit subgradients almost everywhere and thus require no smooth relaxation' and that DUO diverge ratios 'are continuous functions of per-destination vehicle counts and are thus differentiable' is load-bearing for the reported 3000-iteration, 15 000-variable convergence. No verification is supplied that JAX's automatic subgradient selection at the non-differentiable loci matches finite-difference gradients or avoids zero-gradient directions and instability when back-propagated through the full network and route-choice layers.
  2. [Results (Chicago-Sketch toll optimization)] Chicago-Sketch experiment: the statement that a 'high-quality solution' was obtained provides no diagnostics (gradient-norm histories, comparison against a non-differentiable baseline, or sensitivity to subgradient choice) that would confirm the AD gradients, rather than algorithmic heuristics, drove reliable convergence on the 15 000-variable instance.
minor comments (2)
  1. [Abstract] The abstract and methods should explicitly state the precise form of the toll-optimization objective and any regularization terms applied to the 15 000 decision variables.
  2. [Figures] Figure captions and axis labels in the results section would benefit from clearer indication of which curves correspond to the differentiable simulator versus any reference methods.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments correctly identify that the manuscript currently lacks explicit numerical verification of the automatic-differentiation subgradients and supporting convergence diagnostics. We address both points below and will incorporate the requested material in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that LTM min/max operations 'admit subgradients almost everywhere and thus require no smooth relaxation' and that DUO diverge ratios 'are continuous functions of per-destination vehicle counts and are thus differentiable' is load-bearing for the reported 3000-iteration, 15 000-variable convergence. No verification is supplied that JAX's automatic subgradient selection at the non-differentiable loci matches finite-difference gradients or avoids zero-gradient directions and instability when back-propagated through the full network and route-choice layers.

    Authors: We agree that direct numerical verification of the subgradients is necessary to support the central claims. In the revised manuscript we will add a dedicated verification subsection (likely in Section 3 or 4) that compares JAX-computed subgradients against central finite-difference approximations on small synthetic networks (both for isolated LTM min/max operations and for the full LTM+DUO pipeline). We will also report the frequency of non-differentiable points encountered during the Chicago-Sketch run and any safeguards (e.g., subgradient selection rules) used by JAX. These additions will be limited to a few pages and will not alter the core algorithmic contribution. revision: yes

  2. Referee: [Results (Chicago-Sketch toll optimization)] Chicago-Sketch experiment: the statement that a 'high-quality solution' was obtained provides no diagnostics (gradient-norm histories, comparison against a non-differentiable baseline, or sensitivity to subgradient choice) that would confirm the AD gradients, rather than algorithmic heuristics, drove reliable convergence on the 15 000-variable instance.

    Authors: We acknowledge that the current presentation of the Chicago-Sketch results is insufficient to isolate the contribution of the AD gradients. In revision we will augment the results section with (i) a plot of gradient-norm history over the 3000 iterations, (ii) a brief comparison of final objective values obtained with the differentiable simulator versus a derivative-free baseline (e.g., a simple random-search or Nelder-Mead run on a reduced problem), and (iii) a short sensitivity test repeating the optimization with different JAX subgradient modes or with added smoothing. These diagnostics will be presented concisely and will strengthen the claim that the reported convergence is attributable to the end-to-end differentiability. revision: yes

Circularity Check

0 steps flagged

Differentiability and optimization results follow from model properties and empirical demonstration without circular reduction

full rationale

The paper asserts that LTM piecewise-linear min/max operations admit subgradients almost everywhere (allowing direct AD) and that DUO diverge ratios are continuous functions of per-destination vehicle counts (hence differentiable). These are presented as intrinsic mathematical properties of the selected models rather than results derived from fitted parameters, self-referential definitions, or prior self-citations. The central empirical claim—successful convergence of gradient-based toll optimization on the external Chicago-Sketch network after 3000 iterations—is a reported outcome of running the simulator, not a quantity forced by construction or reduced to the inputs. No load-bearing step in the derivation chain (as described in the abstract) equates a prediction to its own fitted or defined inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on existing traffic models plus the mathematical fact that piecewise-linear min/max admit subgradients and that DUO diverge ratios are continuous in vehicle counts. No new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption LTM operates on continuous aggregate state variables through piecewise-linear min/max operations, which admit subgradients almost everywhere and thus require no smooth relaxation for AD.
    Stated directly in the abstract as the reason AD works without modification.
  • domain assumption The DUO shortest-path search yields diverge ratios at each node that are continuous functions of per-destination vehicle counts and are thus differentiable.
    Stated in the abstract as the property that makes the route-choice component compatible with AD.

pith-pipeline@v0.9.0 · 5552 in / 1414 out tokens · 38723 ms · 2026-05-10T15:55:37.329530+00:00 · methodology

discussion (0)

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