FLAT: Smoothing the Rugged Landscape for Learnable, Sample-Efficient Traffic Calibration

Haopeng Deng; Shuo He; Dayuan Wang

physics.soc-phOct 3, 2026

FLAT: Smoothing the Rugged Landscape for Learnable, Sample-Efficient Traffic Calibration

Authors: Haopeng Deng, Shuo He, Dayuan Wang

Organizations: School of Future Transportation Guangzhou Maritime University Guangzhou 510725, China · Department of Civil and Environmental Engineering Imperial College London London SW7 2AZ, United Kingdom · Department of Biostatistics University of Florida Gainesville, FL 32601, USA

Abstract

Calibrating microscopic traffic models for digital twins is an expensive black-box optimization problem: tuning car-following and lane-changing parameters requires a full simulation run, affording only a tight budget per recalibration window. Matching raw trajectories yields a rugged objective that sparse surrogates cannot learn, reducing sequential acquisition to near-random probing. We present FLAT, which couples what to optimize with where to sample next. An eight-dimensional behavioral fingerprint smooths the parameter-error landscape, making the objective learnable from a few dozen samples; annealed lower-confidence-bound (LCB) acquisition then spends each remaining run where it most reduces error. The surrogate, interchangeable among a Gaussian process (GP), random forest (RF), or multi-layer-perceptron (MLP) ensemble, plugs into the same LCB loop. Across six heterogeneous real-world scenes, FLAT-GP achieves the lowest scene-averaged behavioral error, winning 6/6 scenes against SPSA, GA, and CMA-ES and 5/6 against TPE under the matched budget. Some baselines need up to 4.4 times more simulations to match. Ablations show objective choice shifts final behavioral error by 81% on average, removing sequential LCB raises the six-scene mean by 20%, and surrogate choice shifts it by at most 4.2%, confirming gains trace to objective geometry and sequential allocation rather than surrogate capacity.

Figures & tables

Fig. 1: FLAT overview. Black-box calibration of Krauss car-following and LC2013 lane-changing parameters in SUMO–TraCI on heterogeneous scenes. Behavioral fingerprints encode trajectories into a smoother parameter–error landscape, making $J_{b}(\theta)$ learnable for sample-efficient acquisition. Phase A explores $\theta$ by Latin-hypercube sampling (LHS); Phase B refits a GP, RF, or MLP surrogate and selects the next evaluation by annealed LCB.

Fig. 2: Objective landscapes over maximum acceleration

a_{\max}

and desired time headway

\tau

on the XAM expressway, from 400 SUMO runs; surfaces normalized to

[0,1]

. Left:

J_{s}

( 2 ). Right:

J_{b}

( 5 ).

Fig. 3: Sequential acquisition in the

(a_{\max},\tau)

plane (400 SUMO evaluations, XAM expressway);

\tilde{J}(\theta)

normalized to

[0,1]

for shape comparison. Grey: Phase-A LHS. Numbered: Phase-B LCB trials. Star: calibrated

\theta^{\star}

Panel of Figure 4 — Fig. 4: Six heterogeneous evaluation scenes: Left: SIND signalized intersections. Right: UTE urban-expressway segments.

Method	Mean $J_{b}\downarrow$	vs. GP	vs. RF	vs. MLP	Wins/6	$p$
FLAT-GP	0.231 $\pm$ 0.003	—	$-4.0$	$-3.9$	—	—
FLAT-RF	0.241 $\pm$ 0.009	$+4.2$	—	$-0.1$	—	—
FLAT-MLP	0.240 $\pm$ 0.006	$+3.9$	$-0.1$	—	—	—
TPE	0.257 $\pm$ 0.010	$+11.3$	$+6.7$	$+6.9$	5/6	0.0625
GA	0.276 $\pm$ 0.013	$+19.5$	$+14.7$	$+14.8$	6/6	0.0313
CMA-ES	0.280 $\pm$ 0.023	$+21.2$	$+16.3$	$+16.4$	6/6	0.0313

TABLE I: Six-scene mean

J_{b}

at budget

100

(mean

\pm

seed std over five seeds).

\Delta\%

: relative to each FLAT variant’s six-scene mean. Wins/6: FLAT-GP scene wins;

p

: scene-level test.

Fig. 5: Running best-so-far

J_{b}

for seven optimizers, averaged over six scenes and five seeds. Shaded bands:

\pm 1

std across seeds. Dotted line: budget cap

B{=}100

. Insets: zoomed 0–40 and 40–80 simulations.

Fig. 6: Observed vs. FLAT-GP simulated speed distributions (five seeds); simulated mean kernel density estimation (KDE)

\pm 1

std envelope. Per-scene

\rho{=}1{-}D_{\mathrm{KS}}

(mean

\,\pm

seed std); stars/ticks:

v_{\mathrm{mean}}

v_{15}

v_{85}

	vs. GP		vs. RF		vs. MLP
Baseline	Sims $\downarrow$	Ratio	Sims $\downarrow$	Ratio	Sims $\downarrow$	Ratio	Reach $\uparrow$
TPE	221 $\pm$ 162	2.21 $\times$	167 $\pm$ 139	1.67 $\times$	147 $\pm$ 140	1.47 $\times$	28/30
CMA-ES	402 $\pm$ 108	4.02 $\times$	288 $\pm$ 148	2.88 $\times$	276 $\pm$ 108	2.76 $\times$	25/30
GA	414 $\pm$ 142	4.14 $\times$	360 $\pm$ 165	3.60 $\times$	305 $\pm$ 170	3.05 $\times$	15/30
SPSA	442 $\pm$ 131	4.42 $\times$	409 $\pm$ 159	4.09 $\times$	424 $\pm$ 154	4.24 $\times$	9/30

TABLE II: Simulations needed to match FLAT@100, mean

\pm

std over

30

scene

\times

seed pairs, budget capped at

500

. Reach: pairs reaching the quality threshold within cap, vs. FLAT-RF.

Scene	FLAT-RF ( $J_{b}$ )	FLAT-RF ( $J_{s}$ )	TPE ( $J_{b}$ )	TPE ( $J_{s}$ )
Tianjin	0.146 $\pm$ 0.011	0.345 $\pm$ 0.066	0.165 $\pm$ 0.010	0.327 $\pm$ 0.116
Changchun	0.168 $\pm$ 0.016	0.279 $\pm$ 0.041	0.167 $\pm$ 0.013	0.277 $\pm$ 0.068
Xi’an	0.430 $\pm$ 0.005	0.488 $\pm$ 0.014	0.426 $\pm$ 0.013	0.471 $\pm$ 0.017
YTDJ	0.223 $\pm$ 0.014	0.467 $\pm$ 0.129	0.284 $\pm$ 0.067	0.435 $\pm$ 0.102
RML	0.434 $\pm$ 0.022	0.514 $\pm$ 0.016	0.452 $\pm$ 0.008	0.620 $\pm$ 0.024
XAM	0.044 $\pm$ 0.008	0.105 $\pm$ 0.097	0.047 $\pm$ 0.008	0.234 $\pm$ 0.107

TABLE III: Objective ablation: final

J_{b}

when training on

J_{b}

vs.

J_{s}

(always scored by

J_{b}

). Five-seed mean

\pm

std; 6/6 scenes, scene-level exact Wilcoxon

p{=}0.031

per optimizer.

Panel of Figure 10 — Fig. 7: $N_{\mathrm{init}}$ ablation: final $J_{b}$ vs. warm-up size at fixed budget $100$ , five-seed mean $\pm 1$ std. $N_{\mathrm{init}}{=}100$ reduces to pure LHS.

Panel of Figure 11 — Fig. 8: Per-scene best-so-far $J_{b}$ convergence for three surrogates, five-seed mean $\pm 1$ std. $R^{2}$ annotations report surrogate fit at the final training set.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 12 — Fig. 9: Objective landscapes in the $(a_{\max},\tau)$ plane on XAM and Xi’an. Top row: $J_{s}$ and $J_{b}$ . Bottom row: $|\nabla J_{s}|$ and $|\nabla J_{b}|$ .

Panel of Figure 13 — Fig. 10: Local Lipschitz estimate $\hat{L}$ ( 14 ) vs. neighborhood size $k$ on a 10-D $400$ -point LHS design; dotted line marks the reported $k{=}5$ .

Panel of Figure 14 — Fig. 11: Held-out $R^{2}_{\mathrm{ext}}$ vs. training size $N$ for FLAT-GP fit on $J_{s}$ vs. $J_{b}$ ; dotted line marks $N{=}100$ , the calibration budget.

	XAM		Xi’an
	$J_{b}$	$J_{s}$	$J_{b}$	$J_{s}$
NLL $\downarrow$	-1.563 $\pm$ 0.149	0.259 $\pm$ 0.087	-1.258 $\pm$ 0.173	0.093 $\pm$ 0.423
CRPS $\downarrow$	0.026 $\pm$ 0.002	0.169 $\pm$ 0.015	0.035 $\pm$ 0.003	0.106 $\pm$ 0.013

Appendix

TABLE IV: GP probabilistic fit on held-out landscapes (

N{=}100

); five-subsample mean

\pm

std.

TraCI key	Physical meaning	Bounds	Unit
Car-following (Krauss)
accel	Maximum acceleration $a_{\max}$	$[1.0,\,3.5]$	m/s 2
decel	Comfortable deceleration	$[2.0,\,6.0]$	m/s 2
sigma	Driver imperfection	$[0.1,\,0.9]$	—
tau	Desired time headway $\tau$	$[0.5,\,2.5]$	s
minGap	Minimum standing gap	$[1.0,\,4.0]$	m

Appendix

TABLE V: TraCI keys, physical meanings, bounds, and units for six car-following and four lane-changing parameters.

Method	Protocol and budget
FLAT	40-point LHS + 60 sequential LCB evaluations
SPSA	Random start; $a{=}0.22$ , $c{=}0.15$ , $A{=}20$ , $\alpha{=}0.602$ , $\gamma{=}0.101$ ; evaluations capped at 100
GA	Random population of 10; tournament size 3; uniform crossover; mutation std $0.05$ of span; population evaluations count toward cap
CMA-ES	Random start; normalized $[0,1]$ search; $\sigma_{0}{=}0.3$ ; implementation-default adaptation; evaluations capped at 100
TPE	Seeded Optuna TPESampler with implementation-default settings; 100 trials

Appendix

TABLE VI: Optimizer protocols and evaluation-budget accounting for the main comparison.

Model Calibration

Model calibration aligns predicted confidence or model outputs with observed outcomes and measurements to support more reliable decisions.

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Traffic digital twins, which inform policymakers of effective interventions based on large-scale, high-fidelity computational models calibrated to real-world traffic, hold promise for addressing societal challenges in our rapidly urbanizing world. However, conventional fine-grained traffic simulations are non-differentiable and typically rely on inefficient gradient-free optimization, making calibration for real-world applications computationally infeasible. Here we present a differentiable agent-based traffic simulator that enables ultra-fast model calibration, traffic nowcasting, and control on large-scale networks. We develop several differentiable computing techniques for simulating individual vehicle movements, including stochastic decision-making and inter-agent interactions, while ensuring that entire simulation trajectories remain end-to-end differentiable for efficient gradient-based optimization. On the large-scale Chicago road network, with over 10,000 calibration parameters, our model simulates more than one million vehicles at 173 times real-time speed. This ultra-fast simulation, together with efficient gradient-based optimization, enables us to complete model calibration using the previous 30 minutes of traffic data in 455 s, provide a one-hour-ahead traffic nowcast in 21 s, and solve the resulting traffic control problem in 728 s. This yields a full calibration--nowcast--control loop in under 20 minutes, leaving about 40 minutes of lead time for implementing interventions. Our work thus provides a practical computational basis for realizing traffic digital twins.

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