Organizations: Mobility and Industrial Management (MIM), Faculty of Engineering Science, KU Leuven, Leuven, 3001, Belgium · Department of Built Environment, Aalto University, Espoo, 02150, Finland · Department of Mathematics and Systems Analysis, Aalto University, Espoo, 02150, Finland · DTU Management, Technical University of Denmark, Lyngby, 2800, Denmark · The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, 510642, China
Lane-changing (LC) maneuvers are highly challenging vehicle interactions, with substantial implications for traffic safety and efficiency. Developing effective LC strategies for connected and automated vehicles in mixed traffic is complicated by the uncertainty associated with surrounding-vehicle behavior and the computational burden of embedding realistic prediction models within optimization schemes. To capture these complex interactions while retaining tractability, this study proposes a data-driven optimization framework that embeds neural network (NN) surrogates for safety and lateral comfort within a Bézier-curve-based trajectory planner. Specifically, the safety cost during the LC process is quantified via a surrogate composed of an input convex recurrent neural network (ICRNN) encoding the planned trajectory, a fully input convex neural network (FICNN) embedding the LC duration, and a partially input convex neural network (PICNN) decoder fusing them with historical traffic states; separately, another FICNN surrogate approximates the optimal lateral comfort cost as a function of the LC duration. Both surrogates are embedded via linear programming (LP) formulations, avoiding the binary variables required by conventional mixed-integer programming (MIP) approaches for NN-embedded optimization. We further theoretically establish their exactness at optimality and extend the framework to problems with additional analytical objective terms and multiple surrogates. Experiments on a hybrid dataset combining real-world and simulated trajectories demonstrate that the proposed method achieves surrogate accuracy competitive with unconstrained baselines while substantially improving optimization efficiency over conventional MIPs. Closed-loop simulations further show that the proposed planners markedly reduce the weighted planning cost and consistently outperform baseline methods.
Figures & tables
Figure 1: Schematic diagram of LC trajectory planning in mixed traffic environment, reproduced from Yang et al., (2025) .
Figure 2: Architecture of a generic L -layer fully input convex neural network (FICNN).
Figure 3: Architecture of a T -step input convex recurrent neural network (ICRNN).
Figure 4: Architecture of an L -layer partially input convex neural network (PICNN Type I).
Figure 5: Architecture of an L -layer partially input convex neural network (PICNN Type II).
Figure 6: Architecture of the safety surrogate model, comprising a non-convex historical encoder, a convex trajectory encoder, and a skip-connection-free PICNN decoder.
Figure 7: The geographical map of the site for SWIFTraj data collection, reproduced from Han et al., 2026b () .
Figure 8: Distribution of mean speeds of ego vehicles during the LC process, comparing real trajectory data and simulation data.
Figure 9: Distribution of TET-based safety metrics comparing real trajectory data and simulation data
Figure 10: Distribution of TIT-based safety metrics comparing real trajectory data and simulation data
Model
LC Trajectories Encoder
LC Duration Encoder
Decoder
Input Reduced Baseline
–
–
MLP
Baseline
LSTM
MLP
MLP
Proposed Model I
ICRNN
FICNN
PICNN Type I
Proposed Model II
ICRNN
FICNN
PICNN Type II
Table 1: Architectural comparison of the evaluated surrogate models.
Model
MAE
RMSE
R2
Input Reduced Baseline
0.6055±0.0067
1.3785±0.0061
0.8072±0.0017
Baseline
0.1525±0.0236
0.4580±0.0342
0.9786±0.0031
Proposed Model I
0.2865±0.0072
0.7026±0.0070
0.9499±0.0010
Proposed Model II
0.3225±0.0055
0.7626±0.0073
0.9410±0.0011
Table 2: Prediction performance of surrogate models for TET-based safety metric evaluated on the test dataset. Values are presented as mean ± standard deviation.
Model
MAE
RMSE
R2
Input Reduced Baseline
5.7773±0.1306
14.3723±0.0649
0.7650±0.0021
Baseline
2.1371±0.3413
7.2389±1.1846
0.9388±0.0196
Proposed Model I
2.9469±0.0896
7.5187±0.4438
0.9355±0.0081
Proposed Model II
3.5098±0.0882
9.6971±0.2507
0.8930±0.0055
Table 3: Prediction performance of surrogate models for TIT-based safety metric evaluated on the test dataset. Values are presented as mean ± standard deviation.
Model
MAE
RMSE
R2
MLP
0.0306±0.0088
0.0772±0.0188
0.9969±0.0016
FICNN
0.0150±0.0011
0.0408±0.0012
0.9992±0.0001
Table 4: Prediction performance of FICNN and MLP models for the optimal lateral comfort cost as a function of LC duration.
Figure 11: Predicted optimal lateral comfort cost of the FICNN surrogate against the sampled ground truth as a function of the LC duration.
Model
Reformulation Type
Binary Variables
Continuous Variables
Constraints
Solution Time (s)
Proposed Model I
LP
–
255
255
0.0560±0.0156
Proposed Model I
MIP
200
255
655
2.2146±1.0703
Proposed Model II
LP
–
255
255
0.0512±0.0107
Proposed Model II
MIP
200
255
655
1.5420±0.8623
Table 5: Computational performance and structural statistics of the two safety-surrogate architectures, Proposed Models I and II, under LP and MIP reformulations. Values are presented as mean ± standard deviation.
Planner
Weighted Cost (TET)
Weighted Cost (TIT)
Infeasible Scenarios
Collision Scenarios
Baseline 1
103.475
86.514
23
13
Baseline 2
61.957
44.525
2
7
I-TET
51.141
44.046
-
-
I-TIT
60.461
51.078
-
-
II-TET
42.950
35.800
-
-
II-TIT
44.593
35.246
-
-
Table 6: Comparison of six planners across 100 test scenarios. Mean weighted objective values are computed over the 61 scenarios in which all planners completed LC maneuvers without collisions. The numbers of infeasible and collision scenarios are reported over 100 test scenarios. Both weighted-cost columns are evaluated for every planner, independent of the safety metric used during optimization.
Planner
Csafety (TET)
Csafety (TIT)
Clon
Clat
Ceff
Baseline 1
1.020
6.813
4.569
0.650
0.264
Baseline 2
0.961
6.120
1.551
0.369
− 5.276
I-TET
0.493
3.513
1.648
0.599
4.019
I-TIT
0.560
3.725
1.645
0.917
6.833
II-TET
0.513
3.697
1.206
0.608
− 0.829
II-TIT
0.602
4.147
1.220
0.864
− 6.330
Table 7: Mean values of performance metrics for baseline methods and all proposed planners on common successful scenarios.
Figure 12: Longitudinal speed profiles generated by the proposed planners on representative LC scenarios.
Planner
Weighted Cost (TET)
Weighted Cost (TIT)
I-TET
20
20
I-TIT
10
14
II-TET
44
50
II-TIT
26
16
Table 8: Number of test scenarios in which each planner achieves the lowest weighted objective among all four planners.
Model
Label
MAE
RMSE
Proposed Model I
TET
0.6570
1.6297
Proposed Model II
TET
0.6383
1.1963
Proposed Model I
TIT
7.2254
17.1961
Proposed Model II
TIT
6.5026
14.5227
Table 9: Safety-cost prediction errors on each planner’s own planned trajectories over all test scenarios.
Department of Mechanical Engineering National University of Singapore Singapore 117575, Singapore · Department of Electrical Engineering and Computer Science University of Michigan Ann Arbor, MI 48109, USA
Department of Mathematics and Systems Analysis, School of Science, Aalto University, Espoo, Finland · Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden · DTU Management, Technical University of Denmark, Lyngby, Denmark