A Driving-Style-Adaptive Framework  for Vehicle Trajectory Prediction
Authors: Di Wen
Yu Wang
Zhigang Wu
Zhaocheng He
Zhe Wu
Zheng Qingfang
Venue: NeurIPS 2025 poster ()
Keywords: Trajectory prediction
driver style category
Kolmogorov-Arnold Networks
model explanation
basis function
Weierstrass Approximation Theorem
Primary area: deep_learning

Abstract: Vehicle trajectory prediction serves as a critical enabler for autonomous navigation and intelligent transportation systems. While existing approaches predominantly focus on temporal pattern extraction and vehicle-environment interaction modeling, they exhibit a fundamental limitation in addressing trajectory heterogeneity originating from human driving styles. This oversight constrains prediction reliability in complex real-world scenarios.
To bridge this gap, we propose the Driving-Style-Adaptive (\underline{\textbf{DSA}}) framework, which establishes the first systematic integration of heterogeneous driving behaviors into trajectory prediction models. Specifically, our framework employs a set of basis functions tailored to each driving style to approximate the trajectory patterns. By dynamically combining and adaptively adjusting the degree of these basis functions, DSA not only enhances prediction accuracy but also provides \textbf{explanations} insights into the prediction process. Extensive experiments on public real-world datasets demonstrate that the DSA framework outperforms state-of-the-art methods.

A Driving-Style-Aaatiive Framework for Vehicle Trajectory Prediction
Di Wen 12,3 Yu Wang" Zhigang Wul
Zhaocheng Hel23, Zhe Wu Zheng Qingfang
Sun Yat-sen University
PPeggceeng Laboratory
Guangdong Provincial Key Laboratory of Intelligent Transportation System {wend25, wuzhig6 Amagi22mmm11...mmmm sysu. edu. cn hezhch@mail sysu. edu, cn {wangy12, wuzh02, phenggfo11@pccl..ccc
Abstract
Vehicle trajectory prediction serves critical enabler for autonomous navigation and intelligent transportation systems. While existing approaches predominantly focus pattern extraction and vehicle-enviromeen interaction modeling, they exhibit a fundamental limitation in addressing trajectory heterogeneity originat- ing from human driving styles. This oversight constrains prediction reliability in complex real-world scenarios. To bridge this gap, we propose the Driving-Si Style- Adaptive (DSA) framework, which establishes the first systematic integration of heterogeneous driving behaviors into trajectory prediction models. Specifically, our framework employs set of basis functions tailored each driving style approx- imate the trajectory patterns. By dynamically combining and adaptively adjusting the degree of these basis functions, DSA not only enhances prediction accuracy but also provides explanations insights into the prediction process. Extensive experiments on public real-world datasets demonstrate that the DSA framework outperforms state-of-the-ar methods.
Introduction
Vehicle Trajectory Prediction (VTP) serves as fundamental capability for numerous intelli- gent transportation applications, including au- tonomous driving systems [1, 2, 31, motion plan- ning algorithms [4, 5] and adaptive traffic con- trol frameworks [6, 7). Recent advances in VTP achieve notable progress through two pri- mary paradigms: (1) capturing temporal patterns from historical trajectories and modeling vehi- cle interactions [8, 10, 11], and (2) leveraging structured scene representations that incorpo- rate road topology and regulatory constraints [12, 13, 14, 15]. However, these methods often overlook the originator of the trajectory: human drivers [16, 17]. whose diverse behavior leads heterogeneous trajectory patterns.
Normal
Conservative
Aggressive
Figure 1: Illustration of three driving styles: Con- servative drivers typically move slowly or stop to avoid obstacles; Aggressive drivers often travel at high speeds and are prone to overtaking other ve- hicles; Normal drivers maintain constant speed and frequently change lanes to ensure safety.
In this paper, we propose an adaptive VTP framework based on distinct driving styles [18, 19]: Conservative, Aggressive and Normal (CAN). Each driving style manifests in characteristic trajectory
Joint Corresponding Authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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patterns, as illustrated in Figure 1. limited variability (conservative), non-smooth trajectories (aggres- sive) and frequent yet smooth motion changes (normal). For each styles, we employs variable basis functions within Kolmogorov-Arnoke Networks (KANs) [20] to capture these trajectory patterns. In complex real-world scenarios, driver behavior often reflects a probabilistic mixture of weighted driving styles [21].
Our framework comprises two core components: (1) the matching between driving styles and their corresponding basis functions, and (2) the weighted combination and adjustment of the degrees of these functions. Additionally, inspired by the Weierstrass Approximation Theorem [22], our proposed DSA framework extends KANs from a theoretical perspective. Each matching in (1) is further grounded in the mechanical properties of basis functions, thereby providing explanations for our DSA framework.
Our main contributions in this paper are summarized follows:
To address the vehicle trajectory prediction task, we propose for the first time, a novel Driving- Style-Adaptive (DSA) framework tailored to the driving styles of human drivers and effectively leverages trajectory information.
We utilize polynomial approximation operators to approximate and predict trajectorics under different driving styles: Conservative, Aggressive and Normal (CAN). These operators support a mathematical explanation matching mechanism that matches each driving style with a corresponding polynomial form.
The experimental results real-world datasets (nuScenes, Argoverse and Waymo) demonstrate that our model significantly outperforms existing methods in vehicle trajectory prediction.
Preliminary & Related Work
2.1 Task Definition: Vehicle Trajectory Prediction (VTP)
VTP aims to predict the future trajectory of vehicles based on history trajectory or other informations available in given scenario. In recent years, deep learning based VTP methods are categorized into two groups [23]: (i) knowledge-basee methods, which incorporate specific information such as maps [24, 25], vehicles [9, 26] and interactions [27, 28] to represent the environment vehicle behaviour. (ii) knowledge-free methods, which rely on deep learning's s ability to encode complex data features, modeling them using structures such tensors [29, 30] attention mechanisms [31, 32].
Following above works, we analyze traffic scenes involving N vehicles (agents). The trajectory of each vehicle i in historical interval [0, ' is denoted as X; { .... =²) J. Each state s is 5-dimensional vector representing the (z, y) position, velocity, acceleration, and the nearest lane segments ID. The superscript * denotes the time step. Similarly, the future trajectory in interval [0, T] given by Y, {o),....
2.2 Basic Network: Kolmogorov-Arnold Networks (KANs)
KANs [20] are inspired by the mathematical principles [33, 34, 35] of the Kolmogorov- representation theorem [36, 37, 38), stated as follows:
Theorem 2.1 (Kolmogorov-Arnold representation Theorem) For any multivariate continuous func- tion f : [0, 1]"h IR, f can be represented as a finite composition of univariate continuous functions Φij : J0, + R and D R R, with the binary operation of addition such that:
2n+1
F = f(a 2nn
Øi (x:)
The key innovation of KANs lies implementing the residual activation functions ((x) in Equation (1)
((x) = wyb(x) + (x),,
(2)
where Mxx) silu(x)) and (xx sp)ine Unlike Multi-Layer Perceptrons (MIPs [39, 40]), which utilize fixed activation functions associated with nodes ("neurons"), KANs feature Iearnable

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((x) on edges ("weights"). However, due to the inherent complexity of these functions, the speed and scalability of the original KANs are not satisfactory [41]. Consequently, variety of KAN-based applications are emerged in A14Science tasks [42, 43, 44, 45, 46, 47]. To the best of our knowledge, we are the first to extension KANs to the VTP task. We achieve this by expanding the set of basis functions (x:) to match different driving styles and by grounding this matching in both mathematical theory and task-specific behavior.
2.3 Core Theory: Applying Polynomials Basis Functions
As describe in Section 2.2, a fixed basis function has inherent limitations. In the VTP task, such function may fail to adequately approximate diverse trajectory or lane curves. This raises an important question: how can we legitimately expand the class of basis functions? ? In approximation theory, fundamental question is whether polynomials can approximate any given continuous function to an arbitrary degree of precision. Weierstrass [22] provides definitive answer by:
Theorem 2.2 (Weierstrass Approximation Theorem) Let fx) E L 1] for any 0. Then there exists an algebraic polynomial Pn (x) = Em=o Cmar" such that
lim [ 15 Pm
- (x)
This interval can be extended to [a, b]. This demonstrates that polynomials pa. Pn can serve as basis functions (xr) in Equation (2) to approximate the f in Theorem 2.1. In our task, the vehicle trajectories are treated as the function f. with respect to the time step t. For different driving styles, we employ corresponding pr. as basis functions to approximate these trajectories accordance with Theorem 2.2. Furthermore, these trajectories also belong to the LP space", and are thus well-defined.
3 Methodology
3.1 Motivation and Overview
Our Livvin-SttyleAdaptte (DSA) framework (illustrate in 2) models the behavior of the trajec- tory originator: the human driver. The driving style of various vehicle drivers categoried as: Conservative, Aggressive and Normal (CAN) [48, 49, 50], each reflecting distinct trajectory characteristics.
Driving Style Conservative
Confimm
Gating Network
Dony
Driving Style Aggressive
Driving Style Normal L,,
Expert weights
Polynomial combination
Degree
CAN Style Matching
Adaptation
History Trajectory
DSA Framework
Predict Trajectory
An overview of our DSA framework, which performs trajectory prediction based on driving style categories: conservative, aggressive and normal (CAN) to prediction. For clarity, illustrate this process using single vehicle example. the solid line represents trajectories length while the arrow ndicates the direction (history future). The symbol B,, Tç and L,, denote different basis functions Pn. corresponding to each driving style. Our proposed DSA framework dynamic adaps the experts (driving style) weighs w, and the degree n, of selected Pn.
Figure 2:
We match each driving style characteristic corresponding approximation polynomial pr, based on the mathematics properties of p,, as described in Section 3.2. To implement this mechanism, we introduce Pn combination and degree adjustment strategies in Section 3.3.
Taking the p-sosition example, it can be shown that the integral Jq |xc (t) dt oc holds

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3.2 Theoretical Foundations for Matching Polynomials to Driving Styles
In this section, we elucidate the matching between the polynomials Pn (Pn Pn) and driving style, focusing on the mathematical properties of Ps, and analyzing the characteristics of each driver's ' type. Specifically, we address conservative drivers in Section 3.2.1, aggressive drivers in Section 3.2.2 and normal drivers in Section 3.2.3.
3.2.1 Conservative Drivers
Conservative drivers [51] prioritize driving comfort and safety, which leads cautious decisions. Their average speed is typically the slowest and rarely change their behavior. Consequently. their trajectories characterized by smoothness and stability, with minimal abrupt changes in speed.
In this situation, we require a Pa to capture approximating drivers with minimal behavioral changes, that is, ensures the approximation error decreases uniformly across the entire interval. we employ the Bernstein operatore" B, [52] to achieve this:
Definition 3.1 (Bernstein polynomial, Ba) Consider function f(x) E C(O, 1], E [0, 1), B, is specified by the equation:
BB)
Σ
x n-k x (1
k<
k=0
is clear that Bn S Pn thus applies to Theorem 2.2. The primary advantage the Bn is articulated in the following proposition:
Proposition 3.2 For all functions C(O, 1}, the sequence (Balin= 1,2,3,...) converges uniform' to J as B, 5
This proposition demonstrates that the B, exhibits uniform convergence across the entire interval, making it particularly suitable for approximating trajectories with slow travel speeds and few be- havioral changes, such those of conservative drivers. This ensures that the approximation error decreases uniformly throughout the interval.
3.2.2 Aggressive Drivers
Aggressive drivers [53] prioritize their own benefits at the expense of safety and comfort, which leads to higher speeds, abrupt changes acceleration and braking, with frequent tendency to change lanes. As a result, their trajectories display more abrupt motions and are less smooth.
Trigonometric polynomials T, are dense in C(1) on the unit circle according the Stone- Weierstrass Theorem [54]. This implies that trigonometric polynomials T, are particularly ef- fective at approximating functions with discontinuities or sharp features, which define as: T2 (x)
an + Σn= (an, cos (nx) + by sin (næ)) We employ the Chebyshev polynomials [55] Th defined follows:
Definition 3.3 (Chebyshev Polynomials, Ty) For x E [-1, 1), the n-th T of the first kind given by T% (x) = cos In- arccos (z)).
The effectiveness of T is further highlighted by the Chebyshev Minimax Theorem [56]:
Theorem 3.4 (Chebyshev Minimax Theorem) For f E C[-1,, тę minimizes the maximum error in the uniform norm compared to any other approximation of the same degree. Formally, this relationship is expressed as:
|| - - < -ppllllr..
'In this task, we leverage pa approximation operators approximate and predict the vehicle trajectories, Specifically, we instantiate these operators using representative basis such as B,, the same as other two approximation operators.
Uniform Convergence For every there exists N Et, N(). s.t. for all N. there
If (x) - (x) |=
Dense A subset A of a topological space X is said to be dense in X if every point of X either belongs to A else is arbitrarily "close" to member of A.

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Theorem 3.4 explicitly states that T% can minimize the maximum error, effectively reducing the impact of sudden behavioral and speed changes typical of aggressive drivers. Furthermore, the overall prediction error is decreased.
3.2.3 Normal Drivers
Normal drivers [57] strike a balance between conservative and aggressive driving styles, representing relatively common group in driving behavior. Their speed and acceleration typically fall between those of conservative and aggressive drivers, exhibiting moderate speed changes and occasional rapid reactions. Consequently, their trajectories are neither smooth those of conservative drivers nor as abrupt as those of aggressive drivers, but their trajectories may exhibit regular fluctuations.
This driving characteristic is closely related to the application of orthogonal polynomials P% [58]. The Pf has significant flexibility and enables accurately capture the trajectories characterized by gradual changes and moderate fluctuations for normal drivers. The ph with weight function" p and (p%) = n are defined as: (pf. .p{ Ls o (z) pf (x) p (x) dx = òij. where dig equal to 0 iff i # j. The approximation of P. can be effectively described by [59]:
Theorem 3.5 (Least Squares Characterization Theorem) For any function I Cla. b), there exists an orthogonal polynomial ph with @ (p%) & n that minimizes the error in the I.g norm' between fx) and (x))
I/ - PR 11
min |/ 11f-plly -Pll133
PEPa
where Pa denotes the space of all polynomials of degree at most n..
The term "least" here does not denote the non-uniqueness, but rather indicates the possible to select optimal coefficients under best L2 approximating. For instance, Legendre polynomials L,, is typical orthogonal polynomial:
Definition 3.6 (Legendre Polynomial, L,) For x E [-1, with a constant weight function (x) = 1, L,, is defined by
Ln 2d! dr (a=2-1)
d"
This orthogonality under the 2 norm particularly without weight or constant weight, makes an exceptionally efficient tool for approximation [60], which represents specific instance covered by Theorem 3.5. Moreover, Ln also defined by a simple recurrence relation:
(n + 1) (x) (2n I. (x) - n. (x),
This recurrence relation facilitates quick calculations and the optimal square approximation property excels under the 1 norm. These characteristics make it well-suited for handling smooth and continuous trajectory fluctuations, align well with the The normal drivers. Their characterized by gradual changes and smoother transitions.
3.3 Algorithm Realization
3.3.1 Polynomial Combination
In Section 3.2, we utilize different polynomial forms to match different driving styles, thereby fully leveraging trajectory information for prediction. However, assuming single fixed driving style may be inadequate in complex real-world scenarios. Kernel density estimation and latent variable analysis, reveal that driver behavior varies continuously with context and can be characterized as probabilistic mixture [21, 61, 62] of weighted driving styles. Here we employ MoE-TopK [63] approach model multiple driving styles for trajectory prediction.
'Weight Function In open interval (a, b). the defined positive, continuous, and integrable function is called weight function.
The L3 Norm Used to measure the "magnitude" or "error" of a function when combined with a particular
weighting function p(z) in a given interval. It is defined as ||fl = (J. [f dx 1/2
5

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The process of combining the polynomials corresponding to multiple driving styles is presented in the algorithm on the right.. Here X; represents the th history trajectory in N vehicles as described in Section 2.1, which has 5 dimensions as (x. 1) position, velocity, acceleration, and the nearest lane segments ID. Experts represents the polynomials Section 3.2. The output =Com is the feature of combine. In line 2, "SN" and "Sp" denote the Standard Normal and Softplus functions [64, 65], respectively, W, and WR are trainable weight matrices. In line 3, we define (Hi, H2, Hz).
This combination structure of pn al- lows each E, to better extract the trajectory feature in different driving styles, and enables the use of vari- ous basis functions for predict vehi- cle trajectory. To encourage all ex- perts to contribute the combination process, Shazeer N al. [63] in- troduce load balancing loss func- tion L to encourage experts have equal importance
Algorithm: Polynomial Combination
Require: Input vehicle trajectory X;,
Gating network G Ensure: Feature 2; via Polynomial Combination
1: Experts networ { {E} } )
2: Hj + (X W
SN ( Sp [(X WW), for all 3: Gj = Softmax(H)) ())
4: for i = 1 to N do
5: =Com + G2 (Xi) Ej (X:).
LMOE-K Woond CV (loads)? ?, where "CV" denotes the coefficient of variation.
6: end for
7: Com 1=
Com
3.3.2 Degree Adjustment
Different driving style of trajectories can be approximated by corresponding Pn- However, the fixed degree of pn can restrict their ability for prediction entire trajectory of vehicles, which refers to:
Theorem 3.7 (Kolmogorov Theorem) For f Cla, b). there exists polynomial pn such that approximation error is bounded by:
(log (!mm))
If PAll
V(f. [a, b)),
n
where (f,, [a, b]) denotes the total variation" of f over the interval.
From Theorem 3.7, the accuracy of the polynomial approximation is directly related to the degree of the Pa. which applies broadly L"-space. On the other hand, the is bounded when error bounded of ps is know, this assertion is proved in Appendix C.
Adapting n presents a complex non-convex and combinatorial optimization problem. To tackle this issue, we utilize SMAC3 [66] tool, which is particularly suitable for optimizing low-dimensional and continuous functions, suitable for characteristic of vehicle trajectory (Section 2). Specifically, the degree n is treated as hyperparameter optimization problem, aimed at minimizing the loss (L) on validation data D and training data D Drain- This process can be formulated as follows:
VSMAC E arg min e (n) = arg min L (Durin, 1t),
neZ!
než!
The hyperparameter optimization process targets the final degree nsMMAc, corresponding to achieve the least error for the corresponding basis function Pn-
4
Experiments
4.1 Basic Setting
We evaluate our DSA framework on three real-world vehicle trajectory prediction datasets: nuScenes
[67], Argoverse [68] and Waymo [69]. These timestep settings follow the format(history time
prediction time): 2 - 6, 2 - 3 and 1 - 8, respectively. We utilize Toss = X₁ Tois
X LMot-KK with LMBE-K Whond CV (loads)² for model training with balanced weighting parameters
A.. We employ common standard metrics as the Average / Final Displacement Error (ADE / FDE)
for evaluate generate & trajectories. More detail of datasets and metrics, please refer Appendix B.
TToaal Variation A measure of the total amount of variation in function over given interval ja 6) which defined by sup [f) f (v)1 [ yl.

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4.2 Main Results
4.2.1
Quantitative Result
We evaluate our proposed DSA framework against existing methods utilize standard metrics. The best and second-best results are highlighted in Table I for the nuScenes and Argoverse datasets (with a 2- second observation window). Table 2 for Waymo (with a I-second observation window) respectively. The results demonstrate that our method outperforms most existing approaches, achieving superior performance in out of evaluation metries and ranking second others. Specifically, the best results over baseline datasets in Section 4.1 are 5.52% FDDEs (nuScenes), 8.82% -ADEs (Argoverse) and 1. 11%.%%mmiii.... (Waymo).
Table 1: Performance comparison of baseline and our DSA framework the nuScenes (left, N- Method) and Argoverse (right, A-Method) datasets. The best and second best are highlighted.
N Method
ADE
FDE,
ADE;
FDE;
ADE
FDE
A Method
ADE₁
FDB;
ADEg
FDEg
THOMAS [0]]
6.71
1.33
1.014
GOHOME [+]
1.70
3.68
0.89
1.29
PreTraM [72]
4.15
1.45
3.22
LTP [13]
1.62
3.55
0.83
1.30
Goal-Driven [73]
1.85
3,87
1.32
2.50
MP++* [74]
1.62
3.61
0.79
1.21
MUSE-VAE [75]
2.90
1.09
2.10
HiVT [76]
1.60
3.53
0.77
1.17
Real-Time [77]
3.56
8.63
1.60
3,34
1.23
2.32
ADAPT [78]
1.59
3.50
0.79
1.17
Aware [79]
5.58
11.47
1.67
2.66
Aware [79]
1.61
3.54
0.86
1.31
FRM [80]
6.59
1.18
0.88
FRM
0.82
1,27
Context-Aware [81]
3.54
8.24
1.59
3,28
R-Pred [82]
1.58
3.47
0.76
1.12
LAformer [11]
1.19
0.93
LAformer [11]
0.77
1.16
DAMM [83]
2.84
6.59
1.39
3.14
1.02
2.05
DAMM [83]
1.57
3.42
0.76
1.29
CASPNet++ [84]
2.74
6.18
1.16
0.92
ProphNet [85]
1.28
2.77
0.68
0.97
CASPFomer [86]
6.70
1.15
QCNet [87]
0.73
1.07
DSA
2.69
6.47
1.21
2.74
0.85
2.00
DSA
1.17
2.85
0.62
0.95
In the nuScenes dataset, DSA outperforms previous methods in metrics of ADE,, FDE,, ADE and FDE Compared to DAMM [83], which utilizes higher-order patterns to describe interac- tions between agents (vehicles), our model shows significant improvement, achieving a 16.67% enhancement in ADE Moreover, compared with FRM [80], which uses lane information to predict stochastic future relationships among agents, there is only a marginal gap of 0.03 in ADE,, but we achieve 3.41% % improvement in ADE However, DSA is slightly less effective than CASPNet++ [84], which employs interaction modeling and scene understanding for joint prediction of all road users while we only predict vehicle, that leads to minimal gap, measured in the thousandth place.
In the Argoverse dataset, our model achieves on three of four metrics in baseline. Although our FDE, metric gap in 0.08 than the baseline best results ProphNet [85], when predicting 6 samples, DSA shows improvements of 8.82% in ADE and 2.06% in ADE. In addition, while ProphNet utilizes an agent-centric model with anchor informed strategies, our DSA employs global positioning directly.
Table 2: Performance comparison of baseline and DSA framework the Waymo datasets. The best and second-best are highlighted.
Method
minADE
minFDE
MR
MultiPath+ [74]
0.9780
2.3050
0.4400
SceneTransformss [88]
0.6117
1.2116
0.1564
MPA [89]
0.5913
1.2507
0.1603
ReCoAt [90]
0.7703
1.6668
0.2437
In the Waymo dataset (Table 2). our DSA achieve the lowest minFDE and MR. We re- duce the MR by 5.14% and minFDE by 1.88% compared to MotionLM [94], their minADE is slightly higher than ours by 0.015, whereas our framework is based on simpler bascline model The ControLMTR [97] generate scene-compliant intention points and converte into a physics- based model, while DSA is driving and math- ematical based, we reduce the minFDE with 4.10% (value 0.0488).
DIPP [91]
0.6951
1.4678
0.1854
LiMTR [92]
1.3640
0.2156
HDGT [93]
0.5933
1.2055
0.1511
MotionLM [94]
0,5702
1.1653
0.1327
MTR++ [95]
0.5912
1.1986
0.9
TC-Mup [96]
0.6181
1.2375
0,1402
ControIMTR (97]
0.5897
1.1916
0.1262
DSA
0.5852
1.1431
0.1259
MR (Missing Rate) is the proportion of cases in which the Euclidean distance between the prediction and the ground truth at FDE exceeds 2m.
Our DSA framework adaptive design accommo- dates three categories of driving styles and we provide comprehensive explanations. This strategy simplifies the prediction process and enhances the accuracy and adaptability of predictions in complex real-world traffic scenarios.
7

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4.2.2
Qualitative Result
Figure 3 demonstrates the effectiveness for our DSA framework in vehicle trajectories prediction. For more visual content, please refer to Appendix. The the number generation trajectories, ground truth trajectory is actual trajectories. To deseribe specific subfigures in Figure 3, the
b
d
1
01 1
Ground Truth
Predictions
Lanes Line
Target Vehicle
Other Vehlcle
Figure 3: Qualitative results of our DSA framework. The value of l (left) represents the number of generation trajectories while letters (top) are index for clearly describe. Round head lines represents predict and ground truth trajectory, respectively.
position index (k, *) where * denotes the letter shown at the top of each subfigure.
When &: (i.e. the first row of Figure 3), he single prediction samples demonstrate that our DSA framework generally produces accurate results. It effectively handles not only simple road scenarios: such as straight lanes (1,a) and (1, or stop conditions in (1, d). But also complex scenarios including T-junctions in (1,b) and crossroads in (1,
In cases for generates and 10 trajectories, accurate and diverse. In simple scenarios, such as go straight in (5, c), (10, a), or stopping in (5, e), our framework maintains high accuracy while offering a broader range of plausible outcomes. particularly excels in complex road conditions, including Y-crossroads in (10, b). high density crossroads (5, b) and roundabouts (10, e). Moreover, the predicted trajectories effectively conform to curved roads, such as turning maneuvers in (5, d) and (10, d).
DSA framework delivers predictions that are both
4.3 Ablation Studies for DSA Framework
To explore the benefits different components and design choices in our DSA framework, we conduct ablation experiments along several dimensions: Type and combination of polynomials pa. in Section 4.3.1. Degree adjustment of polynomials in Section 4.3.2. Analysis of driving style in Section 4.3.3, examining the relationship between styles and specific Pa. and how expert weights reflect behaviors. In addition, we present sensitivity analyses for different scenarios in Appendix D.
4.3.1 Effects of Approximate Polynomial
We design two experiments to evaluate the combination and type of Pn. on the Argoverse dataset. The number of polynomials. To illustrate the importance of considering all drivers' driving styles instead of the parts of them in trajectory predictions, we simulate a scenario where only one or two driving styles existing and select corresponding matching Pn. (Section 3.2) for prediction.

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Analyzing results from Table 3, con- sider whole driving style outperform almost the best than other combine. We observe that DSA framework in- corporating two driving styles gener- ally outperform those with only one. However, this trend is not universal. For instance, in FDE,, the model
Table The performances of DSA framework with different combinations of basis functions on the Argoverse dataset. C. A and denote Conservative- Aggressive- and Normal- respectively. The best and second-best results are highlighted in table.
Metric
C
N
C+A
A+N
C+N
DSA
ADE
1.61
1,45
1.32
1.54
1.39
1.66
1.17
FDE;
3,46
2.87
2.91
3.09
2.87
2.78
2.85
based solely on the normal driver (1.32) over the combination of con- servative and aggressive styles (C+A, 1.54). Compared to the above com- bine (C+A) in ADEs, DSA further re-
ADEs
1.02
1.31
1.12
0.83
0.92
0.98
0.62
FDE,
1.26
1.29
1.33
1.29
1.04
1.17
0.96
duces 0.21 with 25.3%. This illustrate that the necessity of considering all driving styles in trajectory prediction.
The Type of Polynomials. For in- struction the effects of the Ps, we uti- lize DSA framework, we replace types of Pa. to evaluate it, with results in Table We select Charlier ([98), C.), Hermite ([99), H,)) and second- order (Sn,) polynomials to instead pr. we select in our DSA framework.
Table 4: Evaluate different pn in DSA framework on the Ar- goverse dataset, with original - replace and corresponding style (abbreviate with the first three letters) in the second column. The best and second-best results highlight in table.
Method
Replace
ADE,
FDE;
ADE;
FDF
Cn Ta + Ln
B, + C2. Con
1.48
3.76
1.10
1.53
Bn + Sn + Ln
Ta + S,, Agg
1.59
3.91
0.86
1.41
B. + Tu + II
Ln + H. Nor
1.62
2.81
0.71
1.02
Our DSA yields the best performance on three out of four evaluation met- rics in blod, which most improved is 27.8%. 43.6% and 37.3% respectively.
DSA
1.17
2.85
0.62
0.96
Although the combination Bn Tn + Hn achieves slightly lower FDE, from 2.81 2.85 by 1.4% marginal gap, DSA still ranks second on FDE,
4.3.2 Effects of Polynomial Degree
From Theorem 3.7, we understand that the prediction accuracy is directly related to the degree of the polynomial Pa- We now evaluate the impact of adaptively adjusting n. To clearly illustrate this influence, we analyze the performance of a single driving style with varying degrees, as shown Figure 4. We observe that the error generally decreases with an increasing degree. However, the
Conservative-Bn
Normal-i
Aggressive-T7
1.52
1.30
1.25
1.10
1.34
1.27
1.4
1.40
1.37
1.25
1.28
0.9
0.91
1.0
1.2
1.36
1.20
1.20
1.0
1.2
1,32
1.20
0.7
0.5
0.58
0.70
0.44
0.83
0.59
0.61
0.52
0.4
0.6
0.75
0.71
0.85
0.67
0.85
0.85
0.8
0.6
0.79
0.75
0.2
0.60
0.61
0.85
0.67
10
10
Degree
Dogroo
Degree
Figure 4: Performance of our DSA Framework, with only one single fixed basis function on the waymo dataset, which the lowest error highlighted in yellow.
highest degree does not necessarily yield the best results. For instance, within the aggressive driver style, 0 (T)) 5 outperforms other degrees in minFDE while à (T.) 8 is the best minADE. similar to conservative and normal driving style, the best results from different degrees. Consequently, adjusting rather than maintaining fixed set provides enhanced granularity for the Pn. polynomials, thereby improving their capability to generate accurate and varied predictions across diverse vehicle trajectory styles.

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4.3.3 Analysis of Driving Style
In Section 3.2, we provide the mathematical analysis for matching driving styles to their corresponding Pn- Here, we present experimental results that evaluate both the matching relationships and how expert weights reflect those relationships.
Relationship between driving style and polynomials. We compute the cosine similarity between the trajectory corre- sponding the weigtst- polyno- mial Pr and the predefined driving style standards [48, 49, 50], which include Conservative (Con), Aggressive (Agg), and Normal (Nor). The results the nuScenes (from N-Con) and Argoverse (from A-Con) datasets are shown Table 5.
Table 5: Cosine similarity reflecting consistent match- ing between polynomial based predictions and driving style standards the nuScenes (N) and Argoverse (A) datasets.
Pe
N-Con
Agg
Nor
A-Con
Agg
Nor
B,
0.953
0.158
0.489
0.884
0.129
0.391
T,
0.266
0.987
0.353
0.167
0.927
0.329
Lip
0.310
0.263
0.992
0.206
0.333
0.963
From Table 5, matching scheme (Bn-
Con, Tn-Agg. -Nor) obtains significantly higher similarity scores on both datasets. On the nuScenes dataset, the average similarity of the three correct matches (diagram values) is 0.977, while that of all other matches (non-bold entries) is 0.307. On Argoverse, these values are 0.925 and 0.259, respectively, representing substantial gap.
Expert weights mapping to driving style. To examine how expert weights mapping to driving styles, we compute the correct matching rate, defined as the proportion cases which the highest expert weight (Top-1) 1) matches the pected pn. for each driving style. The results appear in Table 6. The average correct match- ing rates across the two datasets are 86.35% and 88.59%, respectively. The highest matching rate is 94.26% for aggressive drivers on the nuScenes dataset. All styles achieve matching rate above 80%, with fluctuation range of 14.22%.
Table 6: The matching relationship between differ- ent driving styles and their corresponding Pn. The percentage indicates the rate which each pn has the highest weight within each driving style.
Style Argoverse
nuScenes
Top-1
Con 86.15%
87.96%
B,
Agg 92.87%
94.26%
Tn
Nor
80.04%
83.56%
Ln
5
Limitation
We evaluate our method on prediction horizons up to 9 seconds (1 second of history and 8 seconds of future), which is the longest duration available in current open-access vehicle trajectory datasets. minute), direct long-term prediction may
However, for significantly longer horizons (c.g., be unreliable and would likely require segment- modeling or hicrarchical strategies. addition, external factors such strong conditions (e.g.. traffic signals and regulatory constraints) and soft conditions (e.g., weather, which is often unlabeled in current datasets) can also affect trajectory prediction. Incorporating these contextual cues remains important direction for future work.
Conclusion
We propose an adaptive framework for vehicle trajectory prediction that tailored to the driving styles of human drivers. To enable effective matching between polynomials Pa, and driving styles, we analyze the behavioral characteristies corresponding P.. Furthermore, we investigate the effects of pn, combine, the influence of different polynomial types, and the necessity of adaptive parameters such as degree. Experiments results on three real-world datasets demonstrate that our framework significantly outperforms existing methods.
each style alongside the mathematical properties of
10

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Acknowledgements
This research is sponsored by the National Natural Science Foundation of China (U2102090, 62472238, 62576181), the National Key Research and Development Program of China (2023YFB4301900) 900)), the Shenzhen Science and Technology Program (CCYI200400811551455559) and the Science and Technology Planning Project of Guangdong Province 22023B122200000
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Normal
Conservative
Aggressive
The diagram illustrates of three driving styles: Conservative, Aggressive and Normal (CAN), with their corresponding trajectories represented by lines recorded at each time step. The lengths of the lines indicate the driving distance, while the direction is shown by arrows. Conservative drivers typically move at low speeds or stop to avoid obstacles, Aggressive drivers often travel at high speeds and are prone to overtaking other vehicles. Normal drivers maintain constant speed and frequently change lanes to ensure safety.
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Answer: [Yes].
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To handle the vehicle trajectory prediction task, we propose the first time a novel Driving- Stale-Adpptive (DSA) framework tailored to the driving styles human drivers, effectively leverages trajectory information.
We utilize polynomial approximation operators approximate and predict trajectories under different driving styles: Conservative, Aggressive and Normal (CAN). These operators support mathematical explanation matching mechanism between driving style with corresponding polynomial form.
The experimental results on the real- world datasets (nuScenes, Argoverse and Waymo) demon- strate that our model significantly outperforms existing methods in vehicle trajectory prediction.
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Appendix of A Priving-Style-Adaphee Framework for Vehicle Trajectory Prediction
A Problem Background
A.1 Vehicle Trajectory Prediction
The methods for vehicle trajectory prediction can be broadly classified into four categories [100]. These include: (i) Physics-based methods: These employ vehicle dynamics kinematics models, such singles trajectory methods, Monte Carlo, and Kalman filtering methods [101, 102, 103, 104, 105, 106]. These methods are known for their conciseness, efficiency, and computational effectiveness. (ii) Classic machine learning: Unlike physics- based methods that rely several physics models, classic machine learning approaches apply data-drive models and consider additional factors for predicting trajectories. Fxamples include the Hidden Markov Model, Dynamic Bayesian Network, and K-Nearest Neighbors [107, 108, 109, 110, 11 However, these traditional methods are typically only suitable for simple prediction scenarios and short-term prediction tasks.
Recently, with the advancement of modern machine learning, vehicle trajectory prediction methods based on: (iii) Deep learning and (iv) Reinforcement learning become increasingly popular. These methods are capable of considering interaction-related factors, understanding high-dimensional complex policies, and adapting to more complex scenarios. Examples include Graph Convolutional Network, Graph Attention Network, Conditional Variational Auto Encoder, and reinforcement learning techniques such as Inverse Reinforcement Learning. Generative Adversarial Imitation Learning, and Deep IRL [112, 113, 83, 114, 115, 116, 117].
In summary, an increasing number of autonomous vehicle trials are utilizing deep lcarning or reinforcement learning methods to predict future vehicle trajectories. These approaches leverage expert demonstrations and extract interaction information from traffic participants and road conditions, considering broader range of influencing factors.
A.2 Kolmogoroy-Arnold Networks (KANs)
Hilbert's 13th problem [118] famously posits that it is impossible to solve general seventh degree equations using only functions of two variables. Subsequent research by Kolmogorov et [36] has shown that any function involving multiple variables can be represented using finite number of stree-variabl functions. Further studies as detailed by Arnol'd al. [37], establish that even functions of just two variables are sufficient, as described Theorem 2.1 presents significant for machine learning: learning a high-dimensional function essentially reduces to learning limited number of one-dimensional basis functions (xx) in Equation (2).
In reference [20], the authors introduce Kolmogorov-Armole Networks (KANs), which neural network applications based on Theorem 2.1. Unlike Multi-Layer Perceptrons (MLPs) that founded the universal ap- proximation theorem [119. 120, 121. 122]. KANs feature learnable activation functions on what are traditionally referred to as "edges" (neurons) and they utilize fixed activation functions at what are typically called "nodes" (weights). Uniquely, cach weight in KANs is replaced by univariate function parametrized as a spline, mcaning the network contains linear weights whatsoever.
A variety of KANs are used across different tasks noted in [123], such solving ordinary diffcrential equations [44, 124], image classification and reconstruction [125, 126), and time series forecasting [127, 128, 129], among others. These applications demonstrate competitive or superior performance efficiency and predictive power compared to traditional models. However, to the best of our knowledge, we are the first to utilize KANs in vehicle trajectory prediction. This involves approximating and predicting trajectories for different driving styles, expanding the range of basis functions, and providing explanations for specific matches between functions and trajectories.
A.1

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B Dataset Information
B.1 Datasets
We preprocess the three datasets using the official pip packages provided by their respective baselines. The characteristics of each dataset are summarized in Table 7.
Table 7: Characteristics of the evaluation datasets.
Datasets
Collect
Size
Library
Select
Argoverse [68]
Miami and Pittsburgh
23,69G
Argoverse API devkit
Motion Forecasting
nuScenes [67]
Boston and Singapore
4.81G
nuscenes-devkit
Motional-Ful dataset
Waymo [69]
Phoenix, AZ, Kirkland etc.
83,50G
waymo-open-dattset
Motion 1.--sennari
This dataset [67] offers high-definition maps and trajectory data from 1,000 driving scenes in Boston and Singapore, areas noted for dense traffic and complex driving challenges. It comprises 245,414 trajectory instances, each sequence of 2D coordinates over seconds, sampled 2Hz. The nuScenes benchmark requires predicting target agent's 6-second future trajectory from a 2-second historical trajectory. The comprehensive dataset features approximately 1.4 million camera images, 390,000 LIDAR sweeps, 1.4 million RADAR sweeps, and 1.4 million object bounding boxes across 40,000 keyframes.
nuScenes
Argoverse This dataset [68] facilitates research in 3D tracking and motion forecasting for autonomous vehicles. Originating from select areas in Miami and Pittsburgh, it includes 113 scenes with 3D tracking annotations, featuring 324,557 significant vehicle trajectories derived from over 1,000 hours of driving. The forecasting component of Argoverse provides agent trajectories and high-definition maps, requiring a target vehicle's future trajectory at 10Hz. The dataset encompasses 333K real-world driving sequences, primarily at intersections or within dense traffic, each focusing one target vehicle for trajectory prediction.
prediction next 3 seconds, based on past trajectory over two seconds, sampled
Waymo This dataset [69] publicly to aid the research community in investigating wide range of interesting aspects of machine perception and autonomous driving technology. This Dataset object trajectories and corresponding 3D maps for 103.354 segments. Given agents' tracks on a corresponding map, predict the joint future positions of 2 interacting agents for 8 seconds into the future. The ground truth future data for the interactive test set is hidden from challenge participants. The validation sets contain the ground truth future data for use in model development. In addition, the test and validation sets provide interacting object tracks in the scene predicted.
the Motion part, with
past I second
B.2 Metrics
We evaluate the predicted trajectory Y against the ground truth trajectory Ydr using standard error- error-based metrics. Our DSA framework adopts the commonly used Average Displacement Error (ADE) and Final Displacement Error (FDE), as defined in [23]:
T
1 ADE
T Σ y
You 11,,, FDE = (Y Yerlzza.
Here, the superseript & denotes the current time step, and T refers the total number of time steps in the prediction horizon. The metrics we use include ADE;, FDF2, minADE, minFDE, and b-minFDE. The subscript k indicates the Top-k most likely predicted future trajectories. The "min" variants (minADE and minFDE) compute the 2 distance between Y and the closest predicted trajectory Y' across all generated samples. averaged over all agents. The b-minFDE metric extends minFDE by incorporating the Brier score [130], which evaluates calibration the predictive distribution. defined as the sum of the Brier score and minFDE.
A.2

---

C Proof of Bound
As we discuss in Section 3.3.2, Kolmogorov's 's thcorem [131] provides error bounds by evaluating the absolute value of the function and the overall variation in the function value. This is illustrated as follows:
Theorem C.1 (Kolmogorov Theorem) For E C bJ, there exists polynomial Pa such that approximation error is bounded by:
1-PPolee x log
la,,
where V(f, Ja, bl) denotes the total variation 9 of over interval.
From theorem, we conclude that when n. > e, increasing the degree n ps, results a smaller decrease the theoretical upper bound on the approximation enror.
However, in practice, considering computational cost and time, the value of n cannot be arbitrarily large. In this section, provide three proofs corresponding to the three calegories of drivers, demonstrating that under given error limit 6, there exists relationship between the minimum degree and the components of the trajectory approximated. described Section 2.1 (i.e., position (z, y). velocity, and acceleration). clarity, we use f to represent each continuous component with respect to t.
C.1 Conservative Drives: Bernstein Polynomial
the properties of Bernstein polynomials (B,) for their uniform convergence to approximate trajectories of conservative drivers. characterized by low speed minimal motion changes. The minimum degree n of the B, polynomial is obtained by:
In Section 3.2.1,
Theorem C.2 For all > 0, if Bn (f) n, then for given error limit with O < & < oc, then
n > max f (E) /85.
Proef:
First, we calculate the error between fx) and (f)
If (x) Ba (f)l
8
1 # (1 (1-x)" x) (x) -
1:
rn(( -x)
1 5"
(k
-x)
x) 27
Pa (8)
(4)
2
f" (5)
1
PB (1)
&E
J. k
(5)
formula (4), Ps (N:) _ ah (1 x)) n- From Equation (5), we next proceed to prove
1. Σ. - Pg (k) = 0,
2. Σ. 2) 2 Pa (k)
z).
For Equation 1, applying the Central Limit Theorem (CLT) as discussed [132], consider the total difference of weights, represented by [nz/ /k) - x)) Specifically, the probability p satisfies p = x = k/n. this case, as described in [133], we have B(k) n.c Thus, we can derive the expectation as follows:
E (k)
x.
(6)
n
Therefore, Equation simplifies to B(k/n) 0.
For Equation 2, we employ similar method; here (" x)²² represents the squared difference in weights between and alternatively described as the deviation between observation and expectation. According to Equation (6),
k
2 (1-x)) r
D
Total Variation A measure of the total amount of variation in function over given interval ja, 6, which defined by sup [f) (v)1 [ yl.
A.3

---

Equation 2 corresponds to the squared deviation
( x)²
based on weights Pa(k). Moreover
h
B
x
11
x (1
xt)
M2
:5 5 (5)
with M₂ = max f (6)
8n'
EED
Considering the limit 6, have:
M2
M2
< 8 >
8n
86
Theorem C.3 If f E (l(a,),, B (f) la, b) and BB (f)) For all 0.00 0c, i
given error limitation, then:
max (E) w|] (b a)²
86
where w is the weights of weighted B. polynomials B (f)-
Proof:
Here the emror L norm, the definition of (f) is
B% Σ w
x² (1 x) nk
k
Let u= (x - a)/ (b- a), then E [0, 1). So M2 is similar to Theorem C.2, here
M₂ max wC(o.1]
a)² !9" (u)l =b² 15" (5)
where (u) [
The next following prove is similar to Theorem C.2.
C.2
Aggressive Drivers: Chebyshev Polynomial
Scction 3.2.2, aggressive drivers' trajectories are characterized by non non-smooth, high-speed movements during motion changes. We use the Chebyshev polynomials T. and their minimum maximum error properties to approximate these trajectories. The minimum degree n of Tç polynomial is obtained as follows:
kinimum-maximum
Theorem C.4 For f E La bj and given error bound (where 0 < c < & oo), the condition @ [T. (f) is satisfied:
n w-
8
12
where w is the inverse of the modulus of continuity for the function f.
To provide the proof of Theorem C.4, first introduce the definition of the modulus of continuity and lemma related to this proof.
Definition C.5 (Modulus of Continuity L" Space) Let L" (a, bl. > and & m a. The modulus of continuity wi (m, f') is defined as:
wp (m) =wp (m,
(
()(=+))-
sup 0
which represents continuity formfor ff over the interval (a, b).
For T. polynomials belonging to the C2n space !", we use E. (f)) to denote the deviation of the approximation of by trigonometric polynomial T,, of degree n, follow:
En(f) = inf |If T, ll-
(Tw)
This deviation satisfies:
10C2z Space Let f B with period 2m. Define
If ||= sup 1f(x))
T
We call the above set the Car space
A.4

---

Lemma C.6 (Jackson [134]) Let E Cze. then for all v. E N. the following inequality holds:
(nn(f) & 12 w
It is evident that T% C C2x. Based on Lemma C.6, we present the proof of Theorem C.4.
Proof: According to the definition of 8, we have || I - Tn || < 6. Lemma C.6 provides the modulus of continuity under the 0 space, so we need to relate wpp from Definition C,5 to w in Lemma C.6, which relates the 1/" and L norms:
I [0llEp((aN0-)) S (b - a- h) 1-llll ((u,b-A))
For the function difference g(x) f(x +) f(x):
( f(x)] w(f,h), for E (a, h).
Therefore, the wy,m, f) related to w(f. h) as follows:
1/8 wp & {Trm 1w (s.)))
w(f. h) h)
(8)
When h + Equation (8) he approximated as:
Up & w a)
According to Lemma C.6 and satisfy the error limit 8, s.t. En(s)
& 8, we have
8
12'
To obtain the lower bound n., since w(,h) is nondecreasing function with respect h. we take its inverse function Vffv) follows:
$
w
S,
(9)
n
12
Finally, the lower bound for n can be derived from inequality (9) as:
1
Lemma provides a minimum bound related to the value of the modulus continuity. Furthermore, Lipschitz continuity [135] of vehicle trajectories X, and Y; offers a more compact bound for inequality (7):
Corollary C.7 The bound in inequality (7) is satisfied as follows:
12L
n2 >
(10)
where L represents Lipschitz constant.
Proof: The proof Corollary C.7 consists two parts: (i) establishing Lipschitz continuity of vehicle trajectories and (ii) deriving Equation (10).
(i) Lipschitz Continuity Vehicle Trajectories
To demonstrate the Lipschitz continuity vehicle trajectories, it suffices to show that their state information (Section 1). including (z, y) position, velocity u and acceleration satisfies the Lipschitz condition (L- condition). Specifically, there exists constant T., s.t. for any x' x" E [a, b). the following holds:
(fz)) (x
(11)
According the physical relationships among these states, acceleration satisfies L-condition, then by boundedness theorem [136], the other states also satisfy it. Thus, we take as an example, and similar arguments apply to the other states, quod erat demonstrandum.
According to [137, 138J, the variation in vehicle acceleration is constrained by factors such as engine performance, vehicle weight, and braking system, which implies that the jet (the rate of change of acceleration time)
A.5

---

cannot be physically infinite. Therefore, there exists constant M,. s.t. 15 (T) & M,. For any 4, E [a, b]. the following holds:
[ j(r)
{ [T
(a(4) - a
M, dr M, |41 (2).
Therefore, Hence, the acceleration function (t)) Lipschitz continuous with the Lipschitz constant I. M,-
(ii) Derivation of more compact bound.
To obtain tighter bound, we use I-continuity property of f. From the continues of modulus, we have:
wf,h) & LA + (f.D)
(12)
L'
From Jackson's 's inequality in Lemma C.6, let 8/12 in Equation (12). This ensures that the error remains below 8, with 6/12 acting piecewise threshold. Then Equation (12) becomes:
8
(f.y)
(S
(13)
12
12L
Combining inequalities (7) and (13), obtain the more compact bound (10).
C.3 Normal Drivers: Legendre Polynomial
In Section 3.2.3, discuss that the speed and acceleration of normal drivers maintain an intermediate state between conservative and aggressive drivers. Their trajectories do not change dramatically as aggressive drivers, they so slow affect the flow of traffic. To approximate these trajectories, we use the Legendre polynomial L,, The minimum degree n of I., obtain by:
Theorem C88 For all 0, L.. (f) for given error limit with 0 < < 8 & co, then
CH
1/a
11
(14)
where Cn is Hölder constant modulus of continuity and represents Hölder exponent.
From Theorem we apply Jackson's ' incquality (Lemma C.6) to establish a relationship between the approxi-
mation and the modulus of continuity. This incquality also applies to continuous functions defined on the interval [-1, interval. To achieve the bound in Equation (14), further use the Hölder continuous property [139] of vehicle trajectory. Similar L-continues defined in Equation (11), Hölder continuous (Il-continuous) define as follows:
Definition C.9 (Hölder continuous) For function f defines on interval I, if there exits constant C E R, s.t. for V2 I::
If f < Cu 22' 2010 a a (0,1),
then f said be Hölder continuous order
When 1. H co-ootiiuuuus reduces to L-continuous. Reference [140, 141, 142, 143] analyze the Hölder continuity related smoothness of vehicle trajectories and their states (position, velocity, and acceleration). either directly indirectly by examining the smoothness of physical constraints and changes. Therefore, the bound Equation (14) can similarly be derived from incquality (12):
(1)
w(f.h) & Cuh = CH
Thus, we obtain a bound as expressed by incquality (14).
A.6

---

D Effects of Framework Sensitive
In Section 4.3, we evaluate our DSA framework with respect to module component dimensions: namely the type. combination, and degree of polynomials Pa. well the driving style matching. In this section, we further analyze sensitivity our model with regard lo: variations the driving style categories themselves (Section D.I,, external influences such as changing traffic densities (Section D.2) and varying road conditions (Section D.3).
D.1 Number of Driving Style
In reference [48, 49, 50J we know that driving style are categores as three type and each characters are illustrated in Section 3.2. Now we use automatic manner category numbers rather than the predefining for driving styles. Specifically, evaluate multiplc-categon
settings using K-means clustering, and report the corresponding metrics llog-normalized results except for Silhouette metric) on the Argoverse (from A-AWSS, and nuScenes (from N- WSS) Table 8, here for all metrics listed below, larger values indicate better clustering performance. Our chree-category configuration yields the highest number of best scores across the metrics, supporting the
Table Clustering evaluation results on Argoverse (A) and nuScenes (N), value of "1" indicates the best performance among all settings.
A-AWSS
Silhouette
ACHI
N-AWSS
Silhouette
ACHI
2
0.919
0 0.700
0.810
1
1
4
0.743
0.986
1
0.586
1
0.387
0.022
0.283
0.287
0.268
0.324
6
0
0
0.601
0
0.398
0
validity and rationality of our chosen driving style taxonomy. The metrics [144] are defined as follows:
WSS (Within- Wiithin-Cluster higher value suggests tighter grouping samples within clusters after clustering.
Squares): Measures the improvement in intra-cluster compactness.
Silhouette: Reflects both the cohesion within cluster and the separation between clusters. A higher silhouette score indicates that samples well matched their own cluster and poorly matched neighboring clusters.
CHI (Calinski-Harabass Index): Captures the variation in inter-cluster separability and intra-cluster compactness. Higher values indicate better-defined and more distinet clusters.
D.2 Traffic Density
Traffic density is closely related to vehicle speed and traffic flow, and significantly influences trajectory prediction varying interaction patterns among vehicles [145]. clearly present the impact traffic density, we divide the dataset into five levels based on the number of vehicles per unit area. We compare the performance of our DSA framework against the best pestttrerfoingggg baseline with publicly available code: Context-Awan [8] as identified in Table 1. The comparison results are illustrated in Figure 5.
JDBA)
Context
Ours (D8A)
AF=
Contzxé Awore
5.0
5,35
FDE
10.35
FDE
4.32
None
7:34
6.51
Nanber
E440E
1662
641
343
39
355
417
622
ADE
5316
4.35
4.41
255
7.46
225
ADE
20:2:11
3.8
2.45
2.87
2.59
1.99
1.19
195
Deiaity
Densty
Figure 5: Comparison of trajectory prediction performance under different traffic densities with Context-Aware [81]. Our method is represented by solid lines, while Context-Aware is depicted using dashed lines on both sides. The middle subfigure shows the density distribution with circled numbers indicating the corresponding density levels.
Our DSA framework consistently outperforms the baseline across over 75% of the dataset. When averaged over the highest-density and most common case (density level in the middle subfigure of Figure 5), our method achieves improvements of 1.97 in ADE; and 3.26 in FDE1, well as 0.25 in ADEs and 0.13 in FDEs.
A.7

---

The most notable gains appear in density level 2, where DSA reduces prediction enrors by 49.48% and 44.19% 19% for ls 1, and by 33.78% and 25.05% for & 5, compared Context-Aware. Although framework shows slightly lower performance levels and when generating trajectories, these cases together account for only 6.28% of the dataset.
D.3 Road Condition
Road structure significantly influences the motion patterns of agents navigating through urban or highway environments thus essential accurate trajectory prediction [146, 11, 112]. Does road complexity increase the frequency of driving style changes, thereby making prediction more difficult? To investigate this question, we evaluate the performance of our DSA framework under different road conditions, shown in Table 9.
Table 9: The b-minFDE* in different road condition on the nuScenes dataset. The best results are highlighted.
Type
Stationary
Straight
Straight right
Straight
Right U-turn
Right-turm
U-turn
Left-turn
MTR[147]
2.15
2.58
4.85
4.26
8.13
4.82
5.17
4.85
2.86 2.75
DSA
2.03
2.48
4.96
4.17
8.11
4.77
5.28
4.87
Brier-minFDE (b-minFDE): OminFDE-miinDDE((1 p)². where p the probability of probability of the best forecasting trajectory with minimum endpoint error (minFDF)..
Our DSA framework achieves the lowest overall enror of 2.75, improving upon the baseline by 3.85%. It outperforms the baseline in 6 of 8 categories, including reductions in error for Stationary (5.6%) and Straight (3.9%), well complex maneuvers like Straight-Left (2.1%). Right U-tum (0.25%), and Right Turn (1.0%).
scenarios such
Although MTR performs slightly better Straight-Righ and Left U-tum (by 0.11 both cases), DSA matches or surpasses baseline performance in the most frequent and safety-critical trajectory types. These results demonstrate the robustness and adaptability of our framework across diverse road semantics, particularly in non-linear discontinuous motion patterns.
A.8

---

E Detailed Description of the Algorithm
In Sections 3 I and 3.3.2, we introduced the mechanisms for polynomial combination and degree adaptation. In this section, we provide detailed description of the corresponding algorithms.
E.1 Polynomial Combination
To match each trajectory under various driving styles to a suitable polynomial combination, as analyzed in
Section 3.2, employ Mixture of Experts model based Top-K Gating (MoF-TopKK [63]. In this method, tunable Gaussian noise is added to the gating logits, and only the top K values retained for expert selection.
Let denote by G' (az) and Ey (1r) the output of the gating network and the output of the j-th expert network for a given trajectory Xs, for clearly we omit subscript The output ccm the MoE module can written as follows:
com
G(X), Fg (z).
(15)
As shown in [21, 61, 62J. kernel density estimation and latent variable analysis reveal that a driver's behavior evolves continuously across different situations. This implies that driving behavior can be viewed as probabilis- tic mixture of weighted driving styles. Given that drivers may exhibit behaviors characteristic of multiple styles dynamic scenes-such when another agent suddenly appears-ww we adopt the Noisy Top-K Gating network [63] to capture this mixture behavior. This network activates only the top-k best-matching experts, enhancing responsiveness and specificity. Accordingly, this modification adjusts G-(+) in Equation (15) to G (zr,, detailed as follows:
(²(2)= = Softmax (KeepTopK [H Ax), with
((X) =(X Wo,) + SN () Sp [(X W [.ose)]]
Here, "SN" and "Sp" denote Standard Normal [64] and Softplus [65] functions, respectively. The symbol W, denotes the weight matrix corresponding each subscript. The loss function is defined follows:
LABE-K = Whead CV (loads)?,
(16)
where "load" refers to the importance values assigned to each driving style, with Wiedd representing the corre- sponding weight. "CV" stands for the Coefficient of Variation, which assesses the variability of these values. Equation (16) is part of Loss in Section 4. This structure of the MoE model effectively recognizes the diversity trajectories, allowing each expert to specialize different features of driving styles.
E.2 Degree Adaptation
From Theorem 3.7, understand that the accuracy of polynomial p.. approximation is dircctly influenced by the degree n of the Pv. However, adapting the degree of ps. poses a complex, non-convex, and combinatorial optimization challenge, as the relationship between prediction error and polynomial degree is not straightforward. This complexity often leads to the presence of multiple local optima.
To address this issue, we utilize the versatile Bayesian Optimization (BO) tool SMAC3 [66] for its rohustness and flexibility, making it particularly suitable for optimizing low-dimensional and continuous functions (type: SMAC4BB), such those found in vehicle trajectory prediction.
We the adaptive of polynomial degree which leveraging Gaussian Processes with the Matérn kernel and the Expected Improvement acquisition function. iteratively searches the candidate degree set to minimize the loss function. Specifically, the degree n is treated as hyperparameter optimization problem, aimed at minimizing the loss validation data Dea of our model trained on training data Drrin This process can be formulated as follows:
hyperparameter optimization problem, using SMAC3 for BO,
TISMAC E arg min e (m) arg min L (Denin; Deat; n1).
mgZ
The hyperparameter optimization process targets the degree risMAc, which is defined as the optimal degree that achieves the least error for the corresponding basis function Pa. Here L denotes the loss function.
A.9

---

F Limitation and Discussion
Limitation We summarize existing open vehicle trajectory prediction datasets in Table 10, and observe that the maximum available trajectory duration is typically less than 10 seconds. Despite this limited time span, our framework based on thrce driving styles adapts well such settings. We evaluate its performance on both short-term (3 scconds, Table 1) and long-tem long-term (8 seconds, Table 2) prediction tasks, achieving consistently strong or statc-of-the-ee results across all durations.
Table 10: Existing vehicle trajectory datasets. "His" and "Pre" represent the historical and predicted trajectory durations, respectively, while "Total" denotes the overall duration for each vehicle.
Datasets
Pub.
Collect Locations
His
Pre
Total
KITTI [148]
2012 CVPR
Karlsruhe
2
4
6
Apolloscapes [149]
2018 CVPR
Beijing, ShangHai and SHenZhen
6
nuScenes [67]
2020 CVPR
Boston and Singapore
2
6
8
Argoverse [68]
2019 CVPR
Miami and Pittsburgh
INTERACTION [150]
2019 arXiv
China, Germany and Bulgaria
1
4
InD [151]
2020 TIV
German
3.2
4.8
8
RounD [152]
2020 ITSC
Aachen
2
6
HighD [153]
2018 ITSC
German
2.8
2.8
5.6
Waymo [69] 2020 CVPR
USA
8
9
However, longer prediction horizons, complexity driving behavior increases. suggesting that three driving style categories may be insufficient to cover all possible scenarios. Moreover, trajectory patterns are often influenced by external factors, which can be categorized as either soft or strong conditions.
Soft conditions, such as weather, affect driver perception and reaction. For example, sunny days, improved visibility may enhance drivers' responsiveness, leading smoother and more stable trajectories. In contrast, adverse weather conditions such fog, heavy rain, or snow can result in more abrupt irregular driving behavior.
Similarly, strong conditions such as traffic signals or regulatory constraints also significantly influence vehicle trajectories. Unfortunately, most existing datasets lack labels for these contextual factors. We believe that incorporating such labels could further enhance prediction accuracy in future research.
Discussion For longer vehicle trajectories, we can improve our DSA framework from both practical and thcoretical perspectives.
1. Incorporating more driving styles. Our current DSA framework utilizes three representative styles: Conservative, Aggressive and Normal (CAN), which reflect two behavioral extremes and an interme- diate pattem.. However, the temporal length of each driver's trajectory increases, driving behaviors may exhibit greater variability. capture these nuances, the model can extended by defining or integrating additional driving styles. This would allow for a more fine-grad tine-grained charactcrization of driver behavior and potentially lead to improved trajectory prediction accuracy.
2. Expanding the set of basis functions. more complex, a broader set basis functions required to effectively approximate and predict vehicle trajectories. Instead of relying on a single polynomial type, we can extend to set of basis functions of the same class, such as orthogonal trigonometric polynomials. For example, to minimize the L norm in modeling trajectories of normal drivers, other intermediate states between conservative and aggressive behavior. It beneficial to use richer of orthogonal polynomials that better match the dynamics of these nuanced driving styles.
driving styles hecome
diverse and trajectory conditions
A.10

---

G
Future Work
In this paper. we focus on the characteristics of the individuals who generate the data (i.e., trajectories) and leverage the mathematical properties of basis functions approximate these trajectories. This concept be generalized and extended other domains, such as:
Other traffic participants. In addition to drivers, other agents in the traffic scene such as pedestrians and cyclists, also exhibit distinct behavioral characteristics. By modeling these characteristics, select appropriate basis functions tailored each agent type, thereby improving the accuracy of their trajectory prediction.
Multivariety Time Series Forecasting. Our framework can be extended to long-term forecasting tasks in domains such as weather prediction, energy consumption and electroaadioggaphy. For example, one could model temperature and precipitation trends across different climate zones, analyze electricity usage patterns based on consumer hehavior, or study heart rate dynamics as function of individual health conditions.
Additionally, by leveraging the core theoretical foundation (Theorem 2.2), we aim to construct models grounded in the physical characteristics or bchavioral attributes of the data sources, thereby fully exploiting the inherent structure of the data itself.
H
Visualization
Due to space constraints, the number of visualized prediction results in the main text is limited. Here, provide additional visualizations of predicted trajectories for various scenes, with generated trajectories & 1, 10. Fach value of le presented for hoth simple (e.g., straight roads) and complex scenes (c.g., turns conditions), showcasing different types of driving hehavior.
To enhance clarity of the visualization results, we present them on dedicated page and opacity to improve visual contrast. Specific outcomes are accompanied by detailed explanations provided in the corresponding figure captions. In summary, considering various scenario combinations and adjusting the number of generated trajectories lead predictions. Inereasing the number of predicted trajectories improves prediction diversity and realism, while analyzing different scenarios helps adapt to the diversity and complexity of real-world traffic environments. These improvements contribute to making the model both more mathematically grounded and more adaptive.
background
diverse, accurate, and comprehensive vehicle trajectory
A.11

---

k
Ground Truth
Predictions
Lanes Line
Target Vehicle
Other Vehicle
A.12

---

=k=
Ground Truth
Predictions
Lanes Line
Target Vehicle
Other Vehicle
A.13

---

k
Ground Truth
Predictions
Lanes Line
Target Vehicle
Other Vehicle
A.14

---

k
Ground Truth
Predictions
Lanes Line
Target Vehicle
Other Vehicle
A.15

---

k
Ground Truth
Predictions
Lanes Line
Target Vehicle
Other Vehicle
A.16

---

k
Ground Truth
Predictions
Lanes Line
Target Vehicle
Other Vehicle
A.17

---

k
Ground Truth
Predictions
Lanes Line
Target Vehicle
Other Vehicle
A.18

---

k
Ground Truth
Predictions
Lanes Line
Target Vehicle
Other Vehicle
A.19

---

k 10
Ground Truth
Predictions
Lanes Line
Target Vehicle
Other Vehicle
A.20

---

k
10
Ground
Predictions
Target Vehicle
Other Vehicle
A.21

---

k 10
Ground Truth
Predictions
Lanes Line
Target Vehicle
Other Vehicle
A.22

---

k 10
Ground Truth
Predictions
Lanes Line
Target Vehicle
Other Vehicle
A.23


Reviews:
[{"review_id": "P8ZIKRYzEa", "reviewer": "Reviewer_ejCn", "summary": "This paper introduces the DSA framework, a novel approach for vehicle trajectory prediction. The core contribution is the systematic integration of human driving styles—categorized as conservative, aggressive, and normal —into the prediction model. The framework leverages the principles of KANs and the weierstrass approximation theorem. It proposes matching specific polynomial basis functions to each driving style based on their mathematical properties: Bernstein polynomials for smooth, conservative trajectories; Chebyshev polynomials for abrupt, aggressive trajectories; and Legendre polynomials for moderately fluctuating, normal trajectories. The model uses a MoE structure to create a weighted combination of these style-specific predictors. Furthermore, it employs a degree adaptation mechanism to optimize the complexity of the polynomial functions. The authors demonstrate through experiments on the nuScenes, Argoverse, and Waymo datasets that their DSA framework achieves state-of-the-art performance on multiple standard evaluation metrics.", "questions": "See weaknesses.", "limitations": "Yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\nPropose a systematic and theoretically-grounded framework that explicitly matches different classes of polynomial basis functions to distinct, predefined driving styles for trajectory prediction. Extensive experiments on three major datasets demonstrate SOTA performance. Thorough ablation studies to validate their design choices. The experiments effectively demonstrate the benefit of combining all three driving style experts over using just one or two, and validate the specific choices of polynomials against other alternatives.\nWeaknesses:\n•\tLack of clarity regarding the concrete implementation of the core components.  Could you please elaborate on the precise architecture of the Expert networks? Specifically, how are the assigned polynomial basis functions (Bernstein, Chebyshev, Legendre) integrated?\n•\tThe CAN taxonomy ignores continuous/style-blending behaviors (e.g., moderately aggressive) and relies on heuristic categorization. Real driving styles are context-dependent, but DSA uses fixed polynomial mappings. Evaluations focus on highway-like scenarios (nuScenes/Waymo). Performance in unstructured environments (e.g., pedestrians, intersections) is unverified.\n•\tDegree Adaptation is potentially misleading. The method describes using SMAC3 to solve an optimization problem for the degree n based on validation set performance. This sounds like an offline hyperparameter tuning step that finds a single optimal degree for each driving style, which is then fixed. If the degree is tuned offline, calling it \"adaptation\" is an overstatement. If it is adapted dynamically for each sample during inference, the mechanism and its computational overhead are not explained at all.", "was_revised": true, "final_justification": "After reading the rebuttal and comments from other reviewers, my final rating is borderline accept.", "rebuttal": "We thank the reviewer ejCn for raising thought-provoking questions, which explanations and significantly improve the quality of the paper.\n\nWeaknesses & Question 1: The precise architecture of the Expert networks? How are the assigned polynomial basis functions integrated? \n\nA: The combined feature $z$ is integrated by computing the weighted sum of the expert outputs $E$, using the gating network $G$ as weights.\n\nIn Polynomial Combination algorithm (Expert networks), each expert $E_j,(j=1,2,3)$ is implemented as a KAN-based module corresponding to a specific polynomial basis (Bernstein, Chebyshev and Legendre), each associated with a distinct driving style.. \n\nFor any input vehicle trajectory $X_i (i=1,\\cdots,N)$, we first compute its score with respect to each expert using the following equation:\n\n$$\nH_{j}\\leftarrow (X\\cdot W_{g})_{i}+\\mathsf{N} (0,1)\\cdot \\mathrm{Softplus} \\left[ \\left( X\\cdot W_{n}\\right)  \\right]_{i},  \n$$\n\nwhere $W_g$ and $W_n$ are trainable weight matrices (i.e., implemented as \\texttt{nn.Linear} layers), and $\\mathcal{N} (0,1)$ denotes standard Gaussian noise. \n\nNext, we compute the weight for each expert via the Softmax function:\n\n\n$G_{j}= \\text{Softmax}(H)$, where $ H=\\left( H_{1},H_{2},H_{3}\\right)  $. \n \nFinally, the combined feature representation $z^{\\text{Com}}_i$ for vehicle $i$ in the DSA Polynomial Combination block is computed as a weighted sum over the experts:\n \n$$\nz^{\\text{Com} }_{i}=\\sum^{3}_{j=1} G_{j}\\left( X_{i}\\right)  \\cdot E_{j}\\left( X_{i}\\right) .\n$$\n\n2: (1)The CAN taxonomy ignores continuous/style-blending behaviors (e.g., moderately aggressive) and relies on heuristic categorization. \n\nA: (1) We use three typical driving style categories to represent two behavioral extremes (Aggressive, Conservative) and an intermediate type (normal), following the convention established in prior works [1–4]. For more nuanced behaviors that fall between these categories (e.g., moderately aggressive), the weights of the three polynomial experts in the MoE-TopK network can be adaptively adjusted. \n\n[1] Yao X, Calvert S C, Hoogendoorn S P. Driving heterogeneity identification using machine learning: A review and framework for analysis[J]. Transportation Research Interdisciplinary Perspectives, 2025.\n\n[2] Zheng X, Yang P, Duan D, et al. Real-time driving style classification based on short‐term observations[J]. IET Communications, 2022.\n\n[3] Garrosa M, Olmeda E, Fuentes del Toro S, et al. Holistic vehicle instrumentation for assessing driver driving styles[J]. Sensors, 2021.\n\n[4] Xu L, Hu J, Jiang H, et al. Establishing style-oriented driver models by imitating human driving behaviors[J]. IEEE Transactions on Intelligent Transportation Systems, 2015.\n\n\n2: (2) Real driving styles are context-dependent, but DSA uses fixed polynomial mappings. \n\nA: (2) We justify the choice of \"fixed polynomial\" types using both mathematical theory (**theoretical**) and empirical **experiment** results.\n\n## Theoretical Justification \n\nWe establish a theoretical matching between the three driving styles and corresponding polynomial bases based on the behavioral characteristics of each style:\n\n-**Conservative drivers** tend to maintain the lowest average speed and exhibit minimal behavioral changes. Their trajectories are smooth and stable with few abrupt accelerations or decelerations. The uniform convergence property of Bernstein polynomials ($B_n$) ensures that the approximation error decreases consistently across the interval, making them well-suited for approximating conservative trajectories.\n\n-**Aggressive drivers**, in contrast, display higher speeds and more abrupt behavioral changes, resulting in trajectories that are less smooth and more prone to sudden shifts. Chebyshev polynomials ($T_n$) which minimize the maximum approximation error, are particularly effective in handling sharp variations and limiting the influence of extreme points. \n\n-**Normal drivers** represent a balance between conservative and aggressive styles. Their trajectories tend to maintain moderate speeds and accelerations with regular but smooth behavioral changes. Legendre polynomials ($L_n$) known for their orthogonality and optimal approximation under the $L^2$ norm, are computationally efficient and capable of capturing continuous and moderately fluctuating trajectory patterns.\n\n## Experiment Results\nWe replace each original polynomial in ours with alternative types, including Charlier ($C_n$), Hermite ($H_n$) and Second-order ($S_n$) polynomials. The best-performing results (in bold) are highlighted in Table1.\n\nTable 1. Evaluate different p_n in the DSA framework on the Argoverse dataset. The \"Replace\" column indicates the original → replacement mapping and the corresponding driving style.\n\n| Method                  | Replace        | ADE$_1$    |   FDE$_1$    |   ADE$_6$    |   FDE$_6$   |\n|-------------------------|----------------|------|------|------|------|\n| $C_n+T_n+L_n$            | $B_n\\rightarrow C_n$ Con    | 1.48 | 3.76 | 1.10 | 1.53 |\n| $B_n+S_n+L_n$            | $T_n\\rightarrow S_n$ Agg    | 1.59 | 3.91 | 0.86 | 1.41 |\n| $B_n+T_n+H_n$            | $L_n\\rightarrow H_n$ Nor    | 1.62 | **2.81** | 0.71 | 1.02 |\n| **DSA**                 | -              | **1.17** | 2.85 | **0.62** | **0.96** |\n\nFrom Table 1, our DSA framework yields the largest improvements: 27.8%, 43.6% and 37.3% in key metrics when using the original polynomial assignments. Although the combination $B_n+T_n+H_n$ results in a slightly lower FDE$_1$ (from 2.81 to 2.85, a 1.4% difference), the original DSA configuration still ranks second for FDE$_1$, confirming its overall robustness and effectiveness. \n\n3: Evaluations focus on (1) highway-like scenarios (nuScenes/Waymo). Performance in unstructured environments (e.g., (2) pedestrians, intersections) is unverified.\n\nA: (1) We evaluate our framework on three real-world datasets that span both urban and suburban environments: (i) Boston and Singapore (nuScenes [5]), (ii) Miami and Pittsburgh (Argoverse [6]) and (iii) Phoenix, AZ, Kirkland, and other cities (Waymo [7]). The maps provided in these datasets include diverse road geometries, including standard 3-way and 4-way intersections, as well as T-intersections.\n\n[5] https://www.nuscenes.org/nuscenes.\n\n[6] https://www.argoverse.org/av1.html#overview-link.\n\n[7] https://waymo.com/blog/2019/08/waymo-open-dataset-sharing-our-self.\n\n(2) Our current work focuses on vehicle trajectory prediction. Thank you for your valuable suggestion, we plan to extend our framework to predict the trajectories of other traffic participants, such as pedestrians and cyclists in future work.\n\n4: Degree Adaptation is potentially misleading. The method describes using SMAC3 to solve an optimization problem for the degree $n$ based on validation set performance. This sounds like an offline hyperparameter tuning step that finds a single optimal degree for each driving style, which is then fixed. If the degree is tuned offline, calling it \"adaptation\" is an overstatement. If it is adapted dynamically for each sample during inference, the mechanism and its computational overhead are not explained at all.\n\nA: We will revise Section 3.3.2 to be titled \"Degree Adaptation\". During the training process, our framework learns the optimal polynomial degree for each expert (corresponding to a driving style). During inference, the learned degree is used to determine suitable expert weights, analogous to how attention weights are computed in Transformer architectures."}, {"review_id": "PD54CZIVBE", "reviewer": "Reviewer_Hx7U", "summary": "This paper introduces a novel Driving-Style-Adaptive (DSA) framework for vehicle trajectory prediction that accounts for the variability in human driving behavior. Instead of assuming a fixed driving style, the model categorizes drivers into three types of conservative, aggressive, and normal. It then approximates their behaviors using distinct polynomial families: Bernstein polynomials for conservative, Chebyshev polynomials for aggressive, and Legendre polynomials for normal driving. A mixture of experts mechanism dynamically combines these polynomial experts. The proposed method is evaluated on three benchmark datasets (nuScenes, Argoverse, and Waymo) and demonstrates strong performance.", "questions": "1- For Table 5 in Appendix B.2, could the authors clarify the number of training samples corresponding to each motion type (e.g., turn right, turn left)\n\n2- Are the reported results based on the average of multiple runs, or do they reflect a single best run?", "limitations": "Not discussed in the paper.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Weaknesses: \n\n1- The paper contains several typos that should be corrected. \n\nLine 66: \"similarity\".\n\nLine 132: Should be \"at the expense of safety.\"\n\nLine 223: \"follo\" \n\nLine 269:  Missing \"for\" before \"more\"\n\nLine 283: \"effective\" \n\nLine 288: \"for evaluate\"\n\n2- The paper lacks a visual analysis showing whether the system accurately distinguishes between different driving styles and whether the expert weights appropriately reflect those behaviors.\n\n3- The paper does not appear to discuss training time or provide key implementation details such as optimizer settings, batch size, or number of training epochs. Similarly, the evaluation metrics and dataset characteristics (e.g., size, preprocessing) are not clearly described in the main paper or appendix. \n\n4- No significance tests are presented to assess whether the performance improvements are statistically meaningful. Including such tests would strengthen the empirical claims and support the reported gains more robustly.\n\nStrengths:\n\nPaper is well written, with good flow and clarity. The introduction section is laid out well, and experiments are explained clearly. The method is interesting and novel.", "was_revised": true, "final_justification": "My main concern was whether the proposed method is capable of distinguishing between different driving styles. The authors have addressed this (and other concerns) clearly and sufficiently in their response to my review. I find their clarifications satisfactory and, based on this, I am updating my score to accept.", "rebuttal": "We greatly appreciate the Hx7U's detailed and constructive comments. These would help us better position our work.\n\nQuestion 1: Clarify the number of training samples corresponding to each motion type in Table 5 Appendix B.2.\n\nA: We present the rate of each motion class in Table 1, where the motion definitions follow the *ClassifyTrack* class schema of the Waymo dataset*.\n\nTable 1: The b-minFDE values and motion occurrence rates on the nuScenes dataset. The **best** results are highlighted.\n\n| Method Type | Stationary | Straight | Straight right | Straight left | Right U-turn | Right-turn | Left U-turn | Left-turn |\n|:------------|:----------:|:--------:|:--------------:|:-------------:|:------------:|:----------:|:-----------:|:---------:|\n| MTR         | 2.15       | 2.58     | **4.85**        | 4.26          | 8.13         | 4.82       | **5.17**     | **4.85**   |\n| DSA         | **2.03**   | **2.48** | 4.96           | **4.17**      | **8.11**      | **4.77**    | 5.28         | 4.87      |\n| Rate        | 66.94%     | 12.76%   | 6.56%          | 6.25%         | 0.11%        | 3.49%      | 0.16%        | 3.73%     |\n\nThe weighted average b-minFDE is 2.719 for MTR and 2.627 for our framework, indicating that our method improves prediction accuracy by a margin of 0.092.\n\n*Waymo: https://github.com/waymo-research/waymo-open-dataset.\n\nQ2: Are the reported results based on the average of multiple runs, or do they reflect a single best run?\n\nA2: The result are the average value under five (Argoverse, nuScenes) and three (Waymo) seeds. The whole results are shown in Table 2.\n\nTable 2. Multi seeds results in Argoverse (A), nuScenes (N) and Waymo (W). The metrics are listed by average ± standard.\n|     Dataset    |   ADE$_1$    |   FDE$_1$    |   ADE$_5$    |   FDE$_5$    |  ADE$_{10}$   |  FDE$_{10}$   |\n|:-------------:|:---------:|:---------:|:---------:|:---------:|:--------:|:--------:|\n|       N       | 2.69±0.08 | 6.47±0.15 | 1.21±0.07 | 2.74±0.03 | 0.85±0.02 | 2.00±0.01 |\n|    **Dataset**    |   **A-ADE**$_1$    |   **FDE**$_1$    |   **ADE**$_6$    |   **FDE**$_6$    | **W-minADE**   | **W-minFDE**   |\n|     A & W     | 1.17±0.05 | 2.85±0.11 | 0.62±0.02 | 0.95±0.01 | 0.5852±0.0152 | 1.1431±0.0227 |\n\nWeakness 1: Several typos.\n\nA: We will modify it in final version.\n\nW2: The paper lacks a visual analysis showing whether the system accurately (1) distinguishes between different driving styles and whether the (2) expert weights appropriately reflect those behaviors.\n\nA: (1) Based on the characteristics of the three driving styles [1], we compute statistical measures for the datasets (Table 3). Specifically, we report the mean and variance (var) for key motion indicators, including velocity ($v$), acceleration ($a$), space headway ($S_{h}$), and time headway ($T_{h}$).\n\nTable 3. Statistical characteristics of driving styles in the Argoverse (A-Metric) and nuScenes (N-Metric) datasets.\n\n| A-Metric       | Con-Mean | Var    | Agg-Mean | Var    | Nor-Mean | Var    |\n|:-------------:|:--------:|:------:|:--------:|:------:|:--------:|:------:|\n| $v$           | 7.071    | 9.710  | 8.753    | 14.799 | 8.268    | 12.691 |\n| $a$           | 0.010    | 9.095  | 0.067    | 12.710 | -0.022   | 17.481 |\n| $S_{h}$       | 8.600    | 51.961 | 12.827   | 106.715| 11.040   | 86.566 |\n| $T_{h}$       | 1.859    | 4.936  | 2.177    | 15.911 | 2.041    | 9.669  |\n\nNote:\n$S_h$: Spacing provides the distance between the front centre of a vehicle to the front center of the preceding vehicle.\n\n$T_h$: The time to travel from the front center of a vehicle to the front center of the preceding vehicle.\n\n[1] Zheng X, et al. Real‐time driving style classification based on short‐term observations[J]. IET Communications, 2022.\n\n(2) To reflect the relationship between expert weights and driver behaviors (styles), we compute the **correct matching rate**, defined as the proportion of cases where the highest expert weight (Top-1) corresponds to the expected polynomial $p_n$ under each driving style. The results are reported in Table 4, where the first letter denotes each driving style.\n\nTable 4. The matching relationship between different driving styles and their corresponding $p_n$. The percentage indicates the rate at which each $p_n$ has the highest weight within each driving style.\n\n| style | Argoverse | nuScenes | Top-1 |\n|:-----:|:---------:|:--------:|:-----:|\n| Con   | 86.15%    | 87.96%   | $B_n$    |\n| Agg   | 92.87%    | 94.26%   | $T_n$    |\n| Nor   | 80.04%    | 83.56%   | $L_n$    |  \n\nW3: (1) Training time or provide key implementation details. \n\nA: (1) The average training time per epoch (in minutes:seconds) across the three datasets is reported in Table 3. The second column indicates the number of GPUs used and their specific model (NVIDIA).\n\nTable 3. Average training time per epoch for each dataset.\n\n| Datasets   | No.-GPU |    Times     |\n|:----------:|:-------:|:------------:|\n| Argoverse  |  1-4090 |   1min 20s    |\n| nuScenes   |  1-4091 |   6min 52s   |\n| Waymo      |  2-4090 |  56 min 45 s |\n\nKey implementation details: The model is implemented using PyTorch. We employ single-layer KANs with three expert networks (TopK, K=3), and polynomial degrees ranging from 1 to 10. The initial weights for the distance loss and MoE-$K$ loss are both set to 0.5. Further implementation details will be provided in the final version.\n\nW3: The (2) evaluation metrics and (3) dataset characteristics (e.g., size, preprocessing).\n\nA(2): We will include the mathematical definitions in final version. Specifically, \n\n-- Average / Final Displacement Error (ADE / FDE): \n\nADE $=\\frac{1}{T} \\sum\\nolimits^{T}_{t=1} \\| \\widetilde{p_{t}} -p_{t}\\|_{L^{2}}$ , FDE $=\\| \\widetilde{p_{T}} -p_{T}\\|_{L^{2}} .$ \n\nWhere $T$ represents the prediction length. For $\\widetilde{p_{t}}$ and $p$ are the predicted and ground truth $(x,y)$ positions respectively. \n\n-- Missing Rate (MR): The ratio of whether the Euclidean distance between the $\\widetilde{p_{t}}$ and $p$ at the FDE for all predictions is larger than 2m.\n\n-- Brier-minFDE ($b$-minFDE): $\\text{bminFDE} = \\text{minFDE} + (1 - p)^2$, where $p$ is the probability of probability of the best forecasting trajectory with minimum endpoint error (minFDE).\n\n(3) We preprocess the three datasets using the official pip packages provided by their respective baselines. The characteristics of each dataset are summarized in Table 4. The preprocessing procedures are described below.\n\nTable 4. Characteristics of the evaluation datasets.\n\n| Datasets      | Collect                    | Size    | Library                     | Select                 |\n|:-------------:|:--------------------------:|:-------:|:---------------------------:|:----------------------:|\n| Argoverse [2] | Miami and Pittsburgh       |    23.69G     | Argoverse API / devkit      | Motion Forecasting     |\n| nuScenes [3]  | Boston and Singapore       |    4.81G     | nuscenes-devkit             | Motional-Full dataset  |\n| Waymo [4]     | Phoenix, AZ, Kirkland etc. | 83.50G  | waymo-open-dataset [5]      | Motion1.1-scenario     |\n\n[2] https://www.argoverse.org/index.html.\n\n[3] https://www.nuscenes.org/nuscenes.\n\n[4] https://waymo.com/open/.\n\n[5] https://github.com/waymo-research/waymo-open-dataset.\n\nPreprocessing is based on the corresponding official libraries. \nFirst, extract trajectory features (position, velocity, acceleration) and corresponding map elements (e.g., lane lines, traffic lights).\nThen, align them by timestep to ensure data integrity and output.\n\nW4: Significance tests to assess whether the performance improvements are statistically meaningful.\n\nA: To verify the robustness of our improvements over the baseline, we conduct two analyses: (1) comparison against existing methods and (2) self-evaluation under multiple random seeds.\n\n(1) We perform $t$-tests against Context-Aware [6], the most recent baseline method with publicly available code. For fairness, we use the metrics reported in their original paper. The results are presented in Table 5.\n\nTable 5. Multi-seed results on the nuScenes dataset. \"No.\" indicates the seed value. Columns 2–5 show baseline [6] results, and the last four columns report our results.\n\n|  No.  | Baseline [6]-ADE$_1$ | FDE$_1$ | ADE$_5$ | FDE$_5$ | Ours-ADE$_1$ | FDE$_1$ | ADE$_5$ | FDE$_5$ |\n|:-----:|:---------------:|:-------:|:-------:|:-------:|:------------:|:-------:|:-------:|:-------:|\n|   1   | 3.7141          | 1.6779  | 8.7896  | 3.5351  | 2.6286       | 6.6382  | 1.1055  | 2.7691  |\n|   2   | 3.6947          | 1.6733  | 8.7193  | 3.4978  | 2.6908       | 6.2563  | 1.2898  | 2.7164  |\n|   3   | 3.7732          | 1.6842  | 8.9267  | 3.5500  | 2.6556       | 6.4614  | 1.1795  | 2.7256  |\n|   4   | 3.7670          | 1.6646  | 8.8636  | 3.4412  | 2.8260       | 6.4106  | 1.2446  | 2.7754  |\n|   5   | 3.6500          | 1.6701  | 8.5636  | 3.4879  | 2.6475       | 6.5800  | 1.2297  | 2.7115  |\n\nAll four metrics yield $p$-values less than 0.01, indicating that our method significantly outperforms the baseline.\n\n(2) Meanwhile, the evaluation metrics presented in Table 6 are averaged over five random seeds for Argoverse and nuScenes and three seeds for Waymo.\n\nTable 6. Multi-seed results on Argoverse (A), nuScenes (N), and Waymo (W). Metrics are reported as mean ± standard deviation.\n|     Metric    |   ADE$_1$    |   FDE$_1$    |   ADE$_5$    |   FDE$_5$    |  ADE$_{10}$   |  FDE_{10}$   |\n|:-------------:|:---------:|:---------:|:---------:|:---------:|:--------:|:--------:|\n|       N       | 2.69±0.08 | 6.47±0.15 | 1.21±0.07 | 2.74±0.03 | 0.85±0.02 | 2.00±0.01 |\n|    Metric     |   ADE$_1$    |   FDE$_1$    |   ADE$_6$    |   FDE$_6$    | minADE   | minFDE   |\n|     A & W     | 1.17±0.05 | 2.85±0.11 | 0.62±0.02 | 0.95±0.01 | 0.5852±0.0152 | 1.1431±0.0227 |\n\n[6] Pei Xu, et al. Context-aware timewise vaes for real-time vehicle trajectory prediction. IEEE Robotics and Automation Letters, 2023."}, {"review_id": "V5NsoGsQCh", "reviewer": "Reviewer_5qot", "summary": "This paper proposes a Driving-Style-Adaptive (DSA) Framework by leveraging Kernel-based Adaptive Networks (KANs) with polynomial basis operators tailored to different driving styles. Bernstein, Chebyshev, and Legendre polynomials are used to model conservative, aggressive, and normal driving behaviors, respectively. A Mixture-of-Experts (MoE-TopK) strategy enables dynamic combination of these styles, reflecting real-world behavioral complexity. SMAC3 is used for hyperparameter optimization, particularly to choose effective polynomial degrees. Experiments on nuScenes, Argoverse, and Waymo show notable improvements over baselines.", "questions": "1. How are multi-modal predictions generated during experiments? Are multiple hypotheses sampled, or is a single deterministic output used for evaluation?\n2. Is the polynomial degree fixed or adaptively updated during training? How is the chosen degree used during inference?\n3. Additional experiments would strengthen the paper. For instance, it is not demonstrated whether specific polynomial operators (e.g., Bernstein, Chebyshev, Legendre) consistently correspond to particular driving styles in practice. An interpretability analysis that shows the expert contributions in different scenarios—especially those exhibiting distinct styles—would help validate the core claim. Furthermore, it would be valuable to explore whether increasing the diversity of basis operators (e.g., adding more types of polynomials) leads to better performance or simply increases model complexity without clear gains.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. Integrate KANs into vehicle trajectory prediction with explicit modeling of driving styles.\n2. The use of distinct polynomial bases is conceptually linked to characteristics of driving behaviors (e.g., Bernstein’s smoothness for conservative driving).\n3. MoE-TopK allows dynamic combination of styles, enhancing the model’s ability to handle diverse, multimodal behaviors.\n4. The experiments are extensive. The framework is validated on three datasets and includes qualitative and ablation studies.\n\nWeaknesses:\n1. The model's interpretability is limited, as the contribution of each polynomial expert is not visualized or analyzed across scenarios, making it unclear if driving styles are explicitly learned or used during inference.\n2. Despite the focus on modeling driving styles, the evaluation does not include scenarios that clearly reflect variations in driver behavior, missing an opportunity to demonstrate the model’s key advantage.\n3. The justification for using Legendre polynomials to represent \"normal\" driving style is vague and overlaps with the conservative case, weakening the conceptual distinction between the styles.\n4. The paper omits discussion of training and inference time, which is important given the added complexity of the mixture-of-experts architecture and degree selection procedure.", "was_revised": true, "final_justification": "The rebuttal answered most of my questions.", "rebuttal": "We sincerely appreciate the 5qot's insightful comprehensively consider multiple aspects of our vehicle trajectory work. \n\nQuestion1: How are multi-modal predictions generated during experiments? Are multiple hypotheses sampled, or is a single deterministic output used for evaluation?\n\nA: Generating multi-modal predictions is equivalent to producing $k$ distinct trajectory hypotheses. This is reflected by the subscript $k$ in the evaluation metrics. By independently repeating the stochastic generation process $k$ times, the model outputs $k$ diverse trajectories, from which the best prediction is selected based on the evaluation criterion.\n\nQ2: Is the polynomial degree fixed or adaptively updated during training? How is the chosen degree used during inference?\n\nA: The degree is adaptive during training but fixed during inference. In the training phase, our framework learns the optimal degree for each expert (corresponding to a specific driving style). During inference, these learned degrees are used in combination to obtain appropriate expert weights.\n\nQ3 & Weaknesses 2: (1) Despite the focus on modeling driving styles, whether specific polynomial operators consistently correspond to particular driving styles in practice, missing an opportunity to demonstrate the model's key advantage.\n\nA: We calculate the cosine similarity between the trajectory corresponding to the highest-weighted polynomial $p_n$ and the predefined driving style standards [1-4], which include Conservative (Con), Aggressive (Agg) and Normal (Nor) categories. The results on the nuScenes dataset are presented below:\n\nTable 1. Cosine similarity reflecting consistent matching between polynomial based predictions and driving style standards on the nuScenes dataset.\n| $p_n$   |   Con   |   Agg   |   Nor   |\n|:--------:|:-------:|:-------:|:-------:|\n| $B_n$   |  0.953  |  0.158  |  0.489  |\n| $T_n$   |  0.266  |  0.987  |  0.353  |\n| $L_n$   |  0.310  |  0.263  |  0.992  |\n\n[1] Yao X, Calvert S C, Hoogendoorn S P. Driving heterogeneity identification using machine learning: A review and framework for analysis[J]. Transportation Research Interdisciplinary Perspectives, 2025.\n\n[2] Zheng X, Yang P, Duan D, et al. Real-time driving style classification based on short‐term observations[J]. IET Communications, 2022.\n\n[3] Garrosa M, Olmeda E, Fuentes del Toro S, et al. Holistic vehicle instrumentation for assessing driver driving styles[J]. Sensors, 2021.\n\n[4] Xu L, Hu J, Jiang H, et al. Establishing style-oriented driver models by imitating human driving behaviors[J]. IEEE Transactions on Intelligent Transportation Systems, 2015.\n\nQ3: (2) Furthermore, it would be valuable to explore whether increasing the diversity of basis operators (e.g., adding more types of polynomials) leads to better performance or simply increases model complexity without clear gains.\n\nA: We incorporate additional KAN variants: TaylorKAN [5], JacobiKAN [6], FourierKAN [7] and WaveKAN [8] into our KANlayer named \"multi\". The performance of these basis function combinations under the TopK=3 MoE setting is evaluated on the nuScenes dataset. The corresponding results are summarized in Table 2. \n\nTable 2. Comparison of multiple basis operators with our proposed method on the nuScenes dataset under the MoE TopK=3 setting.\n\n| Method | ADE$_{1}$ | FDE$_{1}$ | ADE$_{5}$ | FDE$_{5}$ | ADE$_{10}$ | FDE$_{10}$ |\n|:------:|:---------:|:---------:|:---------:|:---------:|:----------:|:----------:|\n| multi  |   2.81    |   **6.44**    |   1.46    |   **2.72**    |   0.91     |   2.08     |\n| Ours   |   **2.69**    |   6.47    |   **1.21**    |   2.74    |   **0.85**     |   **2.00**     |\n\nFrom Table 2, although the values of FDE$_1$ and FDE$_5$ show only marginal increases (at the thousandth level), all other metrics exhibit noticeable performance degradation compared to our proposed method. Due to the larger pool of experts, some are selected less frequently during training resulting in undertraining. This imbalance further leads to overfitting in the gating network, ultimately compromising the generalization ability of the model.\n\n[5] https://github.com/Muyuzhierchengse/TaylorKAN.\n\n[6] https://github.com/SpaceLearner/JacobiKAN.\n\n[7] https://github.com/GistNoesis/FourierKAN.\n\n[8] https://github.com/cconut/WaveKANet.\n\nWeaknesses 1: The model's interpretability is limited, as the contribution of each polynomial expert is not visualized or analyzed across scenarios, making it unclear if driving styles are explicitly learned or used during inference.\n\nA: We compute the proportions of each polynomial expert p_n being selected as the Top-1 weight as shown in Table 3. Here, $p_n$ consists of the Bernstein $B_n$, Chebyshev $T_n$, and Legendre $L_n$ polynomials.\n\nTable 3. Top-1 selection proportions for each polynomial expert.\n\n| $p_n$  | Argoverse | nuScenes |\n|:------:|:---------:|:--------:|\n| $B_n$ | 45.47% | 39.22% |\n| $T_n$ | 25.52% | 20.57% |\n| $L_n$ | 29.01% | 40.21% |\n\nThese results indicate that all experts are utilized, with none having a selection rate of zero.\n\nW3: The justification for using Legendre polynomials to represent \"normal\" driving style is vague and overlaps with the conservative case, weakening the conceptual distinction between the styles.\n\nA: We justify the use of Legendre polynomials $L_n$ to represent the Normal driving style from two perspectives: i) empirical evidence and 2) mathematical proposition analysis.\n\n## Experimental Results\n\nWe conduct ablation experiments by replacing the $L_n$, originally used for Normal (Nor) drivers with Hermite polynomials $H_n$, and replacing the Bernstein polynomials $B_n$ (for Conservative drivers) with Charlier polynomials $C_n$. The performance comparison is reported in Table 4 (Argoverse, A-Method) and Table 5 (nuScenes, N-Method), respectively.\n\nTable 4. Replacement results on the Argoverse dataset (**best** results in bold).\n\n|   A-Method    | ADE1 | FDE1 | ADE6 | FDE6 |\n|:----------:|:----:|:----:|:----:|:----:|\n| Cn for Con | 1.48 | 3.76 | 1.10 | 1.53 |\n| Hn for Nor | 1.62 | 2.87 | 0.71 | 1.02 |\n|   ours     | **1.17** | **2.85** | **0.62** | **0.96** |\n\nTable 5. Replacement results on the nuScenes dataset (**best** results in bold).\n\n|  N-Method   | ADE1 | FDE1 | ADE5 | FDE5 | ADE10 | FDE10 |\n|:----------:|:----:|:----:|:----:|:----:|:-----:|:-----:|\n| Cn for Con | 2.79 | 6.69 | 1.32 | 2.81 | **0.79**  | 2.08  |\n| Hn for Nor | 2.81 | 6.53 | 1.48 | 2.94 | 1.01  | 2.23  |\n|   **ours** | **2.69** | **6.47** | **1.21** | **2.74** | 0.85  | **2.00** |\n\nAcross the two datasets, our original configuration achieves 9 out of 10 best results. The only exception is a marginal 0.06 gap in ADE$_{10}$ on the nuScenes dataset. These results strongly support the validity of the proposed matching between driving styles and polynomial basis. Specifically, using $L_n$ for Normal drivers and $B_n$ for Conservative drivers.\n\n## Mathematical Proposition analysis \n**Conservative drivers** typically maintain the lowest average speed among all driving styles and exhibit minimal behavioral variation. Their trajectories are characterized by smoothness and stability, with few abrupt changes. The uniform convergence property of $B_n$ ensures that the approximation error diminishes evenly across the interval, making them well-suited for modeling such stable, predictable trajectories.\n\nIn contrast, **normal drivers** exhibit more regular but moderate changes in behavior. Their trajectories are generally smooth with consistent fluctuations, which are neither overly abrupt nor completely static. The $L_n$ known for their orthogonality and optimal approximation under the $L^2$ norm, are particularly effective for capturing such patterns. Their ability to represent smooth variations with computational efficiency makes them a suitable choice for approximating the trajectories of Normal drivers."}, {"review_id": "40H77CPA63", "reviewer": "Reviewer_3sDS", "summary": "The paper proposes an adaptive framework for predicting trajectories of vehicles of interest by incorporating their driving behavior into trajectory prediction. Incorporating driving behavior is achieved by pre-selected basis functions for the KAN-based method. These basis functions capture the driving styles of conservative, aggressive and normal drivers and are polynomially combined by the proposed method to capture “mixed” driving styles. Extensive comparisons with the SOTA and a comprehensive ablation study demonstrate the method's capability for driving-behavior-based prediction of trajectories", "questions": "S1: Add error bars to your results to demonstrate the stability of the method. This will further improve the quality of the evaluation presented.\n\nS2: Integrate a summary of the limitations into the main text for a more conclusive understanding of the presented work. \n\nS3: Figure 2 is crucial for understanding the method. To increase clarity and ensure that the flow from input to output is immediately apparent, the authors may consider improving Figure 3 (e.g., numbering the individual components or reorganizing the arrows, or some better solution) to illustrate the sequence of operations within the method in an improved manner.\n\nQ1: The proposed method seems to use historical trajectories to incorporate the driving situation to some extent. However, the driving style is strongly influenced by a broader context. Have the authors considered incorporate the current situation in a more comprehensive manner?\n\nQ2: The authors point out in the appendix (limitations section) that more driving styles can be included: Have the authors considered scaling the proposed method by identify and incorporate driving styles in an automatic manner rather than predefining the driving styles?", "limitations": "No, although the authors have discussed the limitations in detail in the appendix, in the reviewer's view the limitations should be discussed in the main text (e.g. in the “Summary” section). Including a summary of the limitations discussed in the appendix in the main text with the most important limitations would be appropriate.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "$\\textbf{Quality:}$\n\n$\\textit{Strengths:}$ The submission is presented in a technically sound manner that is further extended in the appendix. Theoretical claims are supported by comprehensive experiments on relevant datasets (Waymo, nuScenes, Agroverse) and are compared to various baselines that capture the field sufficiently. \n\n$\\textit{Weaknesses:}$ Although the evaluation is comprehensive, it lags error bars on the presented results. The authors argue that prior work is followed by not incorporating error bars. However, this should not be an argument for not incorporating error bars to, e.g., assess the stability of the achieved results and the proposed components. The reviewer suggests adding error bars to improve the quality of evaluation further. The review hopes this is achievable in the remaining time but acknowledges the limits of the timeframe. Also, the authors should be a bit more careful about including the weaknesses/limitations of the method in the main text. Limitations are presented in detail in the appendix but should (in the eyes of the reviewer) be incorporated in the main text to present a conclusive picture of the proposed method.\n\n\n$\\textbf{Clarity:}$ \n\n$\\textit{Strengths:}$ The writing of the paper is clear and good to understand. Although the paper is heavier on math, the reasoning behind the method can be followed throughout the entire text. The results are presented in well-organized tables and figures.\n\n$\\textit{Weaknesses:}$ Figure 2 could be improved to increase clarity: The flow from input to output within the DSA framework is not clear at first glance. Numbering the individual components or restructuring the arrows could be a way to improve the flow and to understand when which part of the method is performed.\n\nFurthermore, there are small formatting errors that can be improved. Here are some examples that the reviewer found:\n\n-\tLine 76, Eq (2), Line 77, Line 83, Line 95, Figure 4: Remove the yellow color \n\n-\tLine 77: Additional space after the citation [39,40]\n\n-\tFigure 2 description: Missing character (ndicates)\n\n-\tLine 186:  two times “.”\n\n\n$\\textbf{Significance/Originality:}$ \n\n$\\textit{Strengths:}$ Driving styles have a significant impact on ego vehicles and other traffic participants. Various studies investigate driving behavior for autonomous driving, often showing that the impact of driving behavior should not be neglected. Therefore, the reviewer thinks that incorporating driving behavior into deep learning methods is a solid way to further advance the field. The combination of KANs and basic functions to achieve this incorporation is well-articulated by the authors and it will be interesting to see how further work extends this approach.\n\n$\\textit{Weaknesses:}$ The driving styles of traffic participants depend on the current temporal situation. For the proposed method, this situation is incorporated in a limited way through the historical trajectory of the vehicles (as far as the reviewer understands). A more comprehensive inclusion of the current situation (e.g. through a visual component that encodes the current situation (surroundings, weather, etc.) may have a high impact on driving style prediction. The reviewer understands that this probably lies beyond the scope of the paper. Therefore, this could be discussed in the limitations / future work section.", "was_revised": true, "final_justification": "Due to the thorough rebuttal, which significantly improved the work, and the detailed answers to the questions raised during the discussion, the reviewers' confidence in accepting the article increased during the discussion phase. Issues regarding error bars, the relocation of important limitations from the appendix to the main text, and minor issues regarding the overall clarity of the article were resolved during this phase. The error bars and the additional experiments to clarify the pre-defined driving style using clustering had the greatest impact on the reviewers' positive assessment of the rebuttal.", "rebuttal": "We are very grateful for the 3sDS's professional and meticulous comments. They have been instrumental in refining our presentation and strengthening the paper's scientific contributions.\n\nS1: Add error bars to your results to demonstrate the stability of the method. \n\nA: The evaluation metrics reported in Table 1 represent the mean ± standard deviation over five random seeds for the Argoverse (A) and nuScenes (N) datasets, and over three seeds for the Waymo (W) dataset.\n\nTable 1. Multi-seed evaluation results on Argoverse (A), nuScenes (N), and Waymo (W). Metrics are reported as mean ± standard deviation.\n\n|     Dataset    |   ADE$_1$    |   FDE$_1$    |   ADE$_5$    |   FDE$_5$    |  ADE$_{10}$   |  FDE$_{10}$   |\n|:-------------:|:---------:|:---------:|:---------:|:---------:|:--------:|:--------:|\n|       N       | 2.69±0.08 | 6.47±0.15 | 1.21±0.07 | 2.74±0.03 | 0.85±0.02 | 2.00±0.01 |\n|    **Dataset**    |   **A-ADE**$_1$    |   **FDE**$_1$    |   **ADE**$_6$    |   **FDE**$_6$    | **W-minADE**   | **W-minFDE**   |\n|     A & W     | 1.17±0.05 | 2.85±0.11 | 0.62±0.02 | 0.95±0.01 | 0.5852±0.0152 | 1.1431±0.0227 |\n\nS2: Integrate a summary of the limitations into the main text for a more conclusive understanding of the presented work.\n\n(2) We will add the limitations in the main paper in the final version as follows:\n\n**Limitation**  We evaluate our method on prediction horizons up to 9 seconds (1 second of history and 8 seconds of future), which is the longest duration available in current open-access vehicle trajectory datasets. However, for significantly longer horizons (e.g., over one minute), direct long-term prediction may be unreliable and would likely require segment-wise modeling or hierarchical strategies.\nIn addition, external factors such as strong conditions (e.g., traffic signals and regulatory constraints) and soft conditions (e.g., weather, which is often unlabeled in current datasets) can also affect trajectory prediction. Incorporating these contextual cues remains an important direction for future work. \n\nS3: the authors may consider improving Figure 2 to illustrate the sequence of operations within the method in an improved manner.\n\nA: We will revise Figure 2 by incorporating arrows to enhance clarity. On the left side, arrows will connect the highest-weighted experts in the histogram to their corresponding driving styles. On the right side, arrows will indicate the association between the predicted degrees and their respective polynomial bases.\n\nQ1 & Significance/Originally Weaknesses: Have the authors considered incorporate the current situation in a more comprehensive (the current situation e.g.surroundings, weather, etc.) manner? \n\nA: Driving styles reflect the behavioral characteristics of a driver [1-4], which typically remain stable over short temporal spans. Existing vehicle trajectory prediction datasets record trajectories of less than 10 seconds (Table 2).\nHowever, as you pointed out, how driving styles evolve in more comprehensive or long-term scenarios is indeed a compelling and valuable question. To address this issue, we plan to extend our work in future research as follows:\n\n## Trajectory slicing\nFor long-term vehicle trajectory prediction tasks (e.g., over 1 minute), we propose to segment the full trajectory into multiple shorter slices. Our DSA framework will be applied to each slice individually, and the final trajectory will be reconstructed by composing these predicted slices sequentially.\n\n## Fine-grained learning\nThe trajectory slices in above can be segmented with either equal or variable lengths. For segments exhibiting similar behavioral patterns, we may merge them into longer slices to reduce redundancy. Conversely, irregular or highly dynamic segments can be treated with greater attention, enabling fine-grained feature learning and more accurate predictions in critical scenarios.   \n\n## Surrounding environment\nAlthough existing datasets do not fully align contextual information (e.g., other traffic participants, weather conditions) with vehicle trajectories, we can incorporate available partial contexts and treat them as impact conditions for future feature learning. With the development of more comprehensive datasets, these contextual factors can be leveraged to enhance the accuracy and generalization capabilities of our framework. \n\nTable 2: Comparison of existing vehicle trajectory datasets. \n\"His\" and \"Pre\" denote the durations of the historical and predicted trajectories, respectively, while \"Total\" refers to the overall recorded duration per vehicle.\n\n| Datasets         | Collect Locations                | His | Pre | Total |\n|:----------------:|:--------------------------------:|:---:|:---:|:-----:|\n| KITTI [5]        | Karlsruhe                        |  2  |  4  |   6   |\n| Apolloscapes [6] | Beijing, ShangHai and SHenZhen   |  3  |  3  |   6   |\n| nuScenes [7]     | Boston and Singapore             |  2  |  6  |   8   |\n| Argoverse [8]    | Miami and Pittsburgh             |  2  |  3  |   5   |\n| INTERACTION [9]  | China, Germany and Bulgaria      |  1  |  3  |   4   |\n| InD [10]         | German                           | 3.2 | 4.8 |   8   |\n| RounD [11]       | Aachen                           |  2  |  4  |   6   |\n| HighD [12]       | German                           | 2.8 | 2.8 |  5.6  |\n| Waymo [13]       | USA                              |  1  |  8  |   9   |\n\n[1] Yao X, et al. Driving heterogeneity identification using machine learning: A review and framework for analysis[J]. TRIP, 2025.\n\n[2] Zheng X, Yang P, et al. Real-time driving style classification based on short-term observations[J]. IET Communications, 2022.\n\n[3] Garrosa M, et al. Holistic vehicle instrumentation for assessing driver driving styles[J]. Sensors, 2021.\n\n[4] Xu L, Hu J, et al. Establishing style-oriented driver models by imitating human driving behaviors[J]. IEEE TITS, 2015.\n\n[5] https://www.cvlibs.net/datasets/kitti/.\n\n[6] https://apolloscape.auto/.\n\n[7] https://www.nuscenes.org/nuscenes.\n\n[8] https://www.argoverse.org/.\n\n[9] https://interaction-dataset.com/.\n\n[10] https://levelxdata.com/ind-dataset/.\n\n[11] https://levelxdata.com/round-dataset/.\n \n[12] https://levelxdata.com/highd-dataset/.\n\n[13] https://waymo.com/open/.\n\nQ2: Have the authors considered scaling the proposed method by identify and incorporate driving styles in an automatic manner rather than predefining the driving styles?\n\nA: We consider adopting (1) semi-supervised and (2) unsupervised method for automatic categorization, and (3) evaluating predefined categories under multiple values beyond our proposed three-category setting.\n\n(1) **Semi-supervised approach**. Based on a set of partially predefined driving styles, we define a similarity threshold $c_{\\text{style}}$ for each style. Trajectories with similarity scores below the corresponding $c_{\\text{style}}$ are assigned to that predefined style. In contrast, those exceeding the threshold are categorized as newly discovered or blended styles, enabling further refinement and enrichment of the driving style taxonomy.\n\n(2) **Unsupervised approach**. We extract trajectory features such as speed, acceleration and other motion attributes. Then, we employ deep clustering-based methods, such as SharPer [14] and SMYRF [15] to automatically partition the driving styles into distinct categories. \nThe optimal number of categories is determined using standard clustering evaluation metrics [16].\n\n\n(3) We evaluate multiple $k$-category settings using K-means clustering, and report the corresponding metrics (log-normalized except for Silhouette) in Table 3.\n\nTable 3. Clustering evaluation results on Argoverse (A) and nuScenes (N). A value of “1” indicates the best performance among all settings. \n\n| $k$ | A-$\\Delta$WSS | Silhouette | $\\Delta$CHI | N-$\\Delta$WSS | Silhouette | $\\Delta$CHI |\n|:---:|:-------------:|:----------:|:-----------:|:-------------:|:----------:|:-----------:|\n|  2  |               | 0.919      |             |               | 0          |             |\n|  3  | 1             | 1          | 0.810       | 1             | 0.700      | 1           |\n|  4  | 0.743         | 0.986      | 1           | 0.586         | 1          | 0.387       |\n|  5  | 0.022         | 0.283      | 0           | 0.287         | 0.268      | 0.324       |\n|  6  | 0             | 0          | 0.601       | 0             | 0.398      | 0           |\n\nOur three-category configuration yields the highest number of best scores across the metrics, supporting the validity and rationality of our chosen driving style taxonomy.\n\nNote: For all metrics listed below, larger values indicate better clustering performance.\n\nNote: For all metrics listed below, larger values indicate better clustering performance.\n\n-$\\Delta$WSS (Within-Cluster Sum of Squares): Measures the improvement in intra-cluster compactness. A higher value suggests tighter grouping of samples within clusters after clustering.\n\n-Silhouette: Reflects both the cohesion within a cluster and the separation between clusters. A higher silhouette score indicates that samples are well matched to their own cluster and poorly matched to neighboring clusters.\n\n-$\\Delta$CHI (Calinski–Harabasz Index): Captures the variation in inter-cluster separability and intra-cluster compactness. Higher values indicate better-defined and more distinct clusters.\n\n[14] Amiri M J, Agrawal D, El Abbadi A. Sharper: Sharding permissioned blockchains over network clusters[C]//sigmod. 2021.\n\n[15] Daras G, Kitaev N, Odena A, et al. Smyrf-efficient attention using asymmetric clustering[J]. NIPS, 2020.\n\n[16] Vysala A, Gomes J. Evaluating and validating cluster results[J]. arXiv, 2020.\n\nSmall Formatting Corrections:\n\n-Remove the highlight on $\\psi \\left( x \\right)$, the extra comma (Line 186), and the extra space (Line 77). \n\n-Figure 2: Correct to …while the arrow \"indicates\" the direction (history or future)…"}]

Author rebuttal:

Meta-review: This paper proposes the Driving-Style-Adaptive (DSA) framework for vehicle trajectory prediction, addressing the limitation that existing methods fail to account for trajectory heterogeneity from different driving styles. The core claim is that trajectories can be more accurately predicted by modeling driving behaviors through style-specific polynomial basis functions within a Kolmogorov-Arnold Network (KAN) architecture. The framework categorizes driving into three types: conservative (Bernstein polynomials), aggressive (Chebyshev polynomials), and normal (Legendre polynomials), combined through a mixture-of-experts mechanism. Experimental results show state-of-the-art performance across three datasets with 80-94% correct style-to-expert matching rates and statistically significant improvements over baselines.

The main strengths are as follows: (1) Novelty: the systematic integration of driving style modeling into trajectory prediction represents a conceptually sound advancement. The theoretical matching between polynomial properties and driving characteristics provides mathematical justification for the approach. (2) Comprehensive experimental validation: the evaluation spans three major real-world datasets with diverse geographic and traffic conditions. Multi-seed experiments with error bars demonstrate result stability, and statistical significance testing confirms improvements over baselines. (3) Interpretability: unlike black-box approaches, the framework provides explainable insights through expert weight analysis, showing which driving styles contribute to specific predictions. (4) Thorough ablation studies: The paper includes extensive ablation experiments validating the choice of polynomial types, the three-expert configuration, and the degree adaptation mechanism. (5) Strong empirical results: Consistent improvements across multiple metrics with appropriate statistical validation demonstrate the framework's effectiveness.

The weaknesses are as follows:  (1) Lack of clarity regarding the concrete implementation of the core components. (2) Some descriptions are misleading. For example, "degree adaptation" is potentially misleading. (3) The paper lacks a visual analysis, and affects the model's interpretability. The contribution of each polynomial expert is not visualized or analyzed across scenarios, making it unclear if driving styles are explicitly learned or used during inference. (4) The paper does not appear to discuss training time or provide key implementation details.  (5) Lack of tests to assess whether the performance improvements are statistically meaningful. Including such tests would strengthen the empirical claims and support the reported gains more robustly. (6) Although the evaluation is comprehensive, it lacks error bars on the presented results. 

The reasons for acceptance: (1) The paper addresses a genuine limitation in existing trajectory prediction methods by incorporating driving style heterogeneity. The approach is technically sound and demonstrates measurable improvements. (2)  The comprehensive evaluation methodology, including multi-seed experiments, statistical significance testing, and extensive ablation studies, provides strong evidence for the claims. (3) The interpretability aspect adds significant value for safety-critical autonomous driving applications where understanding predictions rationale is crucial. (4) The main concerns of reviewers have been well addressed in the rebuttal period.

The discussion during rebuttal: (1) Reviewer ejCn raised concerns about implementation details, categorical driving styles, and misleading "degree adaptation" terminology. The authors provided detailed mathematical formulations and theoretical justification for polynomial choices, though fundamental limitations in categorical modeling and terminology issues remain. (2)  Reviewer Hx7U identified issues with typos, missing visual analysis, and lack of significance testing. The authors addressed these with multi-seed results, statistical analysis, and detailed implementation specifications, leading this reviewer to a clear acceptance.  (3) Reviewer 5qot questioned multi-modal prediction mechanisms and interpretability analysis. The authors clarified parallel generation processes, provided expert selection analysis, and included clustering validation supporting their design choices. (4) Reviewer 3sDS requested error bars, limitations discussion, and broader contextual considerations. The authors provided comprehensive multi-seed results and outlined extensive future work directions. The rebuttal period was productive, with authors providing substantial additional experimental results including clustering analysis and significance testing.
Decision: Accept (poster)