Explainable AI for transportation process management: symbolic regression of freight train speeds using Kolmogorov–Arnold networks (KANs)

The modern railway sector operates under conditions of increased freight-flow volatility driven both by global economic trends and by internal requirements for transport system efficiency. Under these conditions, effective train traffic management becomes a key element in ensuring infrastructure throughput and carrying capacity. However, this task belongs to the class of complex optimisation problems (Cacchiani et al., 2014).

One of the most important parameters determining the efficiency of the transportation process is the average speed of freight trains. This indicator is affected by a wide range of interrelated factors: track topology and geometric characteristics (gradients, curve radii), locomotive power, train weight and length, infrastructure load and operating standards for stations and line sections (Wang & Han, 2023).

With the advent of big data and the development of computational machine-learning (ML) methods, algorithms such as gradient boosting and deep neural networks have shown high effectiveness in predicting operational indicators. However, their widespread adoption in safety-critical sectors, including railway transport, is constrained by the “black-box problem,” that is, the lack of transparency and interpretability of their decisions (Rudin, 2019). For engineers and managers, an accurate forecast alone is insufficient; it is also necessary to understand the causal relationships underlying it in order to justify investments and operational changes.

Existing explainable artificial intelligence (XAI) methods offer solutions for post hoc model interpretation, but they remain approximations of the internal logic of the “black box,” may be computationally expensive and are not always reliable. The alternative of using inherently simple and interpretable models (for example, linear regression) is often inapplicable because of their insufficient expressive power for complex data.

Another symbolic regression approach is gplearn, which uses genetic programming to evolve mathematical expressions. However, this method has known limitations: sensitivity to hyperparameters, potential for expression bloat, computational expense on high-dimensional data and stochastic variability across runs. As shown in Section 4, on the present task, gplearn performs comparably to linear regression in terms of RMSE, highlighting the need for alternative inherently interpretable models that offer higher fidelity.

A breakthrough in this area is promised by the Kolmogorov–Arnold network (KAN) architecture proposed by Liu et al. (Liu et al., 2024a; Liu, Wang, Wang, Soljačić, & Tegmark, 2024b). KAN is based on the Kolmogorov–Arnold representation theorem, which states that multivariate functions can be represented as a superposition of univariate functions and addition. Unlike classical MLPs, where nonlinearities are fixed in the nodes (for example, ReLU) and weights are trained on the edges, KAN inverts this approach: trainable parametric activation functions (splines) are placed on the edges, while the nodes perform simple summation. This enables KAN not only to achieve high accuracy with fewer parameters but also to be inherently interpretable: the learned functions can be visualised and analysed, allowing analytical dependencies between features and the target variable to be extracted and effectively turning the neural network into a symbolic regression tool.

The remainder of this paper is organised as follows. Section 2 reviews related work in railway predictive analytics, XAI and KAN. Section 3 describes the dataset, feature engineering and experimental setup, including three cross-validation strategies. Section 4 presents the experimental results, comparing KAN against baseline models and analysing the extracted symbolic formula. Section 5 concludes with a discussion of limitations and directions for future research.

The work makes three specific contributions. First, a methodology for extracting closed-form, physically meaningful equations from a trained KAN is developed and validated for railway operations data, moving beyond a “first application” by demonstrating a complete pipeline from raw data to an analytical formula. Second, a comprehensive experimental protocol is established, including hyperparameter optimisation for all compared models, repeated cross-validation and a temporal robustness check, ensuring that the reported performance differences are statistically sound. Third, through the extracted formula, we reveal and quantitatively describe domain-specific nonlinear patterns – such as the asymptotic effect of block section length on speed – that are not captured by linear models, providing railway practitioners with a new instrument for transparent scenario analysis.

This study lies at the intersection of three key domains: predictive modelling in railway transport, the paradigm of XAI and the latest developments in neural-network architectures, namely KANs. This section systematises the existing work in these areas in order to substantiate the novelty and necessity of the proposed approach.

Forecasting operational indicators such as travel time, speed and train schedules is central to improving transportation management efficiency. Historically, deterministic methods based on physical train-motion models and operating standards were used to solve this problem. However, these methods are often inflexible and unable to fully account for the stochastic nature of the railway system (for example, weather conditions, track geometry, track quality, traffic density, etc.).

With the growth of computing power and the accumulation of large volumes of data, machine learning (ML) has become the dominant approach. Studies demonstrate the superiority of ML methods such as Gradient Boosting Machines (GBM) (Huang, Wen, Fu, Peng, & Li, 2020) and Random Forest (Markovic, Milinkovic, Tikhonov, & Schonfeld, 2015) in predicting travel times and delays due to their ability to capture complex nonlinear interactions among numerous factors (infrastructure, rolling stock, operational environment). More advanced models, including recurrent neural networks (RNNs) and convolutional neural networks (CNNs), are used to account for temporal and spatial dependencies in the data, for example, to predict delays across the network as a whole (Zhang, Sun, Liu, & Wei, 2021).

Despite their high accuracy, these models suffer from a fundamental drawback: they operate as “black boxes” (Rudin, 2019). A track engineer or dispatcher may obtain an accurate speed forecast, yet be unable to understand which factors and to what extent, had a decisive influence on that forecast in a specific situation. This limits their practical usefulness for informed infrastructure-management decisions.

Recognition of the limitations of “black-box” models in critical industries has led to a surge of research in XAI. In the context of railway transport, XAI seeks to make model predictions interpretable and understandable to human experts.

The most common approaches are SHAP (SHapley Additive exPlanations) and LIME (Local Interpretable Model-agnostic Explanations). SHAP is a game-theoretic method for distributing the contribution of each feature to the final prediction (Lundberg & Lee, 2017) and has been successfully applied, for example, to analyse the factors influencing train delays (Oneto, Fumeo, Clavijo, Canepa, & Anguita, 2020). LIME approximates the behaviour of a complex model locally, around a specific prediction, by means of a simple interpretable model such as linear regression.

However, these methods have substantial drawbacks. They are approximations of the model's internal logic rather than its direct reflection, which may lead to incorrect or contradictory explanations (Rudin et al., 2022). In addition, they create extra computational overhead and do not provide global interpretability of the model as a whole.

An alternative line of research is symbolic regression, which aims to extract interpretable mathematical expressions directly from data. One of the most accessible frameworks is gplearn, which implements evolutionary search for analytical formulas using genetic programming. gplearn generates a population of mathematical expressions by combining basic operators (addition, multiplication, division, logarithms, trigonometric functions) through crossover and mutation operations, selecting the best solutions at each generation.

This approach has been successfully applied in materials science and process control. Nevertheless, gplearn has recognised limitations: sensitivity to hyperparameters (population size, number of generations), expression bloat, computational cost for high-dimensional data and non-reproducible results due to stochastic search. In the present freight train speed prediction task (Section 4), gplearn yields results comparable to linear regression. This does not diminish the value of genetic programming but indicates that alternative symbolic regression methods – such as the KAN-based approach proposed here – may offer advantages in certain structured regression problems. Independent studies confirm these shortcomings. For instance, Liu et al. (2024), in their work on explainable traffic signal control, explicitly highlight the trade-off between interpretability and computational scalability when applying genetic programming to high-dimensional transportation data, noting that evolutionary search often produces overly complex expressions that are difficult to deploy in practice. This external evidence supports our empirical finding that gplearn struggles to outperform linear regression on the full 23-feature set, motivating the search for alternative symbolic regression frameworks such as KAN.

An alternative is to use inherently interpretable models, such as linear models, decision trees and their ensembles. However, their expressive capacity is often insufficient for modelling the complex multidimensional dependencies inherent in railway systems.

Introduced in 2024 by Liu et al. (2024a), KANs offer a fundamentally new architecture that may provide a solution to the problems outlined above.

Unlike traditional multilayer perceptrons (MLPs), which use fixed nonlinear activation functions in the nodes and trainable weights on the edges, KANs invert this structure. They place trainable parametric activation functions (for example, splines) on the edges, while the nodes perform simple summation.

This architecture has two key advantages. First, high accuracy: KANs demonstrate higher accuracy with fewer parameters than similarly sized MLPs because of their ability to approximate complex functions more efficiently. Second, inherent interpretability: KANs do not require post hoc methods. The learned activation functions can be visualised and analysed directly, allowing not only understanding of feature importance but also identification of the nature of the dependency (linear, exponential, threshold-based, etc.) and even extraction of analytical formulas.

Within the first months after publication, KANs showed outstanding results in tasks where interpretability is critically important: law discovery in physics, biomedical research (prediction of gene-drug interactions) (Yadalam, Ayyachamy, Barbosa, & Natarajan, 2025) and forecasting in technical systems (state-of-charge estimation for lithium-ion batteries) (Dao, Liu, & Sidorov, 2024). These studies emphasise that KANs not only predict but also provide researchers with new, nontrivial insights into the relationships in the data. To the best of our knowledge, our study is the first to adapt KANs to the complex task of forecasting in the railway sector, with an emphasis on extracting analytically useful and practically meaningful dependencies.

Despite these advantages, KANs also have limitations. Training spline-based edge functions can be computationally more intensive than standard MLP backpropagation and the risk of overfitting increases with the number of spline knots and network depth. For this reason, the present study deliberately restricts the architecture to a single hidden layer, prioritising interpretability and stability over marginal gains in accuracy. To the best of our knowledge, this is the first adaptation of KAN to a comprehensive predictive task in railway operations with a focus on extracting practical analytical dependencies.

The literature review reveals a clear gap: existing railway predictive models are either highly accurate but opaque (deep learning, gradient boosting) or interpretable but insufficiently expressive (linear regression, shallow trees). Post hoc XAI methods provide only local, approximate explanations and do not yield a global analytical description of the model. Symbolic regression via genetic programming (e.g. gplearn) offers a path to analytical formulas but suffers from reproducibility, scalability and bloat issues, particularly in high-dimensional settings.

KANs present a promising synthesis: they achieve accuracy comparable to state-of-the-art black-box models while offering inherent, global interpretability through the extraction of closed-form symbolic expressions. This study bridges the gap by applying KANs to railway operations data and demonstrating a complete pipeline from raw data to an analytically expressed, physically interpretable speed prediction model that can directly inform capacity management and operational decision-making.

This section describes the data collection and processing pipeline, the feature space and target variable, the experimental protocol including three cross-validation strategies, the evaluation metrics and the benchmark models used for comparison.

For this study, a representative dataset was constructed by aggregating information from heterogeneous operational and infrastructure systems. The dataset covered a 12-month period (January–December 2024), which made it possible to take into account seasonal fluctuations in freight traffic and varying weather conditions.

Primary data aggregation was performed at the level of “block section – date – track section identifier.” This level of detail was chosen as optimal for analysing daily operational dynamics without losing essential features of infrastructure performance. As a result of the preliminary processing, a dataset containing 55,508 observation rows was formed, each representing a unique “snapshot” of the state of a block section on a specific day.

The target variable in this study was the average technical speed of freight trains on a block section per day (km/h). This indicator was selected as a key integral measure of infrastructure utilisation efficiency and the quality of transportation-process management.

Figure 1 shows the distribution of the target variable. The histogram reveals a unimodal distribution with a peak in the range of approximately 35 km/h, which corresponds to typical operational speeds for heavy freight traffic on mixed-use lines.

The feature space was formed on the basis of expert judgment provided by railway operation specialists. In total, 23 features were selected and engineered, grouped into four semantic categories:

• Group 1: Operational traffic indicators

• Group 2: Geometric and vertical alignment characteristics of the track

• Group 3: Rolling stock characteristics

• Group 4: Capacity constraint factors

Table 1 provides the complete mapping between variable indices (used in the symbolic formula in Section 4.3) and physical features.

To ensure high data quality and the applicability of machine-learning methods, the following preprocessing pipeline was performed:

1. Outlier removal: anomalous values caused by recording errors (for example, speed ≈ 0 or extremely high speed) were identified using the interquartile range (IQR) method and removed.

2. Missing-value imputation: a small number of missing values were filled with median values for numerical features and the mode for categorical features.

To ensure robust results and assess the impact of the data splitting method on prediction quality, a comprehensive cross-validation strategy comprising three different schemes was employed. This approach is motivated by the fact that standard random cross-validation may fail to account for railway-specific dependencies, such as temporal autocorrelation (date dependency) and structural differences between track section identifiers.

The dataset contains two key identifiers used for stratified splitting: date (daily observations over 12 months) and railway track section identifier.

Three 5-fold cross-validation schemes were implemented:

• Random cross-validation (baseline). The dataset is randomly shuffled and evenly divided into five non-overlapping folds. In each of the five training cycles, one fold serves as the test set and the remaining four serve as the training set. This scheme serves as a standard benchmark.

• Time-based cross-validation with a fixed sliding window. This scheme respects the temporal structure of the observations: the model is trained on fixed-size windows and tested on a non-overlapping period. Unlike classical expanding-window cross-validation, the fixed sliding window maintains a constant training set size across all folds, ensuring comparability with random cross-validation.

• Cross-validation by track section identifier. All unique track sections are divided into five folds. In each iteration, one fold is included in the test set, while the remaining data constitute the training set. This scheme evaluates how well a model trained on one set of track section identifiers can predict on new, unseen track sections – a crucial capability for practical application to new railway segments.

For comprehensive assessment of predictive performance across the three cross-validation schemes, the following base metrics were selected:

1. Root Mean Square Error (RMSE). The primary metric, measures the standard deviation of prediction errors. It has the same units as the target variable (km/h), facilitating interpretation by railway engineers.

2. Coefficient of determination (R²). Indicates the proportion of variance in the target variable explained by the model. R² = 1 indicates perfect prediction; R² = 0 indicates prediction at the level of the mean value.

For each cross-validation type (random, time-based, track-stratified) and each model, RMSE and R² were computed for each of the five folds. The following statistical summaries were then derived:

1. Arithmetic mean (μ) – central tendency of the metric.

2. Standard deviation (σ) – dispersion of the metric across folds.

3. Coefficient of variation (CV = σ/μ × 100%) – relative variability. Low CV values (<10–15%) indicate high model stability and absence of overfitting to a specific data split.

This aggregation approach allows not only comparison of models by average accuracy but also assessment of their robustness – the ability to produce stable results under different data splitting strategies.

To test the hypothesis that the data splitting method (random, time-based, or track-stratified) does not have a statistically significant impact on model quality estimates, the non-parametric Friedman test was applied (Table 6). This test was chosen for the following reasons:

1. It does not assume the normality of metric distributions.

2. It allows simultaneous comparison of three or more related samples (in this case, the three cross-validation schemes).

3. It is robust to outliers and widely used in comparative ML algorithm studies.

Input data for the Friedman test. For each model, a 5 (folds) × 3 (CV types) matrix was constructed, containing the metric values (e.g. RMSE) for each fold of each scheme. Thus, all 15 measurements (5 folds × 3 schemes) were included in the analysis.

Hypotheses:

Interpretation:

• If p-value >0.05, H0 is not rejected. Differences between the three CV types are not statistically significant, supporting high model robustness.

• If p-value ≤0.05, H1 is rejected, and post hoc analysis (e.g. Wilcoxon test with Bonferroni correction) is conducted to determine which CV scheme yields systematically different results.

To comprehensively evaluate the effectiveness of KAN for freight train speed prediction, both in terms of accuracy and interpretability, the set of algorithms and models presented in Table 2 was selected.

The experimental evaluation was designed to assess the effectiveness of KAN according to two key criteria: predictive accuracy and degree of interpretability. The models described in Table 2 were used for a comprehensive comparison. All models were evaluated using the three 5-fold cross-validation strategies described in Section 3.4: random CV, time-based CV with a fixed rolling window, and track-stratified CV. For each fold and each model, RMSE and R² were computed.

For all models except gplearn, hyperparameters were tuned at each cross-validation iteration using Optuna's TPESampler (Akiba, Sano, Yanase, Ohta, & Koyama, 2019). For gplearn, symbolic formulas were evolved using genetic algorithms with dynamic programming within the gplearn framework. A fixed random seed was set for reproducibility where applicable. To determine whether the choice of cross-validation strategy significantly affects the reported metrics, the non-parametric Friedman test was applied (see Section 3.6), with the significance threshold set at α = 0.05.

Stability across cross-validation strategies. Tables 3–5 report the mean RMSE and R² with standard deviations and coefficients of variation for each model across the three CV schemes (see Table 6).

All p-values exceed 0.05, indicating that the null hypothesis (no statistically significant difference between CV schemes) cannot be rejected. This confirms that the reported metrics are robust and not artefacts of a particular data splitting strategy. The models generalise consistently across random, temporal and track-stratified partitions.

CatBoost achieves the lowest RMSE across all three CV schemes (range 3.866–3.979), confirming gradient boosting on decision trees as the state-of-the-art for structured tabular data in this domain. Among neural network architectures, KAN consistently outperforms MLP: KAN achieves RMSE in the range 4.119–4.247, while MLP achieves 4.531–4.675. The relative improvement of KAN over MLP is approximately 9–10% in RMSE, despite identical architecture (same number of layers and neurons) and comparable numbers of trainable parameters. This advantage is attributable to the spline-based edge functions of KAN, which more efficiently approximate the nonlinear relationships inherent in railway operations data.

Tables 3–5 also includes results for the symbolic formula extracted from the trained KAN model (denoted “KAN formula”; see Section 4.3 for the full expression). As expected, replacing the full neural network with its symbolic approximation leads to an increase in RMSE from approximately 4.2 to 6.2 km/h. This degradation represents the price paid for full analytical explainability. Critically, however, the KAN-derived formula outperforms both linear regression and gplearn across all CV schemes: KAN formula RMSE ranges from 6.206 to 6.250, compared to 7.169–7.194 for linear regression and 7.250–7.690 for gplearn. In terms of R², the KAN formula achieves 0.711–0.724, whereas linear regression reaches only 0.638–0.642 and gplearn 0.591–0.633. This confirms that KAN provides a qualitatively richer and more accurate analytical instrument for interpretation than either linear models or genetic programming-based symbolic regression.

While CatBoost achieves the highest predictive accuracy, its fundamental limitation is low interpretability. Post hoc XAI methods such as SHAP provide approximate feature importance estimates but do not yield a global analytical description of the model. The key advantage of KAN lies in its ability to be inherently interpretable (by design), allowing direct extraction of a closed-form mathematical expression that maps input features to the target variable. The methodological feasibility of this extraction has been independently validated in recent work on physics-informed symbolic regression using KAN (Wang, Rabczuk, & Liu, 2025), which demonstrates that the learned spline-based edge functions can be reliably converted into analytical expressions even in the presence of noisy real-world data, without significant loss of fidelity.

Equation (1) presents the symbolic expression extracted from the trained KAN model (after simplification and pruning of negligible terms). This formula enables average speed prediction directly, without any reference to the original neural network:

Equation (1). Extracted KAN formula for the full model. Source: author's own work.

where variable indices correspond to the features defined in Table 1.

Table 7 compares the predictive accuracy of the KAN-derived formula against linear regression and gplearn across varying feature space dimensions (using the most parsimonious formulas obtained during cross-validation).

The KAN formula achieves the lowest RMSE across all feature set sizes. With 23 features, the KAN formula's RMSE is 13.4% lower than linear regression and 14.2% lower than gplearn. While gplearn performs competitively on low-dimensional inputs (1–2 features), its relative performance declines as dimensionality increases – a known limitation of genetic programming in high-dimensional spaces.

Figure 2 shows the dependence of average speed on block-section length (x₂). The KAN formula captures an asymptotic saturation effect: speed increases rapidly for short sections (up to 10–15 km) and then gradually plateaus. Linear regression imposes a constant slope, while gplearn produces a non-physical local anomaly near zero length.

Figures 3–5 present the response surfaces generated by each method when trained on both features (x₁ and x₂). These three-dimensional visualisations reveal fundamental differences in how each approach models the joint influence of infrastructure length and traffic composition on average speed.

The response surface from KAN (Figure 3) demonstrates desirable properties: monotonicity, an interaction plateau reflecting diminishing returns and smoothness. Linear regression (Figure 4) produces a plane that systematically misestimates in certain regions. The gplearn surface (Figure 5), while nonlinear, exhibits irregular oscillations that complicate interpretation – a known consequence of stochastic search in genetic programming.

This study addressed the “black-box” problem in railway predictive modelling by introducing KANs for freight train speed prediction. The core contribution is not merely a point forecast but a fully interpretable analytical formula that quantifies the relationship between operational and infrastructure factors and the average technical speed of freight trains.

The experimental results, obtained on a representative dataset of 55,508 observations with 23 features, support the following conclusions.

First, regarding predictive accuracy, KAN with an architecture [23,47,1] achieved an RMSE of 4.12–4.25 across three cross-validation strategies (random, time-based and track-stratified). This performance is comparable to the state-of-the-art gradient boosting model CatBoost (RMSE = 3.87–3.98) and significantly outperforms a classical multi-layer perceptron (MLP) with identical architecture (RMSE = 4.53–4.68) by approximately 9–10%. The superiority of KAN over MLP confirms that spline-based edge functions more efficiently approximate the nonlinear dependencies inherent in railway operations data.

Second, the key scientific contribution is the successful extraction of a closed-form symbolic expression (Equation 1) from the trained KAN network. This formula contains exponential, logarithmic, square-root and polynomial terms, explicitly revealing the nonlinear, asymptotic and interactive nature of the relationships between input factors and train speed. The extracted formula is not merely a post hoc approximation but a fully functional predictive model that can be evaluated without the original neural network.

Third, the KAN-derived formula achieves an RMSE of 6.11 with 23 features, compared to 7.06 for linear regression and 7.13 for genetic programming-based symbolic regression (gplearn). In terms of R², the KAN formula outperforms linear regression by 13% and gplearn by 14%. These improvements confirm that KAN provides a qualitatively richer and more accurate analytical instrument for interpretation than either linear models or evolutionary symbolic regression.

Fourth, the Friedman test (Table 6) applied to three distinct cross-validation schemes (random, time-based, track-stratified) yielded p-values > 0.05 for all models, indicating that the reported metrics are statistically stable and not artefacts of a particular data split. This robustness is essential for practical deployment, where temporal and structural generalisation is critical.

Fifth, the symbolic expression immediately reveals practical insights: an asymptotic (saturation) effect of block-section length on speed, quantified by the term exp(−0.113x₂), positive contributions from double-headed traction (+0.248x₂₂) and high-share transit trains (+0.146x₁), negative contributions from high energy consumption (−0.045x₂₀) and prolonged station intervals (−0.055x₃, −0.0425x₄) and complex interactions between gradient variability, curvature distribution and section length, captured by a high-dimensional exponential term.

Despite the promising results, this study has several limitations that should be acknowledged:

1. Temporal dependencies are not explicitly modelled. The current approach treats each daily observation as independent, ignoring potential autocorrelation between consecutive days on the same block section. Railway systems exhibit memory effects (e.g. congestion spill-over, maintenance cycles) that could be captured by recurrent or hybrid architectures.

2. Weather and seasonal factors excluded. The dataset includes temporal information (date) but does not incorporate explicit weather variables (precipitation, temperature, visibility) or seasonal effects beyond what is implicitly captured by the date identifier. Incorporating meteorological data could further improve accuracy and interpretability.

3. Complexity of the extracted formula. While interpretable in principle, Equation (1) contains more than 20 terms, including nested exponentials and square roots. This complexity limits direct manual inspection. For operational use, further simplification (e.g. pruning negligible terms) would be desirable, though this may reduce accuracy. The trade-off between formula complexity and interpretability remains an open challenge for KAN-based symbolic regression.

4. Single hidden layer architecture. To maximise interpretability, the KAN architecture was deliberately restricted to one hidden layer. Deeper KANs might achieve higher accuracy but would complicate visualisation and symbolic extraction. The optimal trade-off between depth and interpretability remains an open question.

5. Generalisation beyond the study context. The dataset was collected from specific railway lines over a 12-month period. While the three cross-validation strategies provide evidence of internal validity, external validation on different infrastructure types, geographical regions, or operational regimes has not been performed.

The findings of this study open several avenues for further investigation:

1. Hybrid architectures for temporal dependencies. Extending KAN with recurrent structures (e.g. KAN–RNN or KAN–LSTM) could capture sequential dependencies in train movements while preserving interpretability at the feature–output mapping level.

2. Integration of weather and real-time data. Incorporating exogenous variables such as precipitation, temperature and real-time track condition alerts would enhance both predictive accuracy and the practical utility of the extracted formulas for real-time dispatching.

3. Automatic formula simplification. Developing post-extraction simplification methods that prune negligible terms and approximate complex sub-expressions without significant loss of accuracy would make the resulting formulas more accessible to railway practitioners.

4. Application to other railway prediction tasks. The methodology demonstrated for speed prediction can be transferred to related problems: train delay forecasting, energy consumption prediction and dwell time estimation. Each application would require domain-specific feature engineering but would benefit from the same interpretability–accuracy balance.

5. Deployment and user studies. Evaluating the extracted formulas in real-world settings with railway engineers and dispatchers would assess their practical value for decision support, scenario analysis and infrastructure investment appraisal.

6. Benchmarking against deeper KANs. Systematic comparison of shallow (one-hidden-layer) versus deep (multi-hidden-layer) KANs on railway data would quantify the accuracy gains achievable with greater depth and the corresponding loss in interpretability.

This study demonstrates that KANs offer a viable and promising path toward trustworthy, explainable artificial intelligence in railway transportation. The ability to extract a closed-form analytical expression from a trained neural network transforms the machine learning model from an opaque “black box” into a transparent analytical tool. Railway engineers and planners can examine the extracted formula, identify dominant factors, quantify their influence and conduct scenario analyses – all without trusting a model whose internal logic remains hidden.

While gradient boosting (CatBoost) achieves marginally higher accuracy, its fundamental opacity limits its acceptance in safety-critical decisions where understanding the “why” is as important as knowing the “what.” KAN provides a compelling alternative: accuracy comparable to modern black-box models combined with inherent, global interpretability that post hoc XAI methods cannot match.

The extracted formula (Equation 1) is not merely a scientific artefact; it is a practically usable tool that can inform timetable optimisation, infrastructure investment and capacity management. By opening the black box, KAN enables a new generation of data-driven decision support systems in which human experts and machine learning models collaborate with mutual understanding and trust.