1. INTRODUCTION

Russian mathematician Andrey Kolmogorov, together with his student Vladimir Arnold, demonstrated that any continuous multivariable function can be expressed as a finite composition of univariable functions, as established in the well-known representation theorem [1]. This result laid the theoretical foundations for the development of subsequent models focused on functional decomposition. Kolmogorov-Arnold Networks (KANs) are a very recent proposal that revisits this theoretical framework and was formally introduced in April 2024 [2].

The approximation of multivariate functions is a central challenge in disciplines ranging from engineering, applied statistics and artificial intelligence, where the balance between flexibility and explainability largely determines the practical usefulness of models. Classical methods such as multivariate splines offer transparency [3], although they often become computationally unfeasible in high-dimensional scenarios. On the other hand, multilayer perceptrons (MLPs) sacrifice traceability in favour of their ability to model complex non-linear relationships [4]. This dilemma has motivated the search for hybrid architectures that combine mathematical rigour with data-driven adaptability, especially in contexts where decision-making requires interpretability—for example, in materials design or medical diagnostics [5].

In this context, we present Kolmogorov-Arnold Network Splines (KANS), an innovative framework that rethinks the functional approximation from its foundations. KANS emerge from the Kolmogorov-Arnold theorem [1], which proves that any continuous multivariate function can be decomposed into a finite sum of univariate functions. Taking advantage of this property, KANS implement this decomposition using adaptive splines [6], merging theoretical guarantees with modern deep learning tools.

Its architecture, illustrated in Figure 1, operates in two stages: first, each input variable is transformed using univariate splines (ϕ_i,q(x_i)), which capture local behaviours [7]; then, these transformations are combined using composition functions (Φ_q), also modelled with splines, to generate consistent global predictions [8]. This approach not only mitigates the curse of dimensionality by reducing the problem to univariate spaces, but also allows for a granular interpretation of the impact of each variable, facilitating, for example, the identification of critical thresholds in clinical data or inflection points in industrial performance curves.

In this article, we demonstrate how KANS outperform MLPs and traditional splines in moderately complex tasks (3 to 10 variables), including applications in aerodynamic optimisation, climate modelling, and precision robotics [9],[10]. Our contributions focus on three areas: (1) an open-source implementation in Python/PyTorch that integrates cubic splines with adaptive regularisation; (2) a systematic comparison with MLPs and multivariate splines, evaluating performance (RMSE, training time) and interpretability; and (3) practical guidelines for deciding between KANS and MLPs depending on the nature of the problem, highlighting their advantage in contexts with limited data and transparency requirements.

The results reveal error reductions of up to 30% in optimisation tasks and a distinctive ability to unravel nonlinear interactions, positioning KANS as a viable alternative when the balance between accuracy and interpretability is crucial. The article is organised as follows: Section 1 details the mathematical architecture of KANS; Section 2 describes the experiments and the code developed; Section 3 presents the comparative results; and Section 4 discusses implications and future directions [11].

1.1 Theoretical bases

The design of the Kolmogorov–Arnold Network Splines (KANS) architecture is based on the representation theorems of [1], and has been recently developed in the works of [12]–[14].

These authors propose the use of hierarchically composed univariate functions, modelled using adaptive B-splines, as the basis for constructing highly interpretable neural networks.

KANS are based on the Kolmogorov-Arnold theorem, which guarantees that any continuous function f : [0,1]ⁿ → R can be decomposed into an infinite sum of continuous univariate functions:

Where Φ_q,p and Φ_q are internal and external functions, respectively. In this context, Φ_q,p(x_p) is approximated using parametric cubic spliness, defined as linear combinations of basis functions B_k(x_p; t) over a set of nodes t:

These third-order B-splines guarantee C2 continuity, which allows local non-linear behaviours to be modelled while maintaining overall smoothness. Coefficients α_k,p,q, which are adjustable during training, define the shape of each spline based on the data [2], [15]–[17].

The external functions Φ_q, responsible for combining the outputs of the internal functions, are also modelled with adaptive splines:

Where z = Σ_p=1ⁿ Φ_q,p(x_p) and B_l,q are trainable parameters. The sss nodes of these external splines are optimised to capture multivariate interactions through hierarchical compositions [18]–[21].

2. MATERIALS AND METHODS

The experiments were conducted in an accessible and replicable environment, using Python 3.9 for its flexibility and compatibility with deep learning libraries. The implementation and training of the networks, including Kolmogorov-Arnold Network Splines (KANS), was performed with PyTorch, taking advantage of its GPU efficiency and dynamic architecture.

NumPy, Matplotlib, and Seaborn were used for numerical processing and visualisation, allowing for analysis of model behaviour and the effects of noise on geometric structures. The execution environment was Google Colab, which provided free access to GPUs, facilitating training without the need for specialised hardware.

The tests were run on a computer with an AMD Ryzen 5 4500U processor and integrated Radeon graphics, thus ensuring the replicability of the study even with limited resources. This set of tools allows for efficient implementation and rigorous documentation of each experimental phase.

2.1 Experimental Design and Theoretical Basis of the KANS Architecture

This study is part of an experimental-computational approach aimed at evaluating the performance of the Kolmogorov-Arnold Network Splines (KANS) architecture in synthetic data reconstruction tasks under noisy conditions.

To this end, a comparative experimental design between KANS and multilayer perceptrons (MLPs) is proposed, allowing for the analysis of differences in accuracy, computational efficiency, and generalisation capacity under the same execution environment.

As illustrated in Figure 1, the proposed architecture implements a hybrid structure based on the representation theorem of [1], which states that any continuous multivariate function can be decomposed as a sum of univariate functions. This principle is realised through adaptive cubic splines that transform each input variable and then compose them hierarchically to generate the prediction.

Figure [1] Structure of the KANS architecture, showing the spline-based transformation and composition process. Source: Author’s own elaboration.

To validate the effectiveness of the model, the synthetic dataset known as Swiss Roll was used, a three-dimensional structure that simulates a ribbon rolled into a spiral within a Euclidean space. As shown in Figure 2, this dataset is widely used in the study of non-linear learning algorithms because it has a complex geometry that requires models to identify and reconstruct non-obvious relationships between variables. Its rolled-up shape forces models to “unroll” the internal structure to recover the original topology of the data. In this study, samples ranging from 1,000 to 10,000 points were used, allowing for an adequate balance between computational load and generalisation capacity.

Figure [2] Structure of the original Swiss Roll, visually representing the topological complexity that the model must learn to reconstruct. Source: Author’s own elaboration.

A diffusion process was applied to this dataset, in which the original information is progressively degraded by the controlled addition of isotropic Gaussian noise. The operation of a diffusion model is based on three main stages. In the first stage, known as direct diffusion, the original data is degraded by gradually adding small amounts of random noise. As this process progresses, the signal becomes a completely unstructured pattern, where the initial information is hidden. In the second stage, the neural network is trained to learn how to reverse the previous process. To do this, it analyses samples with different levels of noise and practises predicting which part of the noise was added at each step, generating a statistical “map” that allows it to clean the data step by step. Finally, in the generation stage, the model starts with an input composed solely of pure noise and applies its training to remove it iteratively, thus reconstructing coherent data from total disorder. The key to this procedure is that the network not only removes random disturbances, but in doing so, learns the statistical rules that give rise to the internal structure of the data.

2.3 study variable

The experimental design proposed in this work requires clear identification of the variables involved in the comparative evaluation between the KANS and MLP models. Figure 3 illustrates the diffusion process analysed, where the main output variable corresponds to the mean square error (MSE), which measures the accuracy of reconstruction of the original data from corrupted versions. This indicator quantifies the difference between the actual signal and the signal generated by the neural network during the inverse diffusion process and was used both during the training stage and in the testing phase.

Among the independent variables, multiple factors that can directly affect the performance of the models were considered. One of the most relevant is the type of architecture used, comparing the behaviour of KANS, based on adaptive cubic splines, with MLP, a traditional architecture without explicit interpretability mechanisms. Another important factor is the amount of data available during training, as the aim is to explore the performance of both networks in scenarios with abundant data and in conditions with limited data. The number of adjustable parameters in each network was also taken into account, as well as the total time required to complete the training process.

In addition, the data set and the noise sequence applied remained constant, thus establishing a control variable that guarantees equivalent experimental conditions. This control allowed the comparison between networks to focus exclusively on architectural differences and not on aspects external to the model design. Thanks to this configuration, it was possible to objectively evaluate the impact of each network's internal structure on its ability to reconstruct complex data accurately, efficiently, and interpretably.

Figure [3] Representation of the diffusion process, graphically showing how the signal degrades in the forward phase and how it is progressively recovered during the reverse generation phase. Source: Author’s own elaboration.

2.4 Experimental Procedure

The experimental process was carried out in three main phases: training of the KANS model, training of the MLP model, and subsequent comparative evaluation. As detailed in Figure 4 , the objective was to explore how each neural network learned to reverse the diffusion process and reconstruct the original structure of the Swiss Roll from noisy data.

In the case of the KANS architecture, the model was designed following the functional decomposition principle of the Kolmogorov-Arnold theorem. In its implementation, each input variable is transformed by a univariate cubic spline, and these transformations are then combined using hierarchical splines to produce a coherent prediction. Training was carried out using supervised learning, using mean square error (MSE) as the loss function, optimised with the Adam algorithm. The network was exposed to samples generated by the direct diffusion process, in which Gaussian noise was added to the original data in multiple steps. In each iteration, the model learned to predict the corresponding noise component and progressively reverse it.

Figure [4] KANS network training process, showing the noisy data input flow, its passage through the spline architecture, and the reconstructed output. Source: Author’s own elaboration.

A particular feature of the learning process with KANS was its temporal approach. As shown in Figure 5, in order for the model to adapt to different levels of degradation, sinusoidal positional encodings were incorporated to inform the network of the stage of the process each sample is at. This technique facilitates training by improving the model's sensitivity to noise intensity.

Figure [5] Temporal coding process in KANS network training. Source: Author’s own elaboration.

At the same time, a multilayer perceptrons (MLP) was trained under the same experimental conditions, with the aim of comparing its performance against KANS. Figure 6 illustrates how the MLP was initialised with random weights and exposed to samples generated by the same diffusion process. Over multiple iterations, the network learned to remove noise from the data, adjusting its parameters based on the error between the generated outputs and the original signals.

The progress of the MLP network during training was evaluated through periodic visualisations, which examined how the model was recovering the structure of the original Swiss Roll as it learned to eliminate noise. Figure 7 shows a comparison between the original shape of the Swiss Roll set and the reconstruction obtained by the MLP at an intermediate point in the training.

Figure [7] Comparison between the original shape of the Swiss Roll set and the reconstruction obtained by the MLP at an intermediate point in the training Source: Author’s own elaboration.

To make a fair comparison, the KANS network was retrained, this time using exactly the same reduced amount of data as was used in the MLP. As can be seen in Figure 8 , this decision was made to analyse the behaviour of both architectures under equivalent information conditions. Despite the limitation in data, the KANS network managed to maintain greater structural consistency in the reconstructions generated.

Figura [8] Direct comparison between the results of both networks under these conditions, showing the greater geometric fidelity in the output of the KANS model. Source: Author’s own elaboration.

Additionally, the behaviour of the KANS model during this new limited training was visually recorded. Figure 9 shows a series of samples reconstructed by the network, where a notable reduction in dispersion can be seen.

Figure [9] Series of samples reconstructed by the network, where a notable reduction in dispersion and a closer approximation to the original shape of the Swiss Roll can be seen. Source: Author’s own elaboration.

As can be seen in Figure 10 , the comparison between the original Swiss Roll dataset and one of the final reconstructions generated by the KANS network under reduced data conditions shows significant results.

Figure [10] Comparison between the original Swiss Roll dataset and one of the final reconstructions generated by the KANS network under reduced data conditions Source: Author’s own elaboration

Finally, this second training with reduced data allowed us to evaluate the generalisation capacity of the KANS architecture under limited information conditions. During this phase, we observed that the model maintained a stable and consistent reconstruction, with less dispersion in the generated samples, compared to the MLP network. The procedure showed that the combination of adaptive spline functions, together with a hierarchical structure based on univariate decomposition, gives KANS a significant advantage in capturing complex relationships between variables, even in scenarios with low data availability. This stage concludes the experimental process, establishing the conditions for the quantitative and qualitative analysis developed in the following sections.

3. RESULTS

The experiments carried out allowed for a quantitative comparison of the performance of Kolmogorov-Arnold Network Splines (KANS) versus traditional multilayer perceptrons (MLPs) in the task of reconstructing degraded data through diffusion. The evaluation was based on objective metrics such as mean square error (MSE), total training time, and the number of parameters used by each model.

Under standard training conditions, the KANS architecture achieved a significant reduction in error compared to its MLP counterpart. After 500 training epochs, KANS achieved an MSE of 0.08, while the MLP model required twice as many iterations (1,000 epochs) to achieve an error of 0.21. This difference is explained by the ability of univariate splines to capture local nonlinear relationships more accurately, avoiding the distortions that often occur in regions of high curvature of the Swiss Roll. In addition, the KANS model exhibited more stable behaviour, with less variance in the results during the testing phase. Table 1 summarises these comparative metrics, which also include the training time and total number of parameters for each architecture.

In terms of computational efficiency, the KANS model completed its training in approximately 12 minutes, while the MLP required 45 minutes to achieve similar performance. This difference is due to the spline architecture optimising its structure through more controlled compositions, avoiding excessive use of parameters. In total, KANS used around 1,200 trainable parameters, in contrast to the 15,000 required in the MLP network. This gap reflects not only an improvement in performance, but also greater resource economy, a critical aspect in environments with computational limitations.

From a visual standpoint, the reconstructions generated by both architectures show clear differences in terms of geometric accuracy and structural consistency. As seen in Figure 11, there is a notable difference in the preservation of the topological features of the Swiss Roll.

Figure [11] Visual comparison between KANS and MLP reconstructions. (a) Original Swiss Roll, (b) KANS reconstruction, (c) MLP reconstruction. Source: Author’s own elaboration.

In order to facilitate the visualisation and synthesis of the main quantitative findings, a table summarising the comparative metrics between the two architectures is presented below. This table allows for a direct observation of the differences in performance, efficiency, and complexity between KANS and MLP, providing an overview of the behaviour of each model in the reconstruction task.

Table 1. Comparative performance between KANS and MLPs. Source: Author

Comparison of the performance of KANS and MLP architectures in Swiss Roll reconstruction tasks. Computational efficiency metrics (training time), model complexity (number of parameters) and accuracy (mean square error in training and testing) are reported.

A particularly relevant aspect of the KANS model is its interpretability, derived from the use of adaptive splines that explicitly model the influence of each input variable. As illustrated in Figure 12, unlike MLPs, which act as black boxes, KANS allow the functions learned during training to be visualised.

Figure [12]. Representation of the splines associated with the x and z coordinates of the Swiss Roll. Source: Author’s own elaboration.

These curves identify inflection points that correspond to critical regions of the data structure, which can be useful in applications where understanding the behaviour of the model is required in order to make informed decisions. Overall, the results obtained validate the proposal of Kolmogorov-Arnold Network Splines as an efficient and explainable alternative to traditional neural reconstruction models. Their low error, high stability, lower complexity, and greater transparency position them as a solid option for problems where reconstruction quality and interpretability are key factors.

4. DISCUSSION

The results of this study show that Kolmogorov-Arnold Network Splines (KANS) are an efficient and accurate alternative to traditional neural networks in data reconstruction and noise removal tasks. This finding coincides with previous research that has explored the use of spline functions in neural networks. For example, [22] proposed univariate ReLU neural networks interpreted as splines, which allowed for a more intuitive understanding of the structure of the loss surface and its critical points, thus facilitating the analysis of learning dynamics.

Our experiments confirm that KANS achieve lower mean square error (MSE) and shorter training times than traditional models. However, they also reveal that training stability can be compromised by the number of trainable parameters. Following this trend, [23] addressed the problem using a KANS with free knots, reducing the number of parameters and improving stability, bringing the model closer to the complexity scale of conventional neural networks.

In addition, recent research has explored the integration of KANS with attention mechanisms.

[24] presented a Kolmogorov-Arnold-informed version that demonstrates how this combination can improve performance in tasks that require identifying critical regions in data, suggesting that KANS could dynamically adapt to local information density and thus overcome limitations inherent in traditional models. This adaptability, coupled with their accuracy, reinforces the considerable potential of KANS in contexts that demand high flexibility and accuracy in handling complex data. However, their applicability continues to depend on the domain and nature of the problem: in scenarios with lower computational complexity, traditional models remain efficient solutions. Consequently, the choice between KANS and conventional networks should be based on a careful assessment of the specific needs of each application.

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