# BSRBF-KAN: A combination of B-splines and Radial Basis Functions in
Kolmogorov-Arnold Networks

Hoang-Thang Ta
Department of Information Technology
Dalat University
Lam Dong, Vietnam
<EMAIL_ADDRESS>

###### Abstract

In this paper, we introduce BSRBF-KAN, a Kolmogorov Arnold Network (KAN) that
combines B-splines and radial basis functions (RBFs) to fit input vectors in
data training. We perform experiments with BSRBF-KAN, MLP, and other popular
KANs, including EfficientKAN, FastKAN, FasterKAN, and GottliebKAN over the
MNIST and Fashion-MNIST datasets. BSRBF-KAN shows stability in 5 training
sessions with a competitive average accuracy of 97.55% on MNIST and 89.33% on
Fashion-MNIST and obtains convergence better than other networks. We expect
BSRBF-KAN to open many combinations of mathematical functions to design KANs.
Our repo is publicly available at: https://github.com/hoangthangta/BSRBF-KAN.

_Keywords_ Kolmogorov Arnold Networks $\cdot$ B-splines $\cdot$ Radial Basis
Functions $\cdot$ Gaussian RBFs $\cdot$ MNIST

##  1 Introduction

A recent work of Liu et al. [1] over KANs opened a new paradigm in applying
learnable activation functions as "edges" to fit training data instead of
using fixed ones as "nodes" that are usually used in MLPs. The theory behind
KANs relies on the Kolmogorov-Arnold representation theorem (KART), which
states that a continuous function of multiple variables can be expressed as a
combination of continuous functions of a single variable through additions.
KANs are anticipated to bring a fresh perspective to solving issues that have
been overshadowed by MLPs.

With that inspiration, many scientists have flocked to develop different types
of KANs based on popular polynomial and basis functions. While these works
focus on univariate functions to set up KANs, none have explored their
combination. Therefore, we aim to combine B-splines [2] and Radial Basis
Functions [3] to build a combined KAN named BSRBF-KAN. The reason why we use
them is their successful implementation in two popular KANs,
EfficientKAN111https://github.com/Blealtan/efficient-kan and FastKAN [4].
Additionally, they are continuous basis functions used for function
approximation, providing features such as smoothness, adaptability, and local
control. With the experiments on MNIST [5] and Fashion-MNIST [6], we show the
stability and faster convergence of BSRBF-KAN compared to other networks.

##  2 Related Works

In 1957, Kolmogorov provided proof to solve Hilbert’s 13th problem,
demonstrating that a multivariate continuous function can be represented as a
combination of single-variable functions and additions, known as KART [7, 8].
This theorem has been applied in numerous works to build neural networks [9,
10, 11, 12, 13, 14]. However, it has also sparked debates about its ability in
designing neural networks [15, 16].

Despite the long history of application in neural networks, no works on KART
had gained significant attention within the research community until the
recent work by Liu et al. [1]. They recommended not adhering strictly to the
KART but generalizing it to create KANs with more neurons and layers. As a
result, their KANs can surpass MLPs in both accuracy and interpretability on
small-scale AI + Science tasks.

KANs are a fresh breeze that has blown into the scientific community since May
2024. Many KAN design works rely on popular mathematical functions, especially
ones that can deal with curves such as B-Splines [2] (Original KAN [1],
EfficientKAN222https://github.com/Blealtan/efficient-kan), Gaussian Radial
Basis Functions or GRBFs (FastKAN [4]), Reflection SWitch Activation Function
or RSWAF (FasterKAN [17]), Chebyshev polynomials (TorchKAN [18], Chebyshev KAN
[19]), Legendre polynomials (TorchKAN [18]), Fourier transform
(FourierKAN333https://github.com/GistNoesis/FourierKAN/, FourierKAN-GCF [20])
and other polynomial functions [21]. In this paper, we prefer to work on
combining basic mathematical functions to design KANs instead of single
functions to explore the effects of combined functions on KANs.

##  3 Methodology

###  3.1 Kolmogorov-Arnold Representation Theorem

A KAN relies on KART, which states that any multivariate continuous function
$f$ defined on a bounded domain can be expressed using a finite number of
continuous single-variable functions and additions [22, 23]. Given
$\mathbf{x}={x_{1},x_{2},..,x_{n}}$ consisting of $n$ variables, a
multivariate continuous function $f(\mathbf{x})$ is represented by:

$\displaystyle
f(\mathbf{x})=f(x_{1},\ldots,x_{n})=\sum_{q=1}^{2n+1}\Phi_{q}\left(\sum_{p=1}^{n}\phi_{q,p}(x_{p})\right)$
(1)

which contains two sums: outer sum and inner sum. The outer sum
$\sum_{q=1}^{2n+1}$ will sum $2n+1$ of $\Phi_{q}$ ($\mathbb{R}\to\mathbb{R}$)
terms. The inner one sums up $n$ terms for each $q$, and each term
$\phi_{q,p}$ ($\phi_{q,p}\colon[0,1]\to\mathbb{R}$) represents a continuous
function of a single variable $x_{p}$.

###  3.2 The design of KANs

Liu et al. [1] was inspired by KART to design KANs, but authors advised
generalizing it to more widths and depths. In Equation 1, we must search
proper $\Phi_{q}$ and $\phi_{q,p}$ to solve the problem. A general KAN network
consisting of $L$ layers takes $\mathbf{x}$ to generate the output as:

$\displaystyle\text{KAN}(\mathbf{x})=(\Phi_{L-1}\circ\Phi_{L-2}\circ\cdots\circ\Phi_{1}\circ\Phi_{0})\mathbf{x}$
(2)

which $\Phi_{l}$ is the function matrix of the $l^{th}$ KAN layer or a set of
pre-activations. Let denote the neuron $i^{th}$ of the layer $l^{th}$ and the
neuron $j^{th}$ of the layer $l+1^{th}$. The activation function
$\phi_{l,i,j}$ connects $(l,i)$ to $(l+1,j)$:

$\displaystyle\phi_{l,j,i},\quad l=0,\cdots,L-1,\quad i=1,\cdots,n_{l},\quad
j=1,\cdots,n_{l+1}$ (3)

with $n_{l}$ is the number of nodes of the layer $l^{th}$. So now, the
function matrix $\Phi_{l}$ can be represented as a matrix $n_{l+1}\times
n_{l}$ of activations:

$\displaystyle\Phi_{l}=\begin{pmatrix}\phi_{l,1,1}(\cdot)&\phi_{l,1,2}(\cdot)&\cdots&\phi_{l,1,n_{l}}(\cdot)\\\
\phi_{l,2,1}(\cdot)&\phi_{l,2,2}(\cdot)&\cdots&\phi_{l,2,n_{l}}(\cdot)\\\
\vdots&\vdots&\vdots&\vdots\\\
\phi_{l,n_{l+1},1}(\cdot)&\phi_{l,n_{l+1},2}(\cdot)&\cdots&\phi_{l,n_{l+1},n_{l}}(\cdot)\end{pmatrix}$
(4)

###  3.3 Implementation of the current KANs

Original KAN was implemented by Liu et al. [1] by using the residual
activation function $\phi(x)$ as the sum of the base function and the spline
function with their corresponding weight matrices $w_{b}$ and $w_{s}$:

$\displaystyle\phi(x)=w_{b}b(x)+w_{s}spline(x)$ (5)

where $b(x)$ equals to $silu(x)$ and $s(x)$ is expressed as a linear
combination of B-splines. Each activation function is activated with $w_{s}=1$
and $spline(x)\approx 0$, while $w_{b}$ is initialized by using Xavier
initialization.

EfficientKAN adopted the same approach as Liu et al. [1] but reworked the
computation using B-spline basis functions followed by linear combination,
reducing memory cost and simplifying
computation444https://github.com/Blealtan/efficient-kan. The authors replaced
the incompatible L1 regularization on input samples with L1 regularization on
weights. They also added learnable scales for activation functions and
switched the base weight and spline scaler matrices to Kaiming uniform
initialization, significantly improving performance on MNIST.

FastKAN can speed up training over EfficientKAN by using GRBFs to approximate
the 3-order B-spline and employing layer normalization to keep inputs within
the RBFs’ domain [4]. These modifications simplify the implementation without
sacrificing accuracy. The GRBF has the formula:

$\displaystyle\phi(r)=e^{-\epsilon r^{2}}$ (6)

where $r=\|\mathbf{x}-\mathbf{c}\|$ is the Euclidean distance between the
input vector $\mathbf{x}$ and the center vector $\mathbf{c}$, and $\epsilon$
($epsilon>0$) is a sharp parameter that controls the width of the Gaussian
function. FastKAN uses a special form of RBFs where
$r=\frac{\mathbf{x}-\mathbf{c}}{h}$ and $\epsilon=\frac{1}{2}$, we can rewrite
Equation 6 as [4]:

$\displaystyle\phi(r)=\exp\left(-\frac{1}{2}\left(\frac{\mathbf{x}-\mathbf{c}}{h}\right)^{2}\right)$
(7)

which $h$ controls the width of the Gaussian function. Finally, the RBF
network with $N$ centers can be shown as [4]:

$\displaystyle
RBF(\mathbf{x})=\sum_{i=1}^{N}w_{i}\phi(r_{i})=\sum_{i=1}^{N}w_{i}\exp\left(-\frac{1}{2}\left(\frac{\mathbf{x}-\mathbf{c}_{i}}{h}\right)^{2}\right)$
(8)

where $w_{i}$ represents adjustable weights or coefficients, and $\phi$
denotes the radial basis function as in Equation 6.

FasterKAN outperforms FastKAN in both forward and backward processing speeds
[17]. It uses Reflectional Switch Activation Functions (RSWAFs), which are
straightforward to compute due to their uniform grid structure. The RSWAF
function $\phi(r)$ is shown as:

$\displaystyle\phi(r)=1-\left(\tanh\left(\frac{\mathbf{x}-\mathbf{c}}{h}\right)\right)^{2}$
(9)

Then, the RSWAF network with $N$ centers will be:

$\displaystyle\mathit{RSWAF}(\mathbf{x})=\sum_{i=1}^{N}w_{i}\phi(r_{i})=\sum_{i=1}^{N}w_{i}\left(1-\left(\tanh\left(\frac{\mathbf{x}-\mathbf{c}_{i}}{h}\right)^{2}\right)\right)$
(10)

GottliebKAN showed the best result compared to the other 17 polynomial basic
functions for setting up KANs on MNIST [21]. Gottlieb polynomials are a family
of polynomials that arise in the study of the Bernoulli numbers and have
applications in combinatorics and number theory [21].

###  3.4 BSRBF-KAN

Inspired by successful implementations of previous KAN models, we developed
BSRBF-KAN by combining B-splines and RBFs, which are widely recognized in
computational mathematics for interpolation and approximation tasks. The RBFs
we chose here are Gaussian ones because they are popular. Both B-splines and
RBFs are smooth functions, ensuring continuity and derivatives up to a
specified order, making them suitable for choosing proper $\Phi_{q}$ and
$\phi_{q,p}$ in KANs.

From a variable $x$ in the vector input $\mathbf{x}$, the BSRBF function
$\phi(x)$ is represented as:

$\displaystyle\phi(x)=w_{b}b(x)+w_{s}(\phi_{BS}(x)+\phi_{RBF}(x))$ (11)

where $b(x)$ and $w_{b}$ are the base output and its base matrix.
$\phi_{BS}(x)$ and $\phi_{RBF}$ are B-Spline and RBF functions, and $w_{s}$ is
the spline matrix. The design of BSRBF-KAN is similar to the original KAN,
mentioned in Section 3.2. Given the goal of designing a combined KAN, our
intuition understands that the forward and backward speeds will not surpass
those of individual KANs.

##  4 Experiments

We conducted 5 training sessions for each model on MNIST [5] and Fashion-MNIST
[6] to assess their overall performance more reliably. By taking the best
metric value from all sessions, we aimed to mitigate the effects of
variability in training and ensure that the evaluation reflects the models’
peak capabilities. With the intention of designing simple networks, we only
use activation functions, linear transformation, and layer normalization.

All models have a network structure of (784, 64, 10), comprising 784 input
neurons, 64 hidden neurons, and 10 output neurons corresponding to the 10
output classes (0-9). For GottliebKAN, we used a structure (784, 64, 64, 10)
to make it the same as the original
design555https://github.com/seydi1370/Basis_Functions. We trained the models
with 15 epochs on MNIST and 25 epochs on Fashion-MNIST. Other hyperparameters
are the same in all models, including batch_size = 64, learning_rate = 1e-3,
weight_decay=1e-4, optimize = AdamW, and loss=CrossEntropy.

Figure 1 demonstrates training losses of models represented by logarithm
values in a certain training time. On both MNIST and Fashion-MNIST, the
training loss values of BSRBF-KAN drop sharply continuously, which makes it
the model with the smallest loss after the training process. In contrast,
GottliebKAN lost the least loss, which affected its performance in the next
experiments.

Figure 1: The logarithm values of training losses (left) and validation losses (right) in 15 epochs in a training session. Table 1: The best metric values in 5 training sessions on MNIST and Fashion-MNIST. Dataset |  Model |  Train. Acc. |  Val. Acc. |  F1 |  Time (seconds) |  #Params
---|---|---|---|---|---|---
MNIST |  BSRBF-KAN |  100.0 |  97.63 |  97.6 |  222 |  459040
FastKAN |  99.94 |  97.38 |  97.34 |  102 |  459114
FasterKAN |  98.52 |  97.38 |  97.36 |  93 |  408224
EfficientKAN |  99.34 |  97.54 |  97.5 |  122 |  508160
GottliebKAN |  99.66 |  97.78 |  97.74 |  269 |  219927
MLP |  99.42 |  97.69 |  97.66 |  273 |  52512
Fashion-MNIST |  BSRBF-KAN |  99.3 |  89.59 |  89.54 |  219 |  459040
FastKAN |  98.27 |  89.62 |  89.6 |  160 |  459114
FasterKAN |  94.4 |  89.39 |  89.3 |  157 |  408224
EfficientKAN |  94.83 |  89.11 |  89.04 |  182 |  508160
GottliebKAN |  93.79 |  87.69 |  87.61 |  241 |  219927
MLP |  93.58 |  88.51 |  88.44 |  147 |  52512
Average of MNIST + Fashion-MNIST |  BSRBF-KAN |  99.65 |  93.61 |  93.57 |  220.5 |  459040
FastKAN |  99.11 |  93.50 |  93.47 |  131 |  459114
FasterKAN |  96.46 |  93.39 |  93.33 |  125 |  408224
EfficientKAN |  97.09 |  93.33 |  93.27 |  152 |  508160
GottliebKAN |  96.73 |  92.74 |  92.68 |  255 |  219927
MLP |  96.50 |  93.10 |  93.05 |  210 |  52512
Train. Acc = Training Accuracy, Val. Acc. = Validation Accuracy, #Params = Parameters |
Table 2: The average metric values in 5 training sessions on MNIST and Fashion-MNIST. Dataset |  Model |  Train. Acc. |  Val. Acc. |  F1 |  Time (seconds)
---|---|---|---|---|---
MNIST |  BSRBF-KAN |  100.00 ± 0.00 |  97.55 ± 0.03 |  97.51 ± 0.03 |  231
FastKAN |  99.94 ± 0.01 |  97.25 ± 0.03 |  97.21 ± 0.03 |  101
FasterKAN |  98.48 ± 0.01 |  97.28 ± 0.06 |  97.25 ± 0.06 |  93
EfficientKAN |  99.37 ± 0.04 |  97.37 ± 0.07 |  97.33 ± 0.07 |  120
GottliebKAN |  98.44 ± 0.61 |  97.19 ± 0.22 |  97.14 ± 0.23 |  221
MLP |  99.44 ± 0.01 |  97.62 ± 0.03 |  97.59 ± 0.03 |  181
Fashion-MNIST |  BSRBF-KAN |  99.19 ± 0.03 |  89.33 ± 0.07 |  89.29 ± 0.07 |  211
FastKAN |  98.19 ± 0.04 |  89.42 ± 0.07 |  89.38 ± 0.07 |  162
FasterKAN |  94.40 ± 0.01 |  89.26 ± 0.06 |  89.17 ± 0.07 |  154
EfficientKAN |  94.76 ± 0.06 |  88.92 ± 0.08 |  88.85 ± 0.09 |  183
GottliebKAN |  90.66 ± 1.08 |  87.16 ± 0.24 |  87.07 ± 0.25 |  238
MLP |  93.56 ± 0.05 |  88.39 ± 0.06 |  88.36 ± 0.05 |  148
Average of MNIST + Fashion-MNIST |  BSRBF-KAN |  99.60 |  93.44 |  93.40 |  221
FastKAN |  99.07 |  93.34 |  93.30 |  131.5
FasterKAN |  96.44 |  93.27 |  93.21 |  123.5
EfficientKAN |  97.07 |  93.15 |  93.09 |  151.5
GottliebKAN |  94.55 |  92.18 |  92.11 |  229.5
MLP |  96.50 |  93.01 |  92.98 |  164.5
Train. Acc = Training Accuracy, Val. Acc. = Validation Accuracy |

###  4.1 MNIST results analysis

As shown in Table 1, GottliebKAN is better than the other models, with a
validation accuracy of 97.78% and an F1 score of 97.74%. We assume this
because of its deeper network structure. MLP and BSRBF-KAN achieved second-
best and third-best performances but took a long time to train. Conversely,
FasterKAN had the shortest training time, but its performance slightly
surpassed FastKAN. BSRBF-KAN is the only model with a training accuracy of
100% after 15 epochs. If counting the results based on average values, MLP is
the best model, as displayed in Table 2, followed competitively by BSRBF-KAN.
Despite achieving the highest accuracy values, GottliebKAN is unstable.

###  4.2 Fashion-MNIST results analysis

The ease of achieving high accuracy of models trained on MNIST may not
accurately reflect their actual performance. Thus, we experimented with the
model training on Fashion-MNIST. As shown in Table 1 and Table 2, FastKAN
obtained the highest validation accuracy. At the same time, BSRBF-KAN followed
competitively with and still kept the best convergence ability. MLP is the
second-worst model, only performing better than GottliebKAN, but it has the
fastest training time and the fewest parameters. However, when considering
training time, we see that FasterKAN’s training time is only slightly longer
than MLP’s, while achieving nearly 0.9% higher validation accuracy, according
to the results in Table 1.

###  4.3 MNIST + Fashion-MNIST results analysis

When calculating the average values on MNIST and Fashion-MNIST, we clearly see
that BSRBF-KAN achieved the best performance and also the best convergence,
according to Table 1 and Table 2. Furthermore, all KANs except for GottliebKAN
surpassed MLP. This indicates the positive signal of using KANs to obtain
better accuracy values in model training, at least in these experiments.

###  4.4 Ablation Study

Several ablation experiments were performed to help us understand the impact
of various components within BSRBF-KAN on overall model performance. Table 3
presents the metric values for different combinations of BSRBF-KAN components.
Initially, we trained the model with all components included, followed by
training variants with some components removed. The results show that the base
output and layer normalization are key for the model’s performance, as their
both absence leads to the worst results.

While B-splines and RBFs can enhance performance, removing RBFs have less
impact on the model than B-splines. Interestingly, when both are absent, the
model becomes an MLP, which exhibits slightly better validation accuracy and
F1 scores than the complete BSRBF-KAN model on MNIST, but on Fashion-MNIST,
this is pretty bad. Thus, we believe that retaining B-splines and RBFs remains
beneficial for the model’s performance.

Table 3: The performance of BSRBF-KAN by different components on MNIST and Fashion-MNIST in a training setup. Dataset |  Components |  Train. Acc. |  Val. Acc. |  F1
---|---|---|---|---
MNIST |  Full |  100.0 |  97.53 |  97.49
No BS |  \- 0.01 |  \- 0.21 |  \- 0.21
No RBF |  \- 0.003 |  \- 0.14 |  \- 0.14
No BS + No RBF = MLP |  \- 0.55 |  \+ 0.05 |  \+ 0.06
No BO |  \- 0.01 |  \- 0.41 |  \- 0.41
No LN |  \- 1.7 |  \- 1.5 |  \- 1.52
|  No BO + No LN |  \- 5.16 |  \- 5.97 |  \- 6.08
Fashion-MNIST |  Full |  99.39 |  89.43 |  89.4
No BS |  \- 0.16 |  \- 0.39 |  \- 0.37
No RBF |  \- 0.34 |  \- 0.14 |  \- 0.15
No BS + No RBF = MLP |  \- 5.91 |  \- 1.1 |  \- 1.1
No BO |  \- 0.04 |  \- 0.26 |  \- 0.29
No LN |  \- 6.12 |  \- 1.1 |  \- 1.13
|  No BO + No LN |  \- 5.91 |  \- 2.96 |  \- 3.01
Train. Acc = Training Accuracy, Val. Acc. = Validation Accuracy |
No BS = No B-spline, No RBF = No Radial Basis Function |
No BO = No Base Output, No LN = No Layer Normalization |

##  5 Conclusion

We introduced BSRBF-KAN, a novel KAN that integrates B-splines and Radial
Basis Functions (RBFs) to fit input vectors in the training process. We did
not intend to make BSRBF-KAN better than MLP and other KANs by optimizing
hyperparameters; instead, we built it by default to understand aspects of KANs
combination in training models. In the experiments, we trained BSRBF-KAN, MLP,
and other KANs on the MNIST and Fashion-MNIST datasets. BSRBF-KAN demonstrated
competitive performance and rapid convergence compared to other models.
Furthermore, it is the best model if counting on the average values of both
datasets. However, its high convergence rate can lead to overfitting, which
can be mitigated by using dropout.

All models were generally stable, except for GottliebKAN. Although GottliebKAN
achieved the highest validation accuracy on MNIST, it performed the worst on
Fashion-MNIST. Similarly, while MLP achieved the highest average validation
accuracy on MNIST, it underperformed compared to other KANs on Fashion-MNIST.
Therefore, it is crucial to experiment with our approach on other datasets in
the future to gain a more comprehensive understanding. We will also continue
to explore the combination of KANs and their components when designing new KAN
architectures to enhance model performance across a wider range of problems.

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