# Kolmogorov-Arnold Finance-Informed Neural Network in Option Pricing
## 3 Methodology
### Kolmogorov-Arnold Representation  
Let \(f:\mathbb{R}^{n}\to\mathbb{R}\) be a continuous function. According to the Kolmogorov-Arnold representation theorem, there exist continuous functions \(\phi_{k}:\mathbb{R}^{n-1}\to\mathbb{R}\) for \(k=1,2,\ldots,n\), such that \(f\) can be expressed as:  
\[f(x_{1},x_{2},\ldots,x_{n})=\sum\int_{\mathbb{R}^{n-1}}\phi_{k}(y_{2},\ldots,y _{n})\,dy_{2}\ldots dy_{n}\]  
where each \(\phi_{k}\) depends only on the respective variable \(x_{k}\) and possibly on the other variables \(x_{j}\) for \(j\
eq k\).  
In other words, \(f\) can be written as a superposition of functions of fewer variables:  
\[f(x_{1},x_{2},\ldots,x_{n})=\sum_{k=1}^{n}\phi_{k}(x_{k},x_{-k})\]  
where \(x_{-k}=(x_{1},\ldots,x_{k-1},x_{k+1},\ldots,x_{n})\).  
The Kolmogorov-Arnold representation (KAR) theorem states that any continuous multivariate function can be expressed as a superposition of univariate functions. For a continuous function \(f(x_{1},x_{2},\ldots,x_{n})\), KAR asserts the following representation as  
\[f(x_{1},x_{2},\ldots,x_{n})=\sum_{i=1}^{N}g_{i}(h_{i}(x_{1}),h_{i}(x_{2}), \ldots,h_{i}(x_{n})) \tag{1}\]where \(g_{i}\) refers to univariate functions, \(h_{i}\) refers to mappings that define how each univariate function interacts with each variable, and \(N\) refers to the number of terms in the superposition, which can be finite or infinite. Expanding on the theorem, KAR further specifies that any continuous function \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}\) can be expressed in a more specific form as  
\[f(x_{1},x_{2},\ldots,x_{n})=\sum_{i=1}^{2n+1}g_{i}\left(\sum_{j=1}^{n}h_{ij}(x _{j})\right)\]  
where \(g_{i}\) and \(h_{ij}\) are continuous univariate functions. In this form, each \(g_{i}\) is applied to a sum of transformed input variables \(h_{ij}(x_{j})\), where \(j\) indexes over all the input variables \(x_{1},x_{2},\ldots,x_{n}\). The number of terms \(2n+1\) reflects the minimum number of components needed to represent the multivariate function, based on the structure of the input space and the interaction of the variables.  
With the superposition principle, KAR simplifies the representation of complex multivariate functions into a sum of simpler univariate components of KAR. It simplifies the solving of complex differential equations by breaking them down into simpler parts. In computational contexts, KAR reduces the computational complexity of high-dimensional problems by decomposing them into lower-dimensional operations. It facilitates effective nonlinear approximations, where complex relationships between variables can be approximated by simpler functions, aiding in interpretability and efficiency.  
This representation captures the idea that a complex multivariate function can be constructed by summing over simpler, univariate transformations of the individual variables. The functions \(h_{ij}(x_{j})\) capture the individual contributions of each variable \(x_{j}\) to the overall function, while the functions \(g_{i}\) aggregate these contributions into a final output. The decomposition through KAR is crucial for understanding and modeling complex systems where multiple input variables interact in a nonlinear fashion.