# Kolmogorov-Arnold Finance-Informed Neural Network in Option Pricing
## 4 Kolmogorov-Arnold Finance-Informed Neural Network
### Minimal Viable KAFIN  
Based on the KAR theory, Finance-Informed Neural Networks (FINNs) integrate financial theories and constraints directly into neural network architectures to enhance predictive accuracy and model reliability in financial applications. KAFIN serves as a type of Finance-Informed Neural Network (FINN) with Kolmogorov-Arnold representation (KAR), integrating financial domain knowledge and sophisticated mathematical frameworks to enhance the predictive capabilities and interpretability of financial models. Consider a financial system characterized by a set of differential equations representing financial dynamics, such as option pricing models or portfolio optimization:  
\[\mathcal{F}[\mathbf{u}(\mathbf{x},t);\lambda]=0,\quad\mathbf{x}\in\Omega, \quad t\in[0,T] \tag{2}\]  
where \(\mathcal{F}\) denotes the financial differential operator, \(\mathbf{u}(\mathbf{x},t)\) represents the financial variable of interest (e.g., option price, asset value), \(\lambda\) includes parameters of the financial model, \(\Omega\) represents the domain of financial variables, and \([0,T]\) denotes the time period of interest. Kolmogorov-Arnold representation serves as a theoretical backbone for constructing KAFIN that approximates solutions to complex systems. By minimizing the total loss function, the parameters \(\theta\) are optimized to ensure that network predictions satisfy initial, boundary, and governing conditions. This approach combines mathematical rigor with computational efficiency, offering a robust methodology for addressing financial modeling challenges. In KAFIN, the neural network \(\mathbf{u}_{\theta}(\mathbf{x},t)\) is constructed as a Kolmogorov-Arnold Network (KAN) with parameters \(\theta\) to approximate the solution \(\mathbf{u}(\mathbf{x},t)\). Utilizing KAR, Equation (1) can be represented as a network to reflect the decomposition as  
\[\mathbf{u}_{\theta}(\mathbf{x},t)=\sum_{i=1}^{L_{2}}g_{i,\theta}\left(\sum_{j =1}^{L_{1}}h_{ij,\theta}(x_{j},t)\right) \tag{3}\]  
where \(g_{i,\theta}(\cdot)\) and \(h_{ij,\theta}(\cdot)\) are learned by the network, and \(L_{1},L_{2}\) are the numbers of the function units used in layer 1 and 2. By learning the parameters, KAFIN can decompose any continuous multivariate function into the sum of univariate functions of linear combinations of the input variables. The parameters \(\theta,g_{i,\theta}(\cdot)\) and \(h_{ij,\theta}(\cdot)\) are optimized by minimizing the total loss function, which can be formulated as  
\[\theta^{*}=\arg\min_{\theta}\mathcal{L}(\theta) \tag{4}\]The loss function in the context of financial constraints and system comprises several key components, including initial value loss, boundary loss, and financial governing loss. \[\mathcal{L}(\theta)=\mathcal{L}_{\text{initial}}(\theta)+\mathcal{L}_{\text{ boundary}}(\theta)+\mathcal{L}_{\text{financial}}(\theta) \tag{5}\]  
in which the initial value loss \(\mathcal{L}_{\text{initial}}(\theta)\) ensures that the neural network's prediction at the initial time \(t=0\) matches the given initial condition; the boundary loss \(\mathcal{L}_{\text{boundary}}(\theta)\) ensures that the neural network solution satisfies the boundary conditions of the financial domain; the financial governing loss \(\mathcal{L}_{\text{financial}}\) enforces adherence to the financial models and constraints. The formulation of the loss function is crucial in developing a robust minimal viable KAFIN. Integration of convex learning mechanisms [31] ensures that all loss functions satisfy convex constraints related to initial, boundary, and financial considerations. The mean squared error (MSE) metric is particularly effective in this regard, as it precisely measures the average disparities between estimated outputs and their true references, offering a straightforward and robust framework for defining the loss function.