\begin{algorithm}[htb!]
	Step 1: Initialize the parameters $\boldsymbol{\theta}$ of the neural 
	network $\hat{u}(\boldsymbol{x};\boldsymbol{\theta})$.\\
	Step 2: Set up the training sets 
	$\left\{\boldsymbol{x}_{\Omega}\right\}$ 
	and $\left\{\boldsymbol{x}_{\partial \Omega}\right\}$ for the equation 
	and 
	boundary/initial conditions, respectively.\\
	Step 3: Define the loss function by calculating the weighted $L^2$ 
	norm of 
	both the differential equation and boundary condition residuals.\\
	Step 4: Train the neural network by minimizing the loss function 
	$\mathcal{L}(\boldsymbol{\theta})$ to obtain the optimal parameters 
	$\boldsymbol{\theta}^*$.
	\caption{Physics-Informed Neural Network (PINN) algorithm for solving 
	differential equations.}
	\label{algorithm:1}
\end{algorithm}

\begin{algorithm}
[htb!]
Step 1: Initialize the parameters $\boldsymbol{\theta}$ of the neural
network $\hat{u}(\boldsymbol{x};\boldsymbol{\theta})$.\\
Step 2: Set up the training sets
$\left\{\boldsymbol{x}_{\Omega}\right\}$
and $\left\{\boldsymbol{x}_{\partial \Omega}\right\}$ for the equation
and
boundary/initial conditions, respectively.\\
Step 3: Define the loss function by calculating the weighted $L^2$
norm of
both the differential equation and boundary condition residuals.\\
Step 4: Train the neural network by minimizing the loss function
$\mathcal{L}(\boldsymbol{\theta})$ to obtain the optimal parameters
$\boldsymbol{\theta}^*$.
\caption{Physics-Informed Neural Network (PINN) algorithm for solving
differential equations.}
\end{algorithm}
{
    "title": "computing the gerber-shiu function with interest and a constant dividend   barrier by physics-informed neural networks",
    "id": "2401.04378",
    "abstract": "in this paper, we propose a new efficient method for calculating the gerber-shiu discounted penalty function. generally, the gerber-shiu function usually satisfies a class of integro-differential equation. we introduce the physics-informed neural networks (pinn) which embed a differential equation into the loss of the neural network using automatic differentiation. in addition, pinn is more free to set boundary conditions and does not rely on the determination of the initial value. this gives us an idea to calculate more general gerber-shiu functions. numerical examples are provided to illustrate the very good performance of our approximation.",
    "categories": "math.na cs.na math.pr q-fin.rm",
    "doi": "",
    "created": "2024-01-09",
    "updated": "",
    "authors": [
        "zan yu",
        "lianzeng zhang"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2401.04378"
}
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