Since the liberalization of the energy markets, the storage of energy is decoupled from the production and sales. In Western-Europe the storage of natural gas becomes more and more important because production fields get depleted and governments force companies to slow down their production because of tremors in the ground. Natural gas needs to be imported from countries that are far away, like for example Russia. To provide in security of supply and to ensure there is enough natural gas when the demand is high, it is important to store natural gas nearby.

To determine the value of a gas storage facility in a reliable way we need an efficient market. For an efficient market is needed that the financial instruments, like futures contracts and options on natural gas, are liquidly traded on the exchange. If this condition is met, we are able to determine the value of storage according to market prices.

The COS method was already presented as an efficient method for pricing a broad spectrum of financial derivatives and can be used in combination with all processes for the underlying for which a characteristic function is known. For processes whose characteristic function is not available, the adjoint expansion method can be used to obtain an approximation of the characteristic function. In this work the COS method will be presented as an efficient method for determining the value of gas storage contracts which is competitive with existing valuation methods for natural gas storage contracts.

Recent works have shown that neural networks can be employed to solve partial differential equations, bringing rise to the framework of physics informed neural networks.The aim of this project is to gain a deeper understanding of these novel methods, and to use these insights to further improve them. We show that solving a partial differential equation can be formulated as a multi-objective optimization problem, and use this formulation to propose several modifications to existing methods. These modifications manifest as a scaling parameter, which can improve the accuracy by orders of magnitude for certain problems when it is chosen properly. We also propose heuristic methods to approximate the optimal scaling parameter, which can be used to eliminate the need to optimize this parameter. Our proposed methods are tested on a variety of partial differential equations and compared to existing methods. These partial differential equations include the Laplace equation, which we solve in up to four dimensions, the convection-diffucsion equation and the Helmholtz equation, all of which show that our proposed modifications lead to enhanced accuracy.