Today, machine learning has an accelerated impact in quantitative finance. Current models require large amounts of data, which can be expensive. A notable area of research, physics-informed neural networks (PINNs), has proven to be effective in approximating problems that are described by partial differential equations (PDEs). During training, the PDE is embedded in the loss function and evaluated at the residual points. This allows these types of neural networks to solve problems where data is scarce or noisy. Recent studies have shown that the method for sampling residual points has a great influence on training efficiency. Residual-based adaptive distribution (RAD) sampling is the adaptive sampling method used throughout this paper. This research applies PINNs with RAD sampling to solve the Black-Scholes PDE. Here, the Black-Scholes model is used to determine the price of options in a financial market. The fundamental goal of this paper is to study the difference in training performance between non-adaptive and RAD sampling. The types of options that are being considered in this study, are the European call options and the American put options. The results shown suggest that both types of options benefit from using RAD sampling compared to non-adaptive sampling. With a loss decrease of 39.33\%, American put options improve more using RAD sampling than European call options. Although European call options still show a decrease in loss of 7.57\%.

Self-Adaptive Physics-Informed Neural Networks
(SA-PINNs) are a variation of traditional Physics-Informed Neural Networks (PINNs) designed to
solve the challenges of solving ”stiff” partial differential equations (PDEs). By using adaptive weighting, SA-PINNs are able to focus their attention on areas of the domain with higher errors, therefore improving accuracy. This work investigates the
roles of individual loss components, namely residuals, boundary conditions, and initial conditions, in the performance of SA-PINNs.

Physics-Informed Neural Networks (PINNs) are intended to solve complex problems that obey physical rules or laws but have noisy or little data. These problems are encountered in a wide range of fields including for instance bioengineering, fluid mechanics, meta-material design and high-dimensional partial differential equations (PDEs). Whilst PINNs show promising results, they often fail to converge in the presence of higher frequency components; a problem known as the spectral bias. Multiple studies have explored ways to overcome or minimize spectral bias specifically for PINNs. This paper builds on previous studies by investigating the impact of different gradient descent methods on the spectral bias.

Physics-Informed Neural Networks(PINNs) have emerged as a potent, versatile solution to solving both forward and inverse problems regarding partial differential equations(PDEs), accomplished through integrating laws of physics into the learning process. The applications of this new approach are endless, as these types of equations appear across numerous fields: fluids mechanics, quantum mechanics, electrochemistry and many others. Ever since their conception, researchers have continuously improved the flexibility and performance of PINNs through advancements in the architecture of neural networks, optimization algorithms, creative sampling methods and many more. As computational power and the interest of researchers grow, the revolutionary potential of PINNs is closer to fulfillment than ever. This paper aims to examine a small part of this evolutionary process, specifically the performance and flexibility of different activation functions used in the training of the PINN, as well as potential problems this approach could solve.

This thesis evaluates AI-based weather forecasting models—specifically GraphCast and Pangu-Weather—against traditional numerical weather prediction models like the ECMWF High-Resolution Model (HRES) and the Aspire Meso and LES models, focusing on the Netherlands due to its dense observational network and wind energy infrastructure. The study examines operational considerations, including computational costs and hardware requirements, highlighting that while AI models offer significant computational efficiency during inference, they require substantial resources for training and have limitations in local adaptability and variable inclusion (e.g., Pangu-Weather lacks precipitation data). Performance analysis demonstrates that GraphCast consistently outperforms HRES, Pangu-Weather, and Aspire Meso across various meteorological variables and lead times. Despite the advantages, the models exhibit baseline errors due to shared data sources like ERA5, leading to correlated errors that limit the effectiveness of ensemble forecasting. The research underscores the potential of AI models to enhance forecast accuracy and reduce imbalance costs in wind energy production but also emphasizes challenges related to the black-box nature of AI models, data bottlenecks, and limitations imposed by neural scaling laws, where larger models do not necessarily yield better performance. Recommendations for future work include incorporating more observational data, enhancing model modularity and adaptability, improving temporal resolution, expanding variable ranges, emphasizing validation against observational data, and exploring advanced ensemble techniques. The findings suggest that while AI-based models like GraphCast and Pangu-Weather hold significant promise for advancing weather forecasting, addressing limitations in data quality, model architecture, and operational flexibility is crucial for realizing their full potential in operational settings.