Acknowledgement:

I would like to express my gratitude to Additi Pandey for guiding me throughout this process. Her guidance, support, and feedback were invaluable. I am also grateful to Professor George Karniadakis for his work on physics-informed neural networks, which inspired the methods used in this paper. This research was conducted as part of an independent internship at the Crunch Group during the summer of 2025. 

Trying different schedulers

On employing different learning rate schedulers, we observed that the StepLR scheduler significantly improved model performance, decreasing the relative error to approximately 0.76%. This represents a notable improvement over the baseline model without a scheduler, which yielded an error of 1.17%. The gradual and systematic reduction of the learning rate helped the model avoid overshooting during training and led to smoother convergence of the loss function. 

Exploring the right optimizer

In addition to the learning rate schedulers, we analyzed the impact of different optimizers on training efficiency and PDE accuracy. The Adam optimizer achieved the lowest overall error of 0.81%, followed by Adagrad (0.95%) and AdamW (1.89%). RMSprop, in contrast, showed the highest error of 4.27%. RMSprop exhibited the highest error (4.27%), likely due to slower adaptation to the gradient oscillations in the PDE residual. The superior performance of Adam demonstrates its ability to balance convergence speed and stability, making it the most suitable choice for our model. 

Experiments with different activation functions

We further compared the performance of different activation functions. The model employing the ReLU activation function outperformed the GELU-based network in both convergence speed and prediction accuracy. ReLU achieved a lower residual error of 1.014% compared to GELU's 1.309%. ReLU appeared to capture the kink more effectively, while GELU’s curve occasionally slowed down convergence due to smaller gradient magnitudes.

Testing on real market data:

To demonstrate the effectiveness of PINNs in modeling the Black-Scholes equation, we further tested our model on real market data from yfinance for Tesla (TSLA). The model achieved strong agreement with theoretical values across the asset price grid. For TSLA, which exhibits higher volatility, the error was at 1.61%. These results demonstrate that the trained PINN model can generalize well to real-world market data, maintaining high predictive accuracy for both stable and volatile assets.

In a nutshell:

We modeled the Black–Scholes equation for American-style options using Physics-Informed Neural Networks (PINNs) and compared the results with traditional finite difference methods. By experimenting with different optimizers, activation functions, and learning rate schedulers, we found that the StepLR scheduler and ReLU activation yielded the most accurate and stable results, achieving a relative error below 1%. Using historical stock data for Tesla (TSLA), our PINN model demonstrated strong convergence and realistic option price predictions. These findings highlight the potential of PINNs to bridge financial modeling and machine learning, offering a robust alternative to conventional numerical schemes for complex derivative pricing problems.

Project Report with References:

We have uploaded the entire project report with references in this HTML file below. The code is available here.

Note that the aim of this project was to learn various ML techniques and how to integrate them together to solve a problem. The objective was also to explore PINNs. In this report, I only state my empirical observations and report errors from a single run. With different parameter tuning the results may vary.