Aplicación de redes neuronales a la resolución numérica de problemas diferenciales

Abstract

Neural networks emerged as an inspiration from the functioning of the human brain, which motivated their development and evolution. These networks have demonstrated a remarkable ability to approximate complex functions, a capability grounded in the Universal Approximation Theorem. This property has propelled them to become a powerful tool for the numerical solution of complex mathematical problems, such as ordinary differential equations (ODEs), systems of differential equations (SDEs), and partial differential equations (PDEs), which model a wide variety of phenomena in applied sciences and engineering. This work focuses on the design and tuning of neural networks for approximating solutions to differential problems, with special emphasis on the impact that parameter selection has on the model’s behavior, performance, and fitting capability. It also delves into Physics-Informed Neural Networks (PINNs), which directly incorporate physical laws into the training process, resulting in more accurate solutions. The implementations are carried out in Python using the PyTorch and TensorFlow libraries, applying these techniques to both forward and inverse problems. Finally, the potential of PINNs is highlighted as an innovative and evolving technique that offers a fast and efficient way to approximate solutions to differential equations that describe real-world phenomena in diverse fields such as physics, biology, engineering, and more.