Variational Formulation

Variational formulations provide a powerful framework for solving complex problems across diverse scientific domains, primarily by recasting them as optimization problems involving the minimization of a functional. Current research focuses on integrating variational methods with deep learning, particularly employing neural networks within algorithms like Kalman filters and variational autoencoders to address high-dimensional systems and multiscale phenomena. This approach enhances the efficiency and accuracy of solving partial differential equations and inverse problems, impacting fields such as data assimilation, image processing, and fluid dynamics through improved model accuracy and computational speed.