Reduced Basis

Reduced basis methods aim to efficiently approximate solutions of high-dimensional partial differential equations (PDEs) by projecting them onto a lower-dimensional subspace, significantly reducing computational cost. Current research emphasizes integrating machine learning techniques, such as deep autoencoders, neural ordinary differential equations, and support vector regression, with traditional reduced basis approaches to handle nonlinear dynamics and improve accuracy, particularly for problems with limited or noisy data. This allows for faster simulations and real-time applications in diverse fields like optimal control, fluid dynamics, and structural mechanics, accelerating scientific discovery and engineering design processes.