Parametric PDE

Parametric partial differential equations (PDEs) model systems whose behavior depends on varying parameters, posing challenges for efficient solution and uncertainty quantification. Current research focuses on developing neural network architectures, such as convolutional neural networks (CNNs) and physics-informed neural networks (PINNs), often combined with techniques like proper orthogonal decomposition (POD) and radial basis functions (RBFs), to learn the parameter-to-solution map efficiently. These methods aim to accelerate computations and improve generalization to unseen parameter regimes, addressing limitations of traditional numerical solvers. This work has significant implications for diverse fields, enabling faster and more accurate simulations in areas like fluid dynamics, heat transfer, and plasma physics.