High Dimensional Partial Differential Equation

High-dimensional partial differential equations (PDEs) pose significant computational challenges due to the "curse of dimensionality," where computational cost explodes with increasing dimensions. Current research focuses on developing efficient deep learning-based solvers, employing architectures like Physics-Informed Neural Networks (PINNs), DeepONets, and convolutional neural networks (CNNs), often incorporating techniques such as tensor decompositions, multilevel methods, and latent space representations to mitigate the dimensionality problem. These advancements are crucial for tackling complex scientific and engineering problems across diverse fields, enabling more accurate and efficient simulations of high-dimensional systems previously intractable with traditional numerical methods.