Beijing Institute of Mathematical Sciences and Applications

Shing Toung Yau

Jia Yi Kang

Friday, January 5, 2024 9:30 PM - 10:00 PM

Online

Learning high-dimensional functions (e.g., solving high-dimensional partial differential equations (PDEs) and discovering governing PDEs) is fundamental in scientific fields such as diffusion, fluid dynamics, and quantum mechanics, and optimal control, etc. Developing efficient and accurate solvers for this task remains an important and challenging topic. Traditional solvers (e.g., finite element method (FEM) and finite difference) are usually limited to low-dimensional domains since the computational cost increases exponentially in the dimension as the curse of dimensionality. Neural networks (NNs) as mesh-free parameterization are widely employed in solving regression problems and high-dimensional PDEs. Yet the highly non-convex optimization objective function in NN optimization makes it difficult to achieve high accuracy. The errors of NN-based solvers would still grow with the dimension. Besides, NN parametrization may still require large memory and high computation cost for high-dimensional problems. Finally, numerical solutions provided by traditional solvers and NN-based solvers are not interpretable, e.g., the dependence of the solution on variables cannot be readily seen from numerical solutions. The key to tackle these issues is to develop symbolic learning to discover the low-complexity structures of a high-dimensional problem. Low-complexity structures are applied to transform a high-dimensional task into a low-dimensional learning problem.