Randomized Algorithms for Operator Approximations in Sobolev Spaces

Machine learning and artificial intelligence has been successful in the approximation and prediction of complex physical phenomena. A key aspect is the development of models capable of capturing dependencies on input parameters, domain configurations, boundary conditions, initial states, and spacetime coordinates within one neural network. One approach is operator learning, which encodes the solution operators of parametric partial differential equations into neural networks. However, the size of these neural networks often grows with the complexity of the task, that is, the accuracy of the methods can be limited by available computational resources and memory. This project aims to develop efficient algorithms with rigorous theoretical support for constructing accurate solution operators through the use of randomization techniques and numerical analysis. The tools developed will broaden the scope of machine learning applications in scientific modeling. In addition, the outcomes will contribute to curriculum development in both graduate and undergraduate mathematics as well as the training of graduate students in this research. The project is on the design, implementation, and analysis of randomized operator learning algorithms for enabling efficient and reliable scientific computation. In particular, this project will develop tractable training algorithms for solving parametric partial differential equations in Sobolev spaces with a focus on the noisy and limited data settings. The theoretical analysis will involve a comprehensive study of the algorithms' behavior in various settings, including quantitative bounds on the model's complexity, approximation accuracy, and generalization error. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.