Numerical scheme-guided Deep Learning for scientific computing

In recent years, driven by the impressive advancement in Artificial Intelligence, extremely powerful GPUs and software have become available. Meanwhile, with the increasing availability and importance of drones and robotic devices, the need for efficient algorithms to optimally control these devices has become more pressing. This project will develop novel algorithms with solid mathematical theories, bringing Artificial Intelligence to drones and robotic devices for mission-critical tasks, and pushing the frontier of scientific computing and simulations. In addition to drones and robotic devices, research outcomes will expand simulation capacity for a range of application areas, including nuclear fusion research. Regarding education and human resource development, research outcomes will be integrated into graduate-level courses and a new course series for undergraduates emphasizing the mathematical and numerical analytical foundations for machine learning. This project aims to develop and analyze innovative algorithms that integrate classical numerical schemes and the Deep Learning paradigm, including its software-hardware ecosystem, to address complex scientific computing challenges. By leveraging the flexibility of neural networks, the stability and convergence properties of numerical methods, and the computational power of modern GPUs, the proposed framework will tackle problems in high-dimensional, fully nonlinear differential equations, long-time simulation of Hamiltonian systems, and boundary integral equations. 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.