AM Seminar: Solution Discovery in Fluids with High Precision Using Neural Networks
Presenter: Ching-Yao Lai, Assistant Professor, Stanford University
Description: I will discuss examples utilizing neural networks (NNs) to find solutions to partial differential equations (PDEs) that facilitate new discoveries. Despite being deemed universal function approximators, neural networks, in practice, struggle to fit functions with sufficient accuracy for rigorous analysis. Here, we developed multi-stage neural networks (Wang and Lai, J. Comput. Phys. 2024) that can reduce the prediction error to nearly the machine precision of double-precision floating points within a finite number of iterations. We use accurate NNs to tackle the challenge of searching for singularities in fluid equations (Wang-Lai-Gómez-Serrano-Buckmaster, Phys. Rev. Lett. 2023). Unstable singularities, especially in dimensions greater than one, are exceptionally elusive. With NNs we demonstrate the first discovery of smooth unstable self-similar singularities to unforced incompressible fluid equations (Wang et al., arXiv:2509.14185). The example illustrates how deep learning can be used to discover new and highly accurate numerical solutions to PDEs.
Bio: Ching-Yao Lai (Yao) is an Assistant Professor in the Department of Geophysics and an Affiliated Faculty of the Institute for Computational and Mathematical Engineering (ICME) at Stanford. Before joining Stanford, she was an Assistant Professor at Princeton University. She received an undergraduate degree (2013) in Physics from National Taiwan University and a PhD (2018) in Mechanical and Aerospace Engineering from Princeton University. She completed her postdoctoral research at Columbia University where she received the Lamont Postdoctoral Fellowship. Her current research focuses on enhancing the representation of machine-learning models to tackle multiscale problems. She was the recipient of the 2023 Google Research Scholar Award, the 2024 Sloan Research Fellowship, and the 2025 NSF CAREER Award.
Hosted by: Applied Mathematics