Title: Solution discovery in fluids with high precision using neural networks
Speaker: Ching-Yao Lai, Stanford University
Abstract: 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 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.
Joint work with Yongji Wang, Tristan Buckmaster, Javi Gómez-Serrano, and Google Deepmind
Zoom: https://northwestern.zoom.us/j/96136539606
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