CAD Seminar Series: Zhiqiang Cai - Neural Networks in Numerical Partial Differential Equations
Speaker: Zhiqiang Cai, PhD., Purdue University
Professor, Department of Mathematics
Time: Wednesday, November 6, from 2:30 to 3:30 pm
Title: Neural Networks in Numerical Partial Differential Equations
Abstract: Neural networks (NNs) have achieved astonishing performance in computer vision, natural language processing, and many other artificial intelligence (AI) tasks. This success encourages wide applications to other fields, such as scientific computing. In this talk, I will first give a brief introduction of NNs from numerical analysis perspective and use a simple example to show why NNs are superior to piecewise polynomials on fixed meshes when approximating discontinuous functions with unknown interface. I will then use scalar nonlinear hyperbolic conservation laws as an example to discuss numerical difficulties in designing an accurate NN-based method and strategies on how to overcome those difficulties. In particular, I will present the space-time least-squares neural network (LSNN) method, that does not utilize any penalization such as inflow boundary condition, artificial viscosity, entropy condition, and/or total variation. The method shows a great potential to sharply capture shock without oscillation, overshooting, or smearing. The exceptional approximation powers of NN come with a price: the procedure for determining the values of the nonlinear parameters of NN entails solving a high-dimensional non-convex optimization problem. If time permits, I will describe our newly developed training algorithm for shallow ReLU NN.
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CAD Seminar Series: Advancing Knowledge, Innovation, and Collaboration in Computation, AI, and Data Science (CAD)
The CAD Seminar Series is a dedicated platform for advancing knowledge, fostering innovation, and promoting collaboration across the fields of Computation, Artificial Intelligence, and Data Science. This series brings together leading experts, researchers, and professionals to explore the latest developments, tackle emerging challenges, and drive forward-thinking solutions at the convergence of these critical disciplines.
Objectives:
• Advance Knowledge: Share cutting-edge research and insights that push the boundaries of what is known in CAD.
• Foster Innovation: Encourage the development of novel ideas and solutions through interdisciplinary dialogue and creative thinking.
• Promote Collaboration: Unite expertise across disciplines and build bridges between academia, industry, and government to address complex problems and create opportunities for joint ventures.
Schedule: The seminars will be held on a weekly basis (unless otherwise noted) throughout the academic year. For Fall 2024, the seminars are tentatively scheduled for Wednesdays from 2:30 PM to 3:30 PM, and will be available both in-person at 1146 FAB and online. Each seminar will spotlight a particular topic, providing an in-depth exploration and fostering lively discussions.
Target Audience: The CAD Seminar Series is designed for a diverse audience, including faculty, researchers, students, and professionals in Computation, AI, Data Science, and related fields. It serves as a forum for exchanging ideas, networking, and contributing to the growth of these rapidly evolving areas. We highly recommend in-person attendance to enhance engagement and networking opportunities with speakers and fellow participants.
Call for Participation: We welcome contributions from researchers, practitioners, and students. Whether presenting your work, participating in discussions, or attending as a learner, your involvement is crucial to the success of this collaborative initiative.
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