Recent Advances in Physics-Informed Machine Learning
Yiping Lu ( NYU Courant & Northwestern University), Grant Rostkoff (Stanford University)
Machine learning (ML) is spurring a transformation in the computational sciences by providing a new way to build flexible, universal, and efficient approximations for complex high-dimensional functions and functionals. One area in which the impact of these new tools is beginning to be understood is the physical sciences, where traditionally intractable high-dimensional partial differential equations are now within reach. This tutorial will explore how developments in ML complement computational problems in the physical sciences, with a particular focus on solving partial differential equations, where the challenges of high-dimensionality and data acquisition also arise.
The first important example this tutorial will cover is using Deep Learning Methods for solving high-dimensional PDEs, which have wide application in variational rare events calculations, many-body quantum systems, and stochastic control. Another challenge covered by this tutorial that researchers often face is the complexity or lack of specification of the models they are using when performing uncertainty quantification. Thus another line of research aims to recover the underlying dynamic using observational data.
This tutorial will introduce the well-developed methods and theories for using machine learning in scientific computing. We will first discuss how to incorporate physical priors into machine learning models. Next, we will discuss how these methods can help to solve challenging physical and chemical problems. Finally, we will discuss the statistical and computational theory for scientific machine learning. In this tutorial, we will not focus on the technical details behind these theories, but on how they can help the audience to understand the challenges of using machine learning in differential equation applications and to develop new methods for addressing these challenges.
Slide
piml_aaai.pdf Presenter
Yiping Lu
Yiping Lu is a Courant instructor at Courant Institute of Mathematical Sciences at New York University and an incoming assistant professor at Industrial Engineering and Management Science at Northwestern University. He received his PhD in computational and mathematical engineering from Stanford University. His awards include the CPAL Rising Star Award (2024), Rising Star in Data Science (University of Chicago 2022), a Stanford Interdisciplinary Graduate Fellowship (2021-2024), and a SenseTime Scholarship (2018-2019). He currently serves as an area chair at AISTATS. His current research interests include time series analysis, non-parametric statistics, and machine learning, often set in the context of physics-based systems governed by differential equations.
Grant Rotskoff
Grant Rotskoff is an Assistant Professor of Chemistry at Stanford. He studies the nonequilibrium dynamics of living matter with a particular focus on self-organization from the molecular to the cellular scale. His work involves developing theoretical and computational tools that can probe and predict the properties of physical systems driven away from equilibrium. Recently, he has focused on characterizing and designing physically accurate machine learning techniques for biophysical modeling. Prior to his current position, Grant was a James S. McDonnell Fellow working at the Courant Institute of Mathematical Sciences at New York University. He completed his Ph.D. at the University of California, Berkeley in the Biophysics graduate group supported by an NSF Graduate Research Fellowship.
Reference
[1] Karniadakis G E, Kevrekidis I G, Lu L, et al. Physics-informed machine learning. Nature Reviews Physics, 2021, 3(6): 422-440.
[2] Hao Z, Liu S, Zhang Y, et al. Physics-informed machine learning: A survey on problems, methods and applications. arXiv preprint arXiv:2211.08064, 2022.
[3] https://2prime.github.io/files/SML/SciMLReport.pdf
Physics-informed Neural Network
[1] Wang S, Sankaran S, Wang H, et al. An expert’s Guide to Training Physics-informed Neural Networks, arXiv. preprint.
[2] Krishnapriyan A, Gholami A, Zhe S, et al. Characterizing possible failure modes in physics-informed neural networks[J]. Advances in Neural Information Processing Systems, 2021, 34: 26548-26560.
[3] Lu Y, Chen H, Lu J, et al. Machine learning for elliptic pdes: Fast rate generalization bound, neural scaling law and minimax optimality. ICLR 2022.
[4] Marwah T, Lipton Z, Risteski A. Parametric complexity bounds for approximating PDEs with neural networks. Advances in Neural Information Processing Systems, 2021, 34: 15044-15055.
Operator Learning:
[1] Kovachki N, Li Z, Liu B, et al. Neural operator: Learning maps between function spaces. arXiv preprint arXiv:2108.08481, 2021.
[2] Lu L, Jin P, Pang G, et al. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature machine intelligence, 2021, 3(3): 218-229.
[3] Lu L, Meng X, Cai S, et al. A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data. Computer Methods in Applied Mechanics and Engineering, 2022, 393: 114778.
[4] Lanthaler S, Mishra S, Karniadakis G E. Error estimates for deeponets: A deep learning framework in infinite dimensions. Transactions of Mathematics and Its Applications, 2022, 6(1).
[5] Boullé N, Townsend A. A Mathematical Guide to Operator Learning. arXiv preprint arXiv:2312.14688, 2023.
MLA