Solving PDEs from Noisy Multi-Fidelity Measurements Only!

George Em Karniadakis
Division of Applied Mathematics, Brown University, USA

2016-12-15 ~ 2016-12-15


Science Building No. 1 1479


We present a new paradigm in solving linear and nonlinear PDEs from noisy measurements without the use of any classical numerical discretization. Instead, we infer the solution of PDEs from noisy data, which can represent measurements of variable fidelity. The key idea is to encode the structure of the PDE into prior distributions and train Bayesian nonparametric regression models on available noisy data. The resulting posterior distributions can be used to predict the PDE solution with quantified uncertainty, efficiently identify extrema via Bayesian optimization, and acquire new data via active learning. (Based on the work of Dr. Maziar Raissi and Dr. Paris Perdikaris) References: [1] M. Raissi, P. Perdikaris, and G. E. Karniadakis, Inferring solutions of differential equations using noisy multi-fidelity data," arXiv preprint arXiv:1607.04805, 2016. [2] P. Perdikaris, M. Raissi, A. Damianou, N. D. Lawrence, and G. E. Karniadakis, Nonlinear information fusion algorithms for robust multi-fidelity modeling, Proceedings of the Royal Society A (under review), 2016. Short bio: George Em Karniadakis is the Charles Pitts Robinson and John Palmer Barstow Professor of Applied Mathematics at Brown University, and the director of the DOE Center of Mathematics for Mesoscale Modeling of Materials (CM4). He is Fellow of SIAM (2010), APS (2004) and ASME (2003) and Associate Fellow of AIAA (2006). He has been a Research Scietist at MIT since 2000. His current research interests are in stochastic multiscale modeling, and is widely known for his fundamental contributions on high-dimensional stochastic modeling, spectral element methods, and multiscale simulations of physical and biological systems. His most recent awards include SIAM’s Ralph E Kleinman Award (2015) for “many outstanding contributions to Applied Mathematics in a broad range of areas, including computational fluid dynamics, spectral methods and stochastic modeling”, and the MCS Wiederhielm Award (2015) “for the most highly cited original article in Micro-circulation over the previous five year period for the paper, Blood Flow and Cell-Free Layer in Microvessels.”