Some Thoughts on Physics Informed Neural Networks

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Advanced Numerical Methods for Scientific Computing
Wolfgang Dahmen
Mathematics Department, University of South Carolina, Columbia, SC, USA
Thursday 14th October 2021
Link to recording:
"Learning" complex technological or physical processes usually requires fusing the information provided by observational data with knowledge about a "background model" often formulated in terms of a family of parameter dependent PDEs. Frequent forward simulations as well as related inverse tasks calls for reduced models that facilitate an efficient evaluation of the underlying parameter-to-solution map despite expected obstructions caused by the typical high-dimensionality of parameter domains. Physics informed Neural Networks (PINN) offer a promising framework to that effect. The core issue addressed in this talk is the prediction capability of such methods. Related specific questions concern, for instance, the choice of "model compliant" metrics, the choice of training risks that convey certifiable information about the achieved accuracy in such metrics, the role of a priori versus a posteriori error bounds, connections with Generative Adversarial Networks, as well as related implications on training strategies and network adaptation. Contact:
Wolfgang Dahmen is currently a chaired professor in mathematics (SmartState and Williams-Hedberg-Hedberg Chair) at the University of South Carolina in Columbia, South Carolina, USA. His research interests are in Approximation Theory, Numerical, Applied and Harmonic Analysis as well as interdisciplinary applications. A central thematic thread is the development and analysis of adaptive and nonlinear solution concepts in a variety of contexts such as image and data analysis, machine learning, the numerical solution of singular integral and partial differential equations, and model reduction. Wolfgang Dahmen received his PhD from RWTH Aachen in 1976 and his Habilitation from the University of Bonn. After an IBM Postdoctoral Fellowship at the IBM Research Center in Yorktown Heights, NY, he took (associate and full) professor positions at the University of Bielefeld and the Free University of Berlin before joining RWTH Aachen in 1992. In 2017 he became chaired professor at the University of South Carolina. Among his awards and honors are the Gottfried-Wilhelm-Leibniz Award, Keck Future Award of the US Academies (together with P. Binev, T. Vogt), election to the German National Academy of Sciences, Leopoldina, Robert-Piloti-Prize of the Technical University of Darmstadt, and SIAM Fellow. He has served on numerous Scientific Advisory boards such as the CRM in Barcelona and the Isaac Newton Institute in Cambridge, UK. From 2014 to 2017 he was the Chair of the Board of Directors of the Society Foundations of Computational Mathematics.