## Distinguished Seminar Series in Computational Science and Engineering

**May 13, 2021, 12:00 PM EST**

**Bayesian neural networks for weak solution of PDEs with uncertainty quantification**

Krishna Garikipati

Professor, Department of Mechanical Engineering, and Department of Mathematics

Director, Michigan Institute for Computational Discovery & Engineering

University of Michigan

**Recorded Seminar YouTube Link:
**https://youtu.be/PtiGOkpg3uY

**Abstract:**

We propose a physics-constrained neural network (NN) approach to solve partial differential equations (PDEs) without labels. We express the loss function of these NNs in terms of the residual of PDEs obtained through an efficient, discrete, convolution operator-based, and vectorized implementation. We explore an encoder-decoder NN structure for both deterministic and probabilistic models, with Bayesian NNs (BNNs) for the latter, which allow us to quantify both epistemic uncertainty from model parameters and aleatoric uncertainty from noise in the data. For BNNs, the discretized residual is used to construct the likelihood function. In our approach, both deterministic and probabilistic convolutional layers are used to learn the applied boundary conditions (BCs) and to detect the problem domain. Both Dirichlet and Neumann BCs are specified as inputs to NNs, and we explore whether a single NN can solve for similar physics; i.e., the same PDE, but with different BCs and on a number of problem domains. The trained BNN PDE solvers demonstrate a degree of success at extrapolated predictions for BCs that they were not exposed to during training. We demonstrate the capacity and performance of the proposed framework by applying it to different steady-state and equilibrium boundary value problems with physics that spans diffusion, linear and nonlinear elasticity. Such NN solution frameworks assume particular importance in problems where high-throughput solutions of PDEs with different boundary conditions and on varying domains are desired in support of design and decision-making.

**Bio:**

Krishna Garikipati obtained his PhD at Stanford University in 1996, and after a few years of post-doctoral work, he joined the University of Michigan in 2000, where is now a Professor in the Departments of Mechanical Engineering and Mathematics. Since 2016 he also has served as the Director of the Michigan Institute for Computational Discovery & Engineering (MICDE). His research is in computational science, with applications drawn from materials physics, biophysics, mechanics and mathematical biology. Of recent interest are data-driven approaches to computational science. He has been awarded the DOE Early Career Award for Scientists and Engineers, the Presidential Early Career Award for Scientists and Engineers (PECASE), and a Humboldt Research Fellowship. He is a fellow of the US Association for Computational Mechanics, a Life Member of Clare Hall at University of Cambridge, and a visiting scholar in Computational Biology at the Flatiron Institute of the Simons Foundation.

Bayesian neural networks for weak solution of PDEs with uncertainty quantification

Krishna Garikipati

Professor, MechE & Math

Univ. Michigan