Speaker: Ankit Srivastava, IIT

BIO: Dr. Ankit Srivastava is an Associate Professor in the Mechanical, Materials, and Aerospace Engineering department at the Illinois Institute of Technology. He holds a B.Tech. from IIT Guwahati, and M.S. and Ph.D. (Structural Engineering) from the University of California San Diego where he also worked with Prof. Sia Nemat-Nasser as a post-doctoral researcher. Srivastava has receieved the NSF CAREER award and is currently an Associate Editor of the Mechanics of Materials journal. Srivastava's research interest are in the areas of wave propagation, homogenization theory, phononics and metamaterials, soft robotics, and, recently, PINN based methods for wave problems.

ABSTRACT: In this presentation, we introduce a new approach based on distance fields to exactly impose boundary conditions in physics-informed deep neural networks. The challenges in satisfying Dirichlet boundary conditions in meshfree and particle methods are well-known. This issue is also pertinent in the development of physics informed neural networks (PINN) for the solution of partial differential equations. We introduce geometry-aware trial functions in artifical neural networks to improve the training in deep learning for partial differential equations. To this end, we use concepts from constructive solid geometry (R-functions) and generalized barycentric coordinates (mean value potential fields) to construct ϕ, an approximate distance function to the boundary of a domain. To exactly impose homogeneous Dirichlet boundary conditions, the trial function is taken as ϕ multiplied by the PINN approximation, and its generalization via transfinite interpolation is used to a priori satisfy inhomogeneous Dirichlet (essential), Neumann (natural), and Robin boundary conditions on complex geometries. In doing so, we eliminate modeling error associated with the satisfaction of boundary conditions in a collocation method and ensure that kinematic admissibility is met pointwise in a Ritz method. We will present numerical solutions to boundary-value problems in 1D and in 2D over curved geometries, and we demonstrate that the approach extends to higher dimensions by solving a Poisson problem with homogeneneous Dirichlet boundary conditions over the 4D hypercube. This study provides a pathway for meshfree analysis to be conducted on the exact geometry without domain discretization.

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