Deep capsule encoder–decoder network for surrogate modeling and uncertainty quantification

Abstract

We propose a novel capsule-based deep encoder-decoder model for surrogate modeling and uncertainty quantification of systems in mechanics from sparse data. The proposed framework is developed by adapting Capsule Network (CapsNet) architecture into an image-to-image regression encoder-decoder network. Specifically, the aim is to exploit the benefits of CapsNet over convolution neural network (CNN) -retaining pose and position information related to an entity to name a few. The performance of the proposed approach is illustrated by solving two different variants of elliptic partial differential equations (PDE): a stochastic version without a source term having an input dimensionality of 1024 and a deterministic pressure Poisson equation for flow past a cylinder. The first PDE, that is, the stochastic PDE (SPDE), also governs systems in mechanics such as steady heat conduction, groundwater flow, or other diffusion processes. In this article, the problem definition for this SPDE is such that it does not restrict the random diffusion field to a particular covariance structure, and the more strenuous task of response prediction for an arbitrary diffusion field is solved. Finally, we also evaluate the performance of our model on the uncertainty propagation problem based on this equation. The obtained results from the performance evaluation of the developed model on the mentioned problems indicate that the proposed approach is accurate, data efficient, and robust.