Physics-Informed DeepONet Coupled with FEM for Convective Transport in Porous Media with Sharp Gaussian Sources

We present a hybrid framework that couples finite element methods (FEM) with physics-informed DeepONet to model fluid transport in porous media from sharp, localized Gaussian sources. The governing system consists of a steady-state Darcy flow equation and a time-dependent convection-diffusion equation. Our approach solves the Darcy system using FEM and transfers the resulting velocity field to a physics-informed DeepONet, which learns the mapping from source functions to solute concentration profiles. This modular strategy preserves FEM-level accuracy in the flow field while enabling fast inference for transport dynamics. To handle steep gradients induced by sharp sources, we introduce an adaptive sampling strategy for trunk collocation points. Numerical experiments demonstrate that our method is in good agreement with the reference solutions while offering orders of magnitude speedups over traditional solvers, making it suitable for practical applications in relevant scenarios. Implementation of our proposed method is available at https://github.com/erkara/fem-pi-deeponet.

Deep Learning Based Object Tracking in Walking Droplet and Granular Intruder Experiments

We present a deep-learning based tracking objects of interest in walking droplet and granular intruder experiments. In a typical walking droplet experiment, a liquid droplet, known as \textit{walker}, propels itself laterally on the free surface of a vibrating bath of the same liquid. This motion is the result of the interaction between the droplets and the surface waves generated by the droplet itself after each successive bounce. A walker can exhibit a highly irregular trajectory over the course of its motion, including rapid acceleration and complex interactions with the other walkers present in the same bath. In analogy with the hydrodynamic experiments, the granular matter experiments consist of a vibrating bath of very small solid particles and a larger solid \textit{intruder}. Like the fluid droplets, the intruder interacts with and travels the domain due to the waves of the bath but tends to move much slower and much less smoothly than the droplets. When multiple intruders are introduced, they also exhibit complex interactions with each other. We leverage the state-of-art object detection model YOLO and the Hungarian Algorithm to accurately extract the trajectory of a walker or intruder in real-time. Our proposed methodology is capable of tracking individual walker(s) or intruder(s) in digital images acquired from a broad spectrum of experimental settings and does not suffer from any identity-switch issues. Thus, the deep learning approach developed in this work could be used to automatize the efficient, fast and accurate extraction of observables of interests in walking droplet and granular flow experiments. Such extraction capabilities are critically enabling for downstream tasks such as building data-driven dynamical models for the coarse-grained dynamics and interactions of the objects of interest.