ELPINN: Eulerian Lagrangian Physics-Informed Neural Network

Physics-Informed Neural Networks (PINNs) have gained widespread popularity for solving inverse and forward problems across a range of scientific and engineering domains. However, most existing PINN frameworks are limited to the Eulerian domain, where physical quantities are described at fixed spatial locations. In this work, we propose a novel PINN-based framework that couples Eulerian and Lagrangian perspectives by using particle trajectory data to reconstruct Eulerian velocity and pressure fields. We evaluate the performance of our method across three distinct fluid flow scenarios: two-dimensional external flow past a cylinder, two-dimensional internal flow in a confined geometry, and three-dimensional internal flow inside an airplane cabin. In all three cases, we successfully reconstruct the velocity field from Lagrangian particle data. Moreover, for the 2D external and internal flows, we recover the pressure field solely through the physics-informed learning process, without using any direct pressure measurements. We also conduct a sensitivity analysis to understand the effects of temporal resolution and particle count on the reconstruction accuracy. Our results show that smaller time-step sizes significantly improve the predictions, while the total number of particles has a comparatively smaller influence. These findings establish the potential of our coupled Eulerian-Lagrangian PINN framework as a powerful tool for enhancing experimental methods such as Particle Tracking Velocimetry (PTV). Looking ahead, this approach may be extended to infer hidden quantities such as pressure in three-dimensional flows or material properties like viscosity, opening new avenues for data-driven fluid dynamics in complex geometries.