Three dimensional modeling of filtration process in complex filter structures using a data-free physics-informed neural network coupled with smoothed particle hydrodynamics

Efficient removal of solid particles and liquid droplets is essential in many industrial processes. The primary objective is to achieve high separation efficiency while minimizing the associated pressure drop. This trade-off presents a central challenge in filter media design, where the objective is to optimize filtration performance without compromising energy efficiency or operational stability. Over recent decades, both experimental techniques and numerical modeling have advanced substantially. Nevertheless, the micro-scale nature of filtration still restricts our understanding. Experiments often suffer from limited reproducibility and require costly prototype media. Conventional numerical approaches, such as the finite-volume method, can resolve the relevant transport mechanisms, but the small spatial and temporal scales involved require extremely fine meshes and time steps. These constraints make classical computational fluid dynamics (CFD) simulations expensive and sometimes impractical. In this work, a modeling approach for simulating the filtration process is presented, which integrates physics-informed neural network (PINN) with smoothed particle hydrodynamics (SPH). The PINN approximates the three-dimensional velocity and pressure fields inside the fibrous medium directly from the Navier–Stokes equations, without relying on any training data. The resulting flow field is passed to the SPH solver, which tracks the droplet trajectories and their interactions with the fibers. Because both PINN and SPH are mesh-free, the workflow eliminates complex grid generation and the need to solve the Navier–Stokes equations in small spatial and temporal increments. A bidirectional data exchange links the two solvers. As the deposited droplets grow large enough to influence the flow field, their geometrical information is transferred back to the PINN, which recomputes the flow field through Transfer Learning. Comparison between the flow field predicted by the PINN models and those calculated using CFD solver for four representative filter structures demonstrates a very good agreement between the PINN and CFD.