# Estimating Dense-Packed Zone Height in Liquid-Liquid Separation: A Physics-Informed Neural Network Approach

arXiv: https://arxiv.org/abs/2601.18399 · HTML: https://mlsanity.com/paper/2601.18399/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 54 references that are in the map.
- First posted on arXiv 26 Jan 2026.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [You Only Look Once: Unified, Real-Time Object Detection](https://mlsanity.com/paper/1506.02640/) (2015)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [Sim-to-Real Transfer in Deep Reinforcement Learning for Robotics: a Survey](https://mlsanity.com/paper/2009.13303/) (2020)
- [Model-Based Deep Learning](https://mlsanity.com/paper/2012.08405/) (2020)
- [Transfer learning based multi-fidelity physics informed deep neural network](https://mlsanity.com/paper/2005.10614/) (2020)
- [The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?](https://mlsanity.com/paper/2103.09655/) (2021)
- [Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2107.00940/) (2021)
- [Transfer Learning in Physics-Informed Neural Networks: Full Fine-Tuning, Lightweight Fine-Tuning, and Low-Rank Adaptation](https://mlsanity.com/paper/2502.00782/) (2025)
- [Physics-Informed Neural Networks for High-Frequency and Multi-Scale Problems using Transfer Learning](https://mlsanity.com/paper/2401.02810/) (2024)
- [Physics-Informed Neural Networks for Dynamic Process Operations with Limited Physical Knowledge and Data](https://mlsanity.com/paper/2406.01528/) (2024)
- [Fine-Tuning Hybrid Physics-Informed Neural Networks for Vehicle Dynamics Model Estimation](https://mlsanity.com/paper/2409.19647/) (2024)

## Abstract (arXiv)

Separating liquid-liquid dispersions in gravity settlers is critical in chemical, pharmaceutical, and recycling processes. The dense-packed zone height is an important performance and safety indicator but it is often expensive and impractical to measure due to optical limitations. We propose a framework to estimate phase heights by combining a PINN model with readily available volume flow measurements, without requiring phase height measurements during deployment. To this end, a physics-informed neural network (PINN) is first pretrained on synthetic data and physics equations derived from a low-fidelity (approximate) mechanistic model to reduce the need for extensive experimental data. While the mechanistic model is used to generate synthetic training data, only volume balance equations are used in the PINN, as incorporating droplet coalescence and sedimentation submodels would be computationally prohibitive. The pretrained PINN is then fine-tuned with scarce experimental phase height and flow-rate data to capture the actual dynamics of the separator. We then deploy the differentiable PINN as a predictive model in an Extended Kalman Filter inspired state estimation framework, enabling the phase heights to be tracked and updated using flow-rate measurements only. We first test the two-stage trained PINN by forward simulation from a known initial state against the mechanistic model and a non-pretrained PINN. We then evaluate phase height estimation performance with the filter, comparing the two-stage trained PINN with a two-stage trained purely data-driven neural network. All model types are trained and evaluated using ensembles to account for model parameter uncertainty. In all evaluations, the two-stage trained PINN yields the most accurate phase-height estimates.