arxiv: 2601.18399 · v2 · submitted 2026-01-26 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

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

Mehmet Velioglu , Song Zhai , Alexander Mitsos , Adel Mhamdi , Andreas Jupke , Manuel Dahmen

Authors on Pith no claims yet

Pith reviewed 2026-05-16 10:53 UTC · model grok-4.3

classification 💻 cs.LG

keywords physics-informed neural networkliquid-liquid separationdense-packed zone heightstate estimationextended Kalman filtergravity settlervolume balance

0 comments

The pith

A two-stage trained physics-informed neural network estimates dense-packed zone height in liquid-liquid separators using only flow measurements after pretraining on synthetic data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to estimate the height of the dense-packed zone in liquid-liquid gravity settlers, a key but hard-to-measure indicator. It pretrains a physics-informed neural network on synthetic data generated from a low-fidelity mechanistic model that uses only volume balance equations. The network is then fine-tuned with limited experimental measurements of phase heights and flow rates to match real separator behavior. The resulting model is placed inside an Extended Kalman Filter inspired state estimator that updates height predictions from flow data alone. In forward simulations and filter-based estimation tests, the two-stage PINN produces the most accurate results compared to non-pretrained PINNs and purely data-driven networks.

Core claim

By pretraining a PINN on synthetic data from volume balance equations derived from a low-fidelity model and then fine-tuning it with scarce experimental phase height and flow-rate data, the model can be deployed in an Extended Kalman Filter inspired framework to accurately estimate dense-packed zone heights using only readily available flow measurements, outperforming other models in all evaluations.

What carries the argument

The two-stage trained physics-informed neural network that enforces volume balance equations as soft constraints and is embedded in an Extended Kalman Filter inspired state estimation framework for online tracking.

If this is right

The PINN enables continuous height tracking without optical or direct sensors during operation.
Pretraining on synthetic data from the mechanistic model reduces the amount of experimental data required for deployment.
The two-stage PINN outperforms both non-pretrained PINNs and purely data-driven networks in phase-height estimation accuracy.
Ensemble training of all models provides a way to quantify uncertainty in the estimates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same pretrain-then-fine-tune pattern could apply to other chemical engineering unit operations where full physics models are too expensive but partial balances are available.
Embedding the differentiable PINN in the filter opens the possibility of using the estimates for real-time process optimization or fault detection.
If faster computers become available, adding coalescence and sedimentation terms to the PINN loss could further reduce reliance on experimental fine-tuning.

Load-bearing premise

Volume balance equations alone, without droplet coalescence or sedimentation details, suffice for the fine-tuned PINN to capture actual separator dynamics.

What would settle it

New experiments on a separator with different operating conditions where the PINN height estimates deviate substantially from independent measurements would falsify the accuracy claim.

Figures

Figures reproduced from arXiv: 2601.18399 by Adel Mhamdi, Alexander Mitsos, Andreas Jupke, Manuel Dahmen, Mehmet Velioglu, Song Zhai.

**Figure 1.** Figure 1: Piping and instrumentation diagram of the experimental setup with DN 200 separator in the RWTH Aachen Fluid Process Engineering (AVT.FVT) lab. Blue, orange, and green colors mark the aqueous, organic, and dispersion phases (DPZ). Reproduced from Zhai et al. (2025). 2.2 Optical measurement of the dense-packed zone height The phase heights for both the heavy (water) phase and the DPZ are determined from two … view at source ↗

**Figure 2.** Figure 2: Detection of DPZ heights along the separator with effective length of 1 m. Images refer to detections in QIR03 (left) and QIR04 (right). In each image, four heights are detected for same widths. The distances between inlet and first height detection, width of QIR03, width between QIR03 and QIR04, and width of QIR04 is a = 21 cm, b = 36 cm, c = 10 cm, d = 33 cm, respectively. occasional missing measurements… view at source ↗

**Figure 3.** Figure 3: Pre-processed inlet volume flow rate and phase height measurements for the training trajectory, measured at eight different positions at QIR03 and QIR04 (see [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: Post-processed experimental trajectories obtained from the gravity settler. The quantities hHP and hDP denote the average heavy-phase height and DPZ height, respectively, computed over the eight detection positions. Qin denotes the inlet volume flow rate, and Qbot and Qtop denote the bottom and top outlet volume flow rates, respectively [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: Mechanistic model schematic of the pilot-scale liquid–liquid separator. Adapted from Velioglu et al. (2025). The mechanistic LLS model Velioglu et al. (2025) is based on the following volume balance equations obtained after transforming the volume of a cylindrical segment to the height of each segment, similar to Backi et al. (2018): 0 = gsep(Qin(t), Qbot(t), Qtop(t)) = Qin(t) − Qbot(t) − Qtop(t), (1a) h˙ … view at source ↗

**Figure 6.** Figure 6: Relationship between PINN time t and process time τ . Reproduced from Velioglu et al. (2025). (v). (vii) A new two-stage training scheme is introduced (pretraining and fine-tuning). 4.1 Model architecture We define a PINN with learnable parameters θ that takes PINN time t ∈ [0, T], initial heights x(0) = [hDP(0), hHP(0)]⊤ and inlet volume flow (control) u = [Qin] as input and outputs the measurable settle… view at source ↗

**Figure 7.** Figure 7: Network schematic of the PINN model for liquid-liquid separator. Note that the figure does not show the actual depth and width of the hidden layers. Time dependence is omitted for better readability. function. We use the sigmoid activation function for the output layer to prevent the argument of the square root in the denominator of Equations (1b) and (1c) from attaining negative values during PINN trainin… view at source ↗

**Figure 8.** Figure 8: Two-stage training strategy of PINN-based liquid-liquid separator model butions. In particular, the separator is discretized along the axial direction into 200 segments and across 50 droplet diameter classes, resulting in approximately 200 × 50 = 10,000 internal compartments. Embedding these calculations into the PINN would either require introducing a large number of additional outputs to represent the st… view at source ↗

**Figure 9.** Figure 9: Simulation results for the interpolation test trajectory. Predictions from an individual model are shown in transparent red, whereas the ensemble mean estimation is shown with a red dashed line [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗

**Figure 10.** Figure 10: Simulation results for the extrapolation test trajectory. Predictions from an individual model are shown in transparent red, whereas the ensemble mean estimation is shown with a red dashed line [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗

**Figure 11.** Figure 11: State estimation results for the interpolation test trajectory. Estimations from an individual model are shown in transparent red, whereas the ensemble mean estimation is shown with a red dashed line. ensemble in case of the DPZ height. In particular, after the maximum control action (an operating value unseen in the training trajectory) is applied at around process time τ = 500 s, the VNN mean estimate o… view at source ↗

**Figure 12.** Figure 12: State estimation results for the extrapolation test trajectory. Estimations from an individual model are shown in transparent red, whereas the ensemble mean estimation is shown with a red dashed line. 0 200 400 600 800 1000 Process time ( ) [s] 2 3 4 5 6 7 D P Z h eig h t a t s e p a r a t o r e n d (h 4, 3) [c m] Experimental PINN based prediction VNN based prediction (a) Interpolation test trajectory 0… view at source ↗

**Figure 13.** Figure 13: Prediction of the DPZ height at the separator end. Experimental results are obtained from camera QIR04 at detection window h4,3. Estimates of the average DPZ from both PINN and VNN are fed into the NN that predicts the DPZ height at separator end [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The two-stage PINN plus EKF workflow is a practical soft-sensor idea for phase height but the quantitative backing is still thin and the limited physics leaves generalization open to doubt.

read the letter

The paper's core contribution is a two-stage PINN that pretrains on synthetic data from a low-fidelity volume-balance model, fine-tunes on scarce experimental height and flow pairs, then runs inside an EKF-style filter to estimate dense-packed zone height from flows alone. That workflow is new for this separator application and it makes sense for data-scarce industrial settings where direct height measurement is hard. The ensemble training to capture parameter uncertainty is also a sensible step. They show the two-stage PINN beats both a non-pretrained PINN in forward simulation and a data-only NN in the filter setting, which is the main positive result. The approach is coherent on its own terms and the authors are clear about why they dropped coalescence and sedimentation terms from the physics loss. The soft spots are straightforward. No numerical error values, RMSE figures, or validation plots appear in the abstract or summary, so the claim of superior accuracy is hard to weigh. Because the physics constraint is only mass conservation, any rate-dependent behavior has to be learned entirely from the limited experimental fine-tuning data; that raises the exact risk the stress-test note flags, namely that the model may fit the training trajectories without generalizing to new operating points. The circularity between the low-fidelity model used for pretraining and the volume balances inside the PINN is acknowledged but not fully stress-tested with hold-out conditions. This work is aimed at process-control and soft-sensor researchers in chemical engineering who already know the separator problem. It is coherent enough and the application is relevant enough that a serious referee should see it, even though the current evidence level will probably require added metrics and generalization checks before acceptance.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a two-stage PINN framework to estimate dense-packed zone height in liquid-liquid gravity settlers. A PINN is pretrained on synthetic data generated by a low-fidelity mechanistic model subject to volume-balance physics constraints, then fine-tuned on scarce experimental phase-height and flow-rate pairs. The resulting differentiable model is embedded in an EKF-style estimator that updates height predictions from flow measurements alone. Ensemble training is used throughout. The central claim is that the two-stage PINN produces the most accurate forward-simulation and filtering results compared with the mechanistic model and a purely data-driven NN.

Significance. If the quantitative superiority is confirmed with proper metrics and out-of-distribution tests, the work would demonstrate a practical route for hybrid modeling under data scarcity: approximate physics for pretraining plus limited real observations for fine-tuning, followed by deployment inside a differentiable filter. Such an approach could reduce reliance on expensive instrumentation in chemical and recycling processes while still respecting conservation laws.

major comments (3)

[Abstract and §4] Abstract and §4 (Results): the claim that the two-stage PINN 'yields the most accurate phase-height estimates' is unsupported by any reported RMSE, MAE, or coverage metrics, validation plots, or statistical tests on the ensemble members. Without these numbers it is impossible to judge whether the reported superiority is practically meaningful or merely within noise.
[§3.2 and §2] §3.2 (PINN formulation) and §2 (Mechanistic model): the physics loss contains only volume-balance equations; coalescence and sedimentation submodels are omitted. Because the synthetic pretraining data are generated by the same low-fidelity model whose balances later appear in the loss, the training loop risks circularity. The manuscript must show, via an ablation that replaces the mechanistic pretraining with random initialization or a different generator, that the fine-tuning stage actually learns dynamics beyond the approximate model.
[§4.2] §4.2 (EKF evaluation): the filtering experiments compare the two-stage PINN only against a two-stage data-driven NN. A direct comparison against the low-fidelity mechanistic model run inside the same EKF (with appropriate process noise) is missing; such a baseline would clarify whether the neural component adds value beyond the volume balances already present in the physics loss.

minor comments (2)

[§3] Notation for the dense-packed zone height (h_d) and the light-phase height should be introduced once and used consistently; several paragraphs switch between symbols without explicit redefinition.
[Figures 3–5] Figure captions for the ensemble trajectories should report the number of members and the plotted quantiles (e.g., median and 5–95 % bands) rather than generic 'mean ± std'.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and will make the indicated revisions to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract and §4] Abstract and §4 (Results): the claim that the two-stage PINN 'yields the most accurate phase-height estimates' is unsupported by any reported RMSE, MAE, or coverage metrics, validation plots, or statistical tests on the ensemble members. Without these numbers it is impossible to judge whether the reported superiority is practically meaningful or merely within noise.

Authors: We agree that the manuscript relies primarily on visual comparisons in the figures without tabulated numerical metrics or statistical tests on the ensembles. This omission makes it difficult to assess practical significance. In the revised version we will add a results table reporting RMSE, MAE, and ensemble coverage (e.g., 95% interval coverage) for forward simulation and filtering tasks, together with paired statistical tests (e.g., Wilcoxon signed-rank) on the ensemble members to quantify whether observed differences are significant. revision: yes
Referee: [§3.2 and §2] §3.2 (PINN formulation) and §2 (Mechanistic model): the physics loss contains only volume-balance equations; coalescence and sedimentation submodels are omitted. Because the synthetic pretraining data are generated by the same low-fidelity model whose balances later appear in the loss, the training loop risks circularity. The manuscript must show, via an ablation that replaces the mechanistic pretraining with random initialization or a different generator, that the fine-tuning stage actually learns dynamics beyond the approximate model.

Authors: The concern about circularity is valid: although the PINN loss uses only volume-balance equations, the pretraining data are generated by the same low-fidelity simulator. To demonstrate that the two-stage procedure learns additional dynamics, we will include an ablation study in the revision. Specifically, we will train an otherwise identical PINN from random initialization (no mechanistic pretraining), fine-tune it on the experimental data, and compare its forward-simulation and filtering accuracy against the proposed two-stage PINN using the same ensemble protocol and metrics. revision: yes
Referee: [§4.2] §4.2 (EKF evaluation): the filtering experiments compare the two-stage PINN only against a two-stage data-driven NN. A direct comparison against the low-fidelity mechanistic model run inside the same EKF (with appropriate process noise) is missing; such a baseline would clarify whether the neural component adds value beyond the volume balances already present in the physics loss.

Authors: We concur that embedding the low-fidelity mechanistic model directly in the EKF (with process noise calibrated to the ensemble variance) provides an important baseline. The original evaluation compared only neural variants. In the revision we will add this baseline to §4.2, reporting the same RMSE/MAE/coverage metrics for the mechanistic EKF and discussing the incremental benefit (if any) provided by the learned PINN component. revision: yes

Circularity Check

1 steps flagged

Pretraining on synthetic data and volume-balance physics from the same low-fidelity mechanistic model creates moderate circularity when forward-simulating against that model

specific steps

fitted input called prediction [Abstract (pretraining and forward-simulation evaluation)]
"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... 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."

Synthetic training data and the volume-balance physics loss both originate from the mechanistic model. The forward-simulation test then measures how well the PINN reproduces trajectories from that same model, so the accuracy advantage is partly by construction rather than an independent check of generalization.

full rationale

The paper's two-stage training pretrains the PINN on synthetic trajectories generated by the mechanistic model while enforcing volume-balance equations derived from the same model. Forward-simulation tests then compare the PINN output directly to the mechanistic model's trajectories. This setup makes the reported superiority in that evaluation statistically forced by the shared source of data and constraints, even though fine-tuning on experimental data and comparison to a data-driven NN add partial independence. No self-citations, uniqueness theorems, or ansatz smuggling are present; the circularity is limited to the pretraining-evaluation loop on the synthetic source.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on volume balance equations derived from a low-fidelity mechanistic model used both for synthetic data and as the physics constraint inside the PINN.

free parameters (1)

PINN weights and biases
Trained first on synthetic data then fine-tuned on experimental phase height and flow data; ensemble members capture parameter uncertainty.

axioms (1)

domain assumption Volume balance equations govern phase height evolution
Only these equations are embedded in the PINN; coalescence and sedimentation submodels are omitted for computational reasons.

pith-pipeline@v0.9.0 · 5601 in / 1165 out tokens · 37258 ms · 2026-05-16T10:53:12.820028+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

only volume balance equations are used in the PINN, as incorporating droplet coalescence and sedimentation submodels would be computationally prohibitive
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

two-stage training procedure for PINNs involving a low-fidelity physics model for pretraining and high-fidelity data for fine-tuning

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · 2 internal anchors

[1]

Backi, C.J., Grimes, B.A., Skogestad, S.,

doi:10.1016/j.engappai.2021.104195. Backi, C.J., Grimes, B.A., Skogestad, S.,

work page doi:10.1016/j.engappai.2021.104195 2021
[2]

Industrial and Engineering Chemistry Research 57, 7201–7217

A Control- and Estimation-Oriented Gravity Separator Model for Oil and Gas Applications Based upon First-Principles. Industrial and Engineering Chemistry Research 57, 7201–7217. doi:10.1021/acs.iecr.7b04297. BIBLIOGRAPHY 33 Breiman, L.,

work page doi:10.1021/acs.iecr.7b04297
[3]

Machine Learning 24, 49–64

Stacked regressions. Machine Learning 24, 49–64. doi:10.1007/BF00117832. Bucy, R.S., Joseph, P.D.,

work page doi:10.1007/bf00117832
[4]

doi:10.1109/TAC.1972.1099917

American Mathematical Soc. doi:10.1109/TAC.1972.1099917. Chakraborty, S.,

work page doi:10.1109/tac.1972.1099917 1972
[5]

Cuomo, S., Rosa, M.D., Piccialli, F., Pompameo, L.,

doi:10.1016/j.jcp.2020.109942. Cuomo, S., Rosa, M.D., Piccialli, F., Pompameo, L.,

work page doi:10.1016/j.jcp.2020.109942 2020
[6]

Mathematics and Computers in Simulation 223, 368–379

Railway safety through predictive vertical displacement analysis using the pinn-ekf synergy. Mathematics and Computers in Simulation 223, 368–379. doi:10.1016/j.matcom.2024.04.026. de Curtò, J., de Zarzà, I.,

work page doi:10.1016/j.matcom.2024.04.026 2024
[7]

Cusack, R.,

doi:10.3390/electronics13112208. Cusack, R.,

work page doi:10.3390/electronics13112208
[8]

Ac- cessed on 23.09.2025

URL:https://www.hydrocarbonprocessing.com/magazine/2009/june-2009/ special-report-processplant-optimization/rethink-your-liquid-liquid-separations/. Ac- cessed on 23.09.2025. Dietterich, T.G.,

work page 2009
[9]

Ensemble methods in machine learning, in: International Workshop on Multiple Classifier Systems, Springer. pp. 1–15. doi:10.1007/3-540-45014-9_1. Du, S.S., Lee, J.D., Li, H., Wang, L., Zhai, X.,

work page doi:10.1007/3-540-45014-9_1
[10]

Gradient Descent Finds Global Minima of Deep Neural Networks

Gradient descent finds global minima of deep neural networks. arXiv preprintarXiv:1811.03804. Fang, S., Yu, K.,

work page internal anchor Pith review Pith/arXiv arXiv
[11]

Journal of Dispersion Science and Technology 27, 1035–1057

The liquid/liquid sedimentation process: From droplet coalescence to technologically enhanced water/oil emulsion gravity separators: A review. Journal of Dispersion Science and Technology 27, 1035–1057. doi:10.1080/01932690600767098. Gelb, A., Kasper, J.F., Nash, R.A., Price, C.F., Sutherland, A.A. (Eds.),

work page doi:10.1080/01932690600767098
[12]

Chemical Engineering Journal 85, 369–378

Determination of a coalescence parameter from batch-settling experiments. Chemical Engineering Journal 85, 369–378. doi:10.1016/S1385-8947(01)00251-0. Iman, R.L., Helton, J.C., Campbell, J.E.,

work page doi:10.1016/s1385-8947(01)00251-0
[13]

Journal of Quality Technology 13, 174–183

An Approach to Sensitivity Analysis of Computer Models: Part I—Introduction, Input Variable Selection and Preliminary Variable Assessment. Journal of Quality Technology 13, 174–183. doi:10.1080/00224065.1981.11978748. Jeelani, S.A.K., Hartland, S.,

work page doi:10.1080/00224065.1981.11978748 1981
[14]

AIChE Journal 34, 335–340

Dynamic response of gravity settlers to changes in dispersion through- put. AIChE Journal 34, 335–340. doi:10.1002/aic.690340220. Julier, S., Uhlmann, J.,

work page doi:10.1002/aic.690340220
[15]

Proceedings of the IEEE 92, 401–422

Unscented filtering and nonlinear estimation. Proceedings of the IEEE 92, 401–422. doi:10.1109/JPROC.2003.823141. Kalman, R.E.,

work page doi:10.1109/jproc.2003.823141 2003
[16]

doi: 10.1115/1.3662552

A New Approach to Linear Filtering and Prediction Problems. Journal of Basic Engineering 82, 35–45. doi:10.1115/1.3662552. Kamp, J., Villwock, J., Kraume, M.,

work page doi:10.1115/1.3662552
[17]

Reviews in Chemical Engineering 33, 1–47

Drop coalescence in technical liquid/liquid applications: a review on experimental techniques and modeling approaches. Reviews in Chemical Engineering 33, 1–47. doi:10.1515/revce-2015-0071. Kampwerth, J., Weber, B., Rußkamp, J., Kaminski, S., Jupke, A.,

work page doi:10.1515/revce-2015-0071 2015
[18]

Chemical Engineering Science 227, 115905

Towards a holistic solvent screening: On the importance of fluid dynamics in a rate-based extraction model. Chemical Engineering Science 227, 115905. doi:10.1016/j.ces.2020.115905. Kingma, D.P., Ba, J.,

work page doi:10.1016/j.ces.2020.115905 2020
[19]

Adam: A Method for Stochastic Optimization

Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980. Kraume, M., Gäbler, A., Schulze, K.,

work page internal anchor Pith review Pith/arXiv arXiv
[20]

Chemical Engineering & Technology 27, 330–334

Influence of physical properties on drop size distribution of stirred liquid-liquid dispersions. Chemical Engineering & Technology 27, 330–334. doi:10.1002/ceat. 200402006. Liu, D.C., Nocedal, J.,

work page doi:10.1002/ceat
[21]

Math- ematical programming 45, 503–528

On the limited memory BFGS method for large scale optimization. Math- ematical programming 45, 503–528. doi:10.1007/BF01589116. BIBLIOGRAPHY 35 Maddu, S., Sturm, D., Müller, C.L., Sbalzarini, I.F.,

work page doi:10.1007/bf01589116
[22]

Machine Learning: Science and Technology 3, 015026

Inverse Dirichlet weighting enables reliable training of physics informed neural networks. Machine Learning: Science and Technology 3, 015026. doi:10.1088/2632-2153/ac3712. Markidis, S.,

work page doi:10.1088/2632-2153/ac3712
[23]

Mersmann, A.,

doi:10.3389/fdata.2021.669097. Mersmann, A.,

work page doi:10.3389/fdata.2021.669097 2021
[24]

Chemie Ingenieur Technik 52, 933–942

Zum flutpunkt in flüssig/flüssig–gegenstromkolonnen. Chemie Ingenieur Technik 52, 933–942. doi:10.1002/cite.330521203. Mohamed, A., Schwarz, K.,

work page doi:10.1002/cite.330521203
[25]

Journal of Geodesy 73, 193–203

Adaptive Kalman filtering for INS/GPS. Journal of Geodesy 73, 193–203. doi:10.1007/s001900050236. Mustajab, A.H., Lyu, H., Rizvi, Z., Wuttke, F.,

work page doi:10.1007/s001900050236
[26]

arXiv preprintarXiv:2401.02810

Physics-informedneuralnetworksforhigh-frequency and multi-scale problems using transfer learning. arXiv preprintarXiv:2401.02810. Naumann, U.,

work page arXiv
[27]

Society for Industrial and Applied Mathematics

The Art of Differentiating Computer Programs. Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611972078. Padilla, R., Ruiz, M., Trujillo, W.,

work page doi:10.1137/1.9781611972078
[28]

Chemical Engineering Journal Advances 22, 100727

Towards the digital extraction column: Online-monitoringandanalysisoffluiddynamicsinliquid-liquidextractioncolumns. Chemical Engineering Journal Advances 22, 100727. doi:10.1016/j.ceja.2025.100727. Prantikos, K., Chatzidakis, S., Tsoukalas, L.H., Heifetz, A.,

work page doi:10.1016/j.ceja.2025.100727 2025
[29]

Raissi, M., Perdikaris, P., Karniadakis, G.E.,

doi:10.1038/s41598-023-43325-1. Raissi, M., Perdikaris, P., Karniadakis, G.E.,

work page doi:10.1038/s41598-023-43325-1
[30]

Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,

Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707. doi:10.1016/j.jcp.2018.10.045. Redmon, J., Divvala, S., Girshick, R., Farhadi, A.,

work page doi:10.1016/j.jcp.2018.10.045 2018
[31]

You only look once: Unified, real-time object detection, in: 29th IEEE Conference on Computer Vision and Pattern Recognition, IEEE, Piscataway, NJ. pp. 779–788. doi:10.1109/CVPR.2016.91. BIBLIOGRAPHY 36 Shlezinger, N., Whang, J., Eldar, Y.C., Dimakis, A.G.,

work page doi:10.1109/cvpr.2016.91 2016
[32]

arXiv preprint arXiv:2012.08405

Model-based deep learning. arXiv preprint arXiv:2012.08405. Sibirtsev, S., Thiel, L., Zhai, S., Cai, Y.T., Recke, L., Jupke, A.,

work page arXiv 2012
[33]

The Canadian Journal of Chemical Engineering doi:10.1002/cjce.25563

Experimental and model–based investigation of the droplet size distribution during the mixing process in a batch–settling cell. The Canadian Journal of Chemical Engineering doi:10.1002/cjce.25563. Sibirtsev, S., Zhai, S., Neufang, M., Seiler, J., Jupke, A.,

work page doi:10.1002/cjce.25563
[34]

Chemical Engineering Journal 473, 144826

Mask r-cnn based droplet detection in liquid–liquid systems, part 2: Methodology for determining training and image processing parameter values improving droplet detection accuracy. Chemical Engineering Journal 473, 144826. doi:10.1016/ j.cej.2023.144826. Tan, C., Cai, Y., Wang, H., Sun, X., Chen, L.,

work page arXiv 2023
[35]

Velioglu, M., Zhai, S., Rupprecht, S., Mitsos, A., Jupke, A., Dahmen, M.,

doi:10.3390/s23156665. Velioglu, M., Zhai, S., Rupprecht, S., Mitsos, A., Jupke, A., Dahmen, M.,

work page doi:10.3390/s23156665
[36]

Computers & Chemical Engineering 192, 108899

Physics-informed neural networks for dynamic process operations with limited physical knowledge and data. Computers & Chemical Engineering 192, 108899. doi:10.1016/j.compchemeng.2024.108899. Wackerly, D., Mendenhall, W., Scheaffer, R.L.,

work page doi:10.1016/j.compchemeng.2024.108899 2024
[37]

7th ed., Brooks/Cole

Mathematical Statistics with Applications. 7th ed., Brooks/Cole. Wang, Y., Bai, J., Eshaghi, M.S., Anitescu, C., Zhuang, X., Rabczuk, T., Liu, Y., 2025a. Transfer learning in physics-informed neural networks: Full fine-tuning, lightweight fine-tuning, and low-rank adaptation. arXiv preprintarXiv:2502.00782. Wang, Y., del Río Chanona, E.A., Quintanilla, P....

work page doi:10.1021/acs.iecr.5c00660 2023
[38]

Chemical Engineering Science 285, 119611

Impact of feeding conditions on continuous liquid-liquid gravity separation, part ii: Inlet/outlet drop size distribution and fractional separation efficiency. Chemical Engineering Science 285, 119611. doi:10.1016/j.ces.2023.119611. Zhai, S., Bartkowiak, N., Sibirtsev, S., Jupke, A.,

work page doi:10.1016/j.ces.2023.119611 2023
[39]

Separation and Purification Technology 377, 134177

Experimental determination and model-based prediction of flooding points in a pilot-scale continuous liquid-liquid gravity separator. Separation and Purification Technology 377, 134177. doi:10.1016/j.seppur.2025.134177. Zhang, L., Sidoti, D., Bienkowski, A., Pattipati, K.R., Bar-Shalom, Y., Kleinman, D.L.,

work page doi:10.1016/j.seppur.2025.134177 2025
[40]

IEEE Access 8, 59362–59388

On the identification of noise covariances and adaptive Kalman filtering: A new look at a 50 year-old problem. IEEE Access 8, 59362–59388. doi:10.1109/ACCESS.2020.2982407. Zhao, W., Queralta, J.P., Westerlund, T.,

work page doi:10.1109/access.2020.2982407 2020
[41]

Sim-to-real transfer in deep reinforcement learning for robotics: a survey, in: 2020 IEEE Symposium Series on Computational Intelligence (SSCI), pp. 737–744. doi:10.1109/SSCI47803.2020.9308468

work page doi:10.1109/ssci47803.2020.9308468 2020