Physics-informed neural network (PINN) approximations of capillary menisci in polygonal channels
Abstract
In a zero-gravity cylindrical container, a meniscus can span a square or equilateral-triangular cross-section only at contact angles at or above the respective Concus-Finn critical angles of 45 degrees and 60 degrees. We study strong-form physics-informed neural networks (PINNs) across these thresholds under fixed collocation and optimization budgets. Classical capillary flux balance yields a lower bound on the combined interior and wall residual loss for arbitrary positive sampling densities. In the tested nonexistence regime, several degrees below the critical angles, independently evaluated network losses are 1.6-1.9 times this bound. A polynomial least-squares comparison supports the same geometric interpretation. The PINNs still return smooth graphs with finite axial extent, even though the prescribed equilibrium problem has no spanning graph solution. Their small interior residuals coexist with wall-flux deficits near corners, where adjoining contact-angle conditions demand more flux than a finite-slope graph can supply. Residuals vary on both sides of the critical angle, while the bound tends uniformly to zero at threshold for fixed wall weight. These results give a geometric scale for interpreting the computed loss and locate the incompatibility that a small interior residual can conceal.
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BibTeX
@misc{liu2026physicsinformed,
title = {{Physics-informed neural network (PINN) approximations of capillary menisci in polygonal channels}},
author = {Zijian Liu and Julian Olszewski and Jie Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31963},
eprint = {2610.31963},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31963},
}
RIS
TY - UNPB
TI - Physics-informed neural network (PINN) approximations of capillary menisci in polygonal channels
AU - Zijian Liu
AU - Julian Olszewski
AU - Jie Xu
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31963
UR - https://arxiv.org/abs/2610.31963
AB - In a zero-gravity cylindrical container, a meniscus can span a square or equilateral-triangular cross-section only at contact angles at or above the respective Concus-Finn critical angles of 45 degrees and 60 degrees. We study strong-form physics-informed neural networks (PINNs) across these thresholds under fixed collocation and optimization budgets. Classical capillary flux balance yields a lower bound on the combined interior and wall residual loss for arbitrary positive sampling densities. In the tested nonexistence regime, several degrees below the critical angles, independently evaluated network losses are 1.6-1.9 times this bound. A polynomial least-squares comparison supports the same geometric interpretation. The PINNs still return smooth graphs with finite axial extent, even though the prescribed equilibrium problem has no spanning graph solution. Their small interior residuals coexist with wall-flux deficits near corners, where adjoining contact-angle conditions demand more flux than a finite-slope graph can supply. Residuals vary on both sides of the critical angle, while the bound tends uniformly to zero at threshold for fixed wall weight. These results give a geometric scale for interpreting the computed loss and locate the incompatibility that a small interior residual can conceal.
ER -