Pai Zhang | Searcharxiv

arXiv2026

A phase-field neural solver for moving contact line problems with dynamic boundary conditions

Phase-field models based on the Cahn--Hilliard equation coupled with dynamic boundary conditions provide a thermodynamically consistent framework for moving contact line (MCL) problems. Although physics-informed neural networks (PINNs) offer a mesh-free approach for solving partial differential equations, their direct application to MCL problems remains challenging due to long-time error accumulation, sharp interfacial profiles, localized contact line dynamics, and complex contact angle evolution. In this work, we propose MCL-PINNs, a specialized phase-field neural solver designed for MCL problems with dynamic boundary conditions. The method is built on a discrete-time formulation and incorporates several key techniques, including a multi-network time-marching scheme, a relaxed distribution constraint on the neural network outputs, variable scaling for sharply varying solution features, adaptive loss weighting, adaptive collocation sampling with interface extraction, and, when applicable, symmetry preservation through neural network inputs. These techniques improve the capability of the neural solver in resolving sharp interfacial profiles and contact line motion. The proposed method is validated through three numerical examples involving droplet coalescence, shear-induced droplet deformation, and dynamic wetting in a heterogeneous channel. The numerical results show that MCL-PINNs significantly improve prediction accuracy and robustness compared with standard PINNs formulations, enabling reliable resolution of complex interfacial evolution and moving contact line dynamics.

math.NA↗

arXiv2026

A domain decomposition online-learning-enhanced nonlinear elimination preconditioner

Nonlinearly preconditioned inexact Newton methods form an effective class of solvers for large-scale nonlinear algebraic systems arising from the discretization of partial differential equations. A central challenge in nonlinear elimination (NE) preconditioning is the reliable identification of the slowly converging components to be eliminated. Existing selection strategies often rely on problem-specific physical information or user-tuned thresholds applied directly to the raw nonlinear residual, which may contain irregular oscillatory structures near stagnation regions, making the selected bad subset highly sensitive to threshold parameters. In this work, we propose an online-learning-enhanced NE preconditioner that identifies the bad subset from the dominant structure of the nonlinear residual rather than from the raw residual itself. Residual snapshots are collected online during the stagnation phase of the current Newton solve, and an unsupervised extraction model is trained to capture the principal nonlinear imbalance. We consider both a linear extractor based on principal component analysis and nonlinear extractors based on autoencoder neural networks. Moreover, we integrate the approach into a parallel domain decomposition framework, which trains a local extraction model independently on each subdomain. The learned residual reconstruction is then used to define the bad subset and guide the nonlinear elimination process. Numerical experiments on lid-driven cavity flows at Reynolds numbers up to 10,000 show that the proposed method produces more reliable and coherent bad subsets, is robust with respect to both NE and learning parameters, and outperforms the baseline NE preconditioner in terms of the convergence.

math.NA↗