Qijia Zhai

Qijia Zhai, Pengtao Sun, Xiaoping Xie et al.

In this paper, a meshfree method using physics-informed neural networks (PINNs) is developed for solving two-phase flow problems with moving interfaces, where two immiscible fluids bearing different material properties, are separated by a dynamically evolving interface and interact with each other through interface conditions. Two kinds of distinct scenarios of interface motion are addressed: the prescribed interface motion whose moving velocity is explicitly given, and the solution-driven interface motion whose evolution is determined by the velocity field of two-phase flow. Based upon piecewise deep neural networks and spatiotemporal sampling points/training set in each fluid subdomain, the proposed PINNs framework reformulates the two-phase flow moving interface problem as a least-squares (LS) minimization problem, which involves all residuals of governing equations, interface conditions, boundary conditions and initial conditions. Furthermore, approximation properties of the proposed PINNs approach are analyzed rigorously for the presented two-phase flow model by employing the Reynolds transport theorem in evolving domains, moreover, a comprehensive error estimation is provided to account for additional complexities introduced by the moving interface and the coupling between fluid dynamics and interface evolution. Numerical experiments are carried out to illustrate the effectiveness of the proposed PINNs approach for various configurations of two-phase flow moving interface problems, and to validate the theoretical findings as well. A practical guidance is thus provided for an efficient training set distribution when applying the proposed PINNs approach to two-phase flow moving interface problems in practice.

Qijia Zhai

We investigate the learning dynamics of shallow ReLU neural networks on the unit sphere \(S^2\subset\mathbb{R}^3\) in polar coordinates \((τ,φ)\), considering both fixed and trainable neuron directions \(\{w_i\}\). For fixed weights, spherical harmonic expansions reveal an intrinsic low-frequency preference with coefficients decaying as \(O(\ell^{5/2}/2^\ell)\), typically leading to the Frequency Principle (FP) of lower-frequency-first learning. However, this principle can be violated under specific initial conditions or error distributions. With trainable weights, an additional rotation term in the harmonic evolution equations preserves exponential decay with decay order \(O(\ell^{7/2}/2^\ell)\) factor, also leading to the FP of lower-frequency-first learning. But like fixed weights case, the principle can be violated under specific initial conditions or error distributions. Our numerical results demonstrate that trainable directions increase learning complexity and can either maintain a low-frequency advantage or enable faster high-frequency emergence. This analysis suggests the FP should be viewed as a tendency rather than a rule on curved domains like \(S^2\), providing insights into how direction updates and harmonic expansions shape frequency-dependent learning.