Hybrid Weak Galerkin and Physics-Encoded Neural Networks: a sparse local-to-global extension framework for implicit multiphase flow simulation

A. Laadhari, Applied Mathematics and Computation (Elsevier). 2026. In production

Abstract

An efficient, conservative hybrid weak Galerkin finite element methodology is proposed to simulate multiphase flows. An advection-oriented weak gradient yields a flow-sensitive stabilized weak Galerkin method. The directional stabilization reduces numerical dissipation, improving mass conservation and preserving interface features. High-order numerical experiments for steady advection confirm the theoretical analysis and exhibit the expected optimal convergence behavior. To achieve a balance between computational efficiency and local accuracy in full-domain settings,
a hybrid strategy resolves advection with high accuracy near the interface.
A Physics-Informed Neural Network (PINN), calibrated with sparse high-fidelity weak Galerkin data from a localized subdomain,
provides a global surrogate of the solution guided by these data, with the weak Galerkin approximation retained near the interface through a blending procedure. The framework extends naturally to time-dependent problems, including level-set transport, and is applied to capillarity-dominated interface dynamics in Newtonian fluids. We also introduce an implicit coupling scheme based on a multidimensional generalization of a fifth-order multi-stage Newton-type scheme. Benchmark tests and numerical comparisons demonstrate the reliability and accuracy of the proposed approach.

Highlights

• Novel advection-oriented weak gradient yields a flow-sensitive stabilized WG method.
• Stability and error analysis prove optimal convergence of high-order WG schemes.
• Scientific ML combines sparse high-fidelity WG data with PINNs for global prediction.
• Hybrid WG–PINN preserves interface accuracy while reconstructing the global field.
• Fifth-order Newton iterations accelerate implicit multiphase flow simulations.