A Helicity-Conservative Domain-Decomposed Physics-Informed Neural Network for Incompressible Non-Newtonian Flow

TL;DR

This paper introduces a helicity-preserving physics-informed neural network framework for simulating incompressible non-Newtonian flows, emphasizing geometric fidelity and long-time stability.

Contribution

It proposes a novel neural formulation computing vorticity via automatic differentiation and combines domain decomposition with slab-wise temporal continuation for robust, long-term flow simulations.

Findings

The framework accurately preserves helicity in flow simulations.

The method demonstrates stability and scalability for long-time transient flows.

Vorticity is computed directly from neural velocity fields, reducing compatibility errors.

Abstract

This paper develops a helicity-aware physics-informed neural network framework for incompressible non-Newtonian flow in rotational form. In addition to the energy law and the incompressibility constraint, helicity is a fundamental geometric quantity that characterizes the topology of vortex lines and plays an important role in the physical fidelity of long-time flow simulations. While helicity-preserving discretizations have been studied extensively in finite difference, finite element, and other structure-preserving settings, their realization within neural network solvers remains largely unexplored. Motivated by this gap, we propose a neural formulation in which vorticity is computed directly from the neural velocity field by automatic differentiation rather than learned as an independent output, thereby avoiding compatibility errors that pollute the helicity balance. To improve robustness and scalability, we combine two algorithmic ingredients: an overlapping spatial domain decomposition inspired by finite-basis physics-informed neural networks (FBPINNs), and a causal slab-wise temporal continuation strategy for long-time transient simulations. The local subnetworks are blended by explicitly normalized super-Gaussian window functions, which yield a smooth partition of unity, while the temporal evolution is advanced sequentially across time slabs by transferring the converged solution on one slab to the next. The resulting spatiotemporal framework provides a stable and physically meaningful approach for helicity-aware simulation of incompressible non-Newtonian flows.