Helicity-conservative Physics-informed Neural Network Model for Navier-Stokes Equations

TL;DR

This work addresses preserving fluid helicity in the incompressible Navier-Stokes system by introducing helicity-conservative physics-informed neural networks (PINNs) built on the strong form of the PDE. It develops two PINN variants, up_NN and omega_NN, to enforce helicity conservation and compares them with helicity-preserving finite element methods. The authors provide theoretical justification and numerical evidence showing that up_NN preserves helicity in the ideal, non-dissipative limit and demonstrates divergence-free velocity and energy-like behavior, while omega_NN can fail to conserve helicity due to vorticity-related terms, though adjustments to the vorticity definition can improve results. The findings highlight the potential of structure-preserving, data-driven solvers for complex fluid invariants and suggest practical routes for extending physics-informed learning to other conserved quantities.

Abstract

We design the helicity-conservative physics-informed neural network model for the Navier-Stokes equation in the ideal case. The key is to provide an appropriate PDE model as loss function so that its neural network solutions produce helicity conservation. Physics-informed neural network model is based on the strong form of PDE. We compare the proposed Physics-informed neural network model and a relevant helicity-conservative finite element method. We arrive at the conclusion that the strong form PDE is better suited for conservation issues. We also present theoretical justifications for helicity conservation as well as supporting numerical calculations.

Figures (9)

• Figure 1: DOFs of the finite element de Rham sequence of lowest order
• Figure 2: Seq2seq PINN. The blue line is initial condition for each sequence. The red line is boundary condition for each sequence. The domain will be uniformly sectioned. When training of the first sequence finished, the solution at $t=0.01$ will be calculated and used as the initial condition for the next sequence.
• Figure 3: Top view, or projection (onto $xy$ plane) of initial $\bm{u}$
• Figure 4: Maximum divergence of $\bm{u}$ for $T=1s$ for $up_{NN}$ Network
• Figure 5: Energy conserves when $T=1s$ for $up_{NN}$ Network

Theorems & Definitions (15)

• Theorem 1
• Definition 1
• Theorem 2
• proof
• Lemma 1
• Theorem 3
• Theorem 4
• proof
• Theorem 5
• proof