Data-Driven Adaptive Gradient Recovery for Unstructured Finite Volume Computations

TL;DR

This paper introduces a data-driven neural network approach to improve gradient reconstruction in unstructured finite volume methods for hyperbolic PDEs, achieving higher accuracy and efficiency.

Contribution

It extends structured-grid neural methods to unstructured meshes using a geometry-aware DeepONet architecture with physics-informed regularization.

Findings

Achieves 20-60% accuracy improvements over traditional schemes.

Enables high-fidelity simulations on coarser grids.

Demonstrates improved mesh convergence rates.

Abstract

We present a novel data-driven approach for enhancing gradient reconstruction in unstructured finite volume methods for hyperbolic conservation laws, specifically for the 2D Euler equations. Our approach extends previous structured-grid methodologies to unstructured meshes through a modified DeepONet architecture that incorporates local geometry in the neural network. The architecture employs local mesh topology to ensure rotation invariance, while also ensuring first-order constraint on the learned operator. The training methodology incorporates physics-informed regularization through entropy penalization, total variation diminishing penalization, and parameter regularization to ensure physically consistent solutions, particularly in shock-dominated regions. The model is trained on high-fidelity datasets solutions derived from sine waves and randomized piecewise constant initial conditions with periodic boundary conditions, enabling robust generalization to complex flow configurations or geometries. Validation test cases from the literature, including challenging geometry configuration, demonstrates substantial improvements in accuracy compared to traditional second-order finite volume schemes. The method achieves gains of 20-60% in solution accuracy while enhancing computational efficiency. A convergence study has been conveyed and reveal improved mesh convergence rates compared to the conventional solver. The proposed algorithm is faster and more accurate than the traditional second-order finite volume solver, enabling high-fidelity simulations on coarser grids while preserving the stability and conservation properties essential for hyperbolic conservation laws. This work is a part of a new generation of solvers that are built by combining Machine-Learning (ML) tools with traditional numerical schemes, all while ensuring physical constraint on the results.