# Data-Driven Adaptive Gradient Recovery for Unstructured Finite Volume Computations

arXiv: https://arxiv.org/abs/2507.16571 · HTML: https://mlsanity.com/paper/2507.16571/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 52 references that are in the map.
- First posted on arXiv 22 Jul 2025.

## Builds on (most cited first, 16 of 16)

- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [An Exponential Learning Rate Schedule for Deep Learning](https://mlsanity.com/paper/1910.07454/) (2019)
- [DiscretizationNet: A Machine-Learning based solver for Navier-Stokes Equations using Finite Volume Discretization](https://mlsanity.com/paper/2005.08357/) (2020)
- [TensorFlow Eager: A Multi-Stage, Python-Embedded DSL for Machine Learning](https://mlsanity.com/paper/1903.01855/) (2019)
- [Enhancement of shock-capturing methods via machine learning](https://mlsanity.com/paper/2002.02521/) (2020)
- [FiniteNet: A Fully Convolutional LSTM Network Architecture for Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2002.03014/) (2020)
- [Constraint-Aware Neural Networks for Riemann Problems](https://mlsanity.com/paper/1904.12794/) (2019)
- [A data-driven learned discretization approach in finite volume schemes for hyperbolic conservation laws and varying boundary conditions](https://mlsanity.com/paper/2412.07541/) (2024)
- [Finite Volume Features, Global Geometry Representations, and Residual Training for Deep Learning-based CFD Simulation](https://mlsanity.com/paper/2311.14464/) (2023)

## Built on by (most cited first, 2 of 2)

- [An End-to-End PyTorch Interface for Differentiable PDE Solvers: A RANS Model-Correction Study](https://mlsanity.com/paper/2605.28858/) (2026)
- [Guarantees by Construction for Learned Finite Volume Schemes on Steady Supersonic Flow](https://mlsanity.com/paper/2607.20171/) (2026)

## Abstract (arXiv)

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.