Machine learning-based WENO5 scheme - Neural network for weighting functions

Aug 3, 2026

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Machine learning is increasingly used in Computational Fluid Dynamics to improve the accuracy, efficiency, and automation of simulations. For shock-capturing methods, nonlinear components such as smoothness indicators and weighting functions often depend heavily on author expertise. This work introduces a neural-network approach that directly computes the nonlinear weighting functions in the WENO5 scheme. The WENO5-NN method generalizes across different resolutions and, in most tests, outperforms the classical WENO5-JS scheme.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages16

Reading Time40 min

AccessFree to download

Introduction

Motivation: ML in CFD and WENO schemes

Proposed approach: neural-network weighting for WENO5

Computers and Mathematics with Applications 168 (2024) 84–99 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal [homepage: www.elsevier.com/locate/camwa](homepage: www.elsevier.com/locate/camwa) Machine learning-based WENO5 scheme Xesús Nogueira a,∗ , Javier Fernández-Fidalgob, Lucía Ramos c,d, Iván Couceiro a, Luis Ramírez a a Group of Numerical Methods in Engineering-GMNI, Center for Technological Innovation in Construction and Civil Engineering-CITEEC, Civil Engineering School, Universidade da Coruña, Campus de Elviña, 15071, A Coruña, Spain b Departamento de Ingeniería Geológica y Minera, ETSI de Minas y Energía, Universidad Politécnica de Madrid, Calle Ríos Rosas 21, 28003, Madrid, Spain c CITIC Research Center, Universidade da Coruña, Campus de Elviña s/n, 15071, A Coruña, Spain d VARPA Group, Biomedical Research Institute of A Coruña (INIBIC), Universidade da Coruña, Xubias de Arriba, 84, 15006, A Coruña, Spain | A R T I C L E | I N F | O | A B S T R A C T | | --- | --- | --- | --- | | Keywords: WENO
Neural networks
Machine learning Euler equations Finite diﬀerence | | | Machine learning (ML) is becoming a powerful tool in Computational Fluid Dynamics (CFD) to enhance the accuracy, eﬃciency, and automation of simulations. Currently, in the design of shock-capturing methods, thereis still a heavy reliance on the expertise and scientiﬁc knowledge of each author, particularly in nonlinear components such as smoothness indicators and weighting functions. ML has the potential to reduce this dependency, since by leveraging large datasets, they can learn intricate patterns and make accurate predictions of these functions. In this work we present a neural network that compute the weighting functions in the WENO5 scheme. The proposed WENO5-NN scheme generalizes well for diﬀerent resolutions, and in most of the cases tested, it outperforms the classical WENO5-JS scheme. | 1. Introduction In recent years, Machine learning (ML) has emerged as a promising tool in the ﬁeld of Computational Fluid Dynamics (CFD), with the potential to improve the accuracy, eﬃciency, and automation of CFD simulations. ML techniques have been applied in various aspects of CFD, ranging from turbulence modeling and ﬂow prediction [1,2] to optimization and uncertainty quantiﬁcation [3]. Machine learning has also been utilized for the development of numerical schemes in CFD, with the aim of improving the accuracy and robustness of numerical simulations. ML-based approaches have shown promise in various aspects of numerical scheme development, including adaptive mesh reﬁnement [4] and numerical stabilization [5–8]. Adaptive Mesh Reﬁnement (AMR) involves dynamically reﬁning or coarsening the computational mesh based on the solution’s features. However, practical strategies for AMR lack a systematic framework and often rely on domain-speciﬁc expertise, experimentation, or manual adjustments, which makes attractive the use of ML approaches. ML techniques, such as decision trees, support vector machines, and deep learning, can be used to predict the regions of interest in the ﬂow ﬁeld and guide the adaptive mesh reﬁnement process. ML has also been used for numerical stabilization in CFD simulations, since traditional numerical schemes, such as ﬁnite diﬀerence, ﬁnite volume, or ﬁnite element methods, may face stabil- ity issues, especially for challenging ﬂows or complex geometries. In this context, ML-based approaches, such as artiﬁcial neural networks, can learn from a large dataset to estimate the cells where stabilization is required. These ML-based stabilization techniques can help improve the stability and accuracy of numerical schemes, particularly in regions with sharp gradients or high-frequency oscillations. Weighted Essentially Non-Oscillatory (WENO) schemes [9–14] are known for their ability to capture sharp discontinuities in the ﬂow ﬁeld. There are currently numerous variants

FAQs

Why use machine learning for WENO shock-capturing methods?

WENO variants differ in nonlinear components like smoothness indicators and weighting functions, which typically require substantial author expertise. ML can learn patterns from data to reduce that dependence while improving predictive accuracy.

How does the proposed WENO5-NN scheme compute the WENO weights?

A neural-network regressor directly outputs the nonlinear weighting-function values for the WENO5 scheme, instead of perturbing coefficients derived from the original WENO5-JS method.

How does WENO5-NN perform compared with classical WENO5-JS?

The method generalizes well for different resolutions and, in most tested cases, produces results that outperform the classical WENO5-JS scheme while remaining stable in the presence of shocks.

Aug 3, 2026

Upload by Aria