An interpretable convolutional neural network framework for fluid dynamics

Abstract

Modelling fluid dynamics with machine learning (ML) has advanced rapidly, yet most data driven approaches remain opaque because they rely on complex architectures to capture nonlinear flow behaviour. This lack of interpretability limits their reliability and hinders understanding of when and why they succeed or fail. To address this, we present a transparent approach that provides insights into how data-driven fluids dynamics and machine learning (ML) work. This is achieved by training a convolutional neural network (CNN), on data from a simple laminar fluid flow, to behave as an operator that exactly matches the finite-difference numerics, providing a direct link between well-established theory and this new world of ML models. Importantly, the model demonstrates strong generalisation capability by reproducing the dynamics for a wide range of distinct and unseen flow conditions within the same flow category. The CNN learns the forward Euler three-point stencil weights, capturing physical principles such as consistency and symmetry despite having only three tuneable weights. This interpretable ML model goes beyond pure numerical training (numCNN), the approach is shown to work when trained on analytical (anCNN) and even molecular dynamics (mdCNN) data. In some cases, the physics is not captured, and thanks to the simple and interpretable form, these CNNs provide insight into the limits, pitfalls and best practice of data-driven fluid models. Because the approach is based on finite-difference operators, it naturally extends to many structured-grid computational fluid dynamics (CFD) problems, including turbulent, multiphase and multiscale flows as well as systems beyond the continuum such as molecular dynamics (MD). To support reproducibility and accelerate adoption, all simulation code, training pipelines, pretrained models, and processed datasets are available open source on GitHub under kwamea-b/CNN-numerical-schemes.