An Interpretable Convolutional Neural Network Framework for Fluid Dynamics

TL;DR

This paper presents an interpretable CNN framework that directly links data-driven models to classical fluid dynamics operators, enabling transparent, generalizable, and insightful fluid flow modeling across various conditions.

Contribution

The authors introduce a simple, interpretable CNN architecture trained on fluid data that reproduces finite-difference schemes and generalizes across flow regimes, bridging numerical analysis and machine learning.

Findings

CNN learns the forward-Euler three-point stencil with three weights.

The approach generalizes to analytical solutions and molecular dynamics data.

The method reveals when and why physics is or isn't captured in models.

Abstract

Fluid dynamics spans phenomena from the Cheerios effect to cosmic evolution and has been called the 'queen mother' of science. Traditional modelling relies on numerical methods, including finite differences, volumes, and elements, that simulate flows across scales. Recent advances in machine learning have enabled data-driven fluid models, but these approaches are often complex and opaque. We introduce a transparent framework that links data-driven models directly to classical fluid-dynamics operators. A simple convolutional neural network (CNN) is trained on laminar-flow data to reproduce the exact behaviour of a finite-difference scheme, providing an interpretable bridge between numerical analysis and machine learning (ML). The CNN generalises across a wide range of unseen flow conditions and learns the forward-Euler three-point stencil, capturing principles such as consistency and symmetry with only three trainable weights. The approach extends beyond numerical data: the same architecture works when trained on analytical solutions and even molecular-dynamics trajectories. Its simplicity reveals when and why physics is or is not captured, offering insight into the limits and best practices of data-driven fluid modelling. Because it is grounded in finite-difference operators, the method naturally generalises to many structured-grid CFD problems, including turbulent, multiphase, and multiscale flows.