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Multi-Fidelity Machine Learning Applied to Steady Fluid Flows

Kazuko W. Fuchi, Eric M. Wolf, David S. Makhija, Christopher R. Schrock, Philip S. Beran

TL;DR

The paper introduces MF-LEIF, a data-efficient ML framework that predicts steady external flow fields by mapping elliptic input features derived from potential-flow BVPs to flow-field discrepancies against a low-fidelity baseline. It employs a near-body training window with a Partition-of-Unity extension to extend predictions to freestream and uses a quadtree adaptive sampling strategy, complemented by Sobolev training, to maximize accuracy with few training points. Demonstrations on flow around a circular cylinder and a Joukowski airfoil show that elliptic input features enable generalization to nearby geometries and AoAs, while the MF-LEIF predictions can warm-start CFD runs to accelerate solver convergence. The approach offers a practical route to reducing high-fidelity CFD data requirements in design optimization, with potential extensions to 3D, compressible, and turbulent flows.

Abstract

A machine learning method to predict steady external fluid flows using elliptic input features is introduced. Using data from as few as one high-fidelity simulation, the proposed method produces models generalizable under changes to boundary geometry by using solutions to elliptic boundary value problems over the flow domain as the model input, instead of Cartesian coordinates of the domain. Training data is generated through pointwise evaluation of flow features at points selected through a quad-tree adaptive sampling method to concentrate training points in areas with large field gradients. Models are trained within a training window around the body, while predictions are smoothly extended to freestream conditions using a Partition-of-Unity extension. Predictive capabilities of the machine learning model are demonstrated in steady-state flow of incompressible fluid around a cylinder and a Joukowski airfoil. The predicted flow field is used to warm-start CFD simulations to achieve acceleration in solver convergence.

Multi-Fidelity Machine Learning Applied to Steady Fluid Flows

TL;DR

The paper introduces MF-LEIF, a data-efficient ML framework that predicts steady external flow fields by mapping elliptic input features derived from potential-flow BVPs to flow-field discrepancies against a low-fidelity baseline. It employs a near-body training window with a Partition-of-Unity extension to extend predictions to freestream and uses a quadtree adaptive sampling strategy, complemented by Sobolev training, to maximize accuracy with few training points. Demonstrations on flow around a circular cylinder and a Joukowski airfoil show that elliptic input features enable generalization to nearby geometries and AoAs, while the MF-LEIF predictions can warm-start CFD runs to accelerate solver convergence. The approach offers a practical route to reducing high-fidelity CFD data requirements in design optimization, with potential extensions to 3D, compressible, and turbulent flows.

Abstract

A machine learning method to predict steady external fluid flows using elliptic input features is introduced. Using data from as few as one high-fidelity simulation, the proposed method produces models generalizable under changes to boundary geometry by using solutions to elliptic boundary value problems over the flow domain as the model input, instead of Cartesian coordinates of the domain. Training data is generated through pointwise evaluation of flow features at points selected through a quad-tree adaptive sampling method to concentrate training points in areas with large field gradients. Models are trained within a training window around the body, while predictions are smoothly extended to freestream conditions using a Partition-of-Unity extension. Predictive capabilities of the machine learning model are demonstrated in steady-state flow of incompressible fluid around a cylinder and a Joukowski airfoil. The predicted flow field is used to warm-start CFD simulations to achieve acceleration in solver convergence.
Paper Structure (14 sections, 22 equations, 20 figures, 4 tables, 8 algorithms)

This paper contains 14 sections, 22 equations, 20 figures, 4 tables, 8 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. Elliptic Input Features
  4. Training Window and Partition-of-Unity (POFU) Extension
  5. Training Data Sampling
  6. Results
  7. Circular Cylinder
  8. Joukowski Airfoil
  9. CFD Initialization for Joukowski Airfoil
  10. Conclusions
  11. Algorithms
  12. Derivation of Potential Flow Solutions
  13. Circular cylinder
  14. Joukowski airfoil

Figures (20)

  • Figure 1: Flow charts of the MF-LEIF training and prediction processes.
  • Figure 2: Examples of elliptic input features for canonical problems.
  • Figure 3: ML model training window and POFU window functions.
  • Figure 4: Illustration of a quadtree derefinement and refinement process (Algorithm \ref{['alg:qdtree']}, Appendix \ref{['sec:appendix-alg']}).
  • Figure 5: Quadtree sampling points for flow around a cylinder.
  • ...and 15 more figures