[Paper Review] DeepONet prediction of linear instability waves in high-speed boundary layers

Patricio Clark Di Leoni, Lipeng Lu+3 more|arXiv (Cornell University)|May 18, 2021

Fluid Dynamics and Turbulent Flows1 references30 citations

TL;DR

The paper trains DeepONet to learn the operator governing linear instability waves in compressible boundary layers, enabling fast downstream predictions and inverse reconstruction from wall data, with data assimilation applications.

ABSTRACT

Deep operator networks (DeepONets) are trained to predict the linear amplification of instability waves in high-speed boundary layers and to perform data assimilation. In contrast to traditional networks that approximate functions, DeepONets are designed to approximate operators. Using this framework, we train a DeepONet to take as inputs an upstream disturbance and a downstream location of interest, and to provide as output the perturbation field downstream in the boundary layer. DeepONet thus approximates the linearized and parabolized Navier-Stokes operator for this flow. Once trained, the network can perform predictions of the downstream flow for a wide variety of inflow conditions, without the need to calculate the whole trajectory of the perturbations, and at a very small computational cost compared to discretization of the original equations. In addition, we show that DeepONets can solve the inverse problem, where downstream wall measurements are adopted as input and a trained network can predict the upstream disturbances that led to these observations. This capability, along with the forward predictions, allows us to perform a full data assimilation cycle: starting from wall-pressure data, we predict the upstream disturbance using the inverse DeepONet and its evolution using the forward DeepONet.

Motivation & Objective

• Motivate accurate, efficient prediction of linear instability wave amplification in high-speed boundary layers to aid transition prediction and control.
• Develop a DeepONet framework that treats the governing dynamics as an operator mapping upstream disturbances to downstream perturbation fields.
• Demonstrate forward prediction of downstream flow fields for varied inflow disturbances with low computational cost.
• Show inverse capability: infer upstream disturbances from downstream wall measurements to enable data assimilation.

Proposed method

• Adopt the DeepONet architecture with a branch network handling discretized input functions and a trunk network handling evaluation points downstream.
• Formulate the operator G† that maps an upstream disturbance f to downstream perturbation G(f)(ζ), and train the network to approximate this operator via a loss L comparing G(f)(ζ) to G†(f)(ζ).
• Train multiple DeepONets to emulate the parabolized stability equations (PSE) that describe linearized perturbation evolution within a developing boundary layer.
• Use forward models to predict downstream fields (e.g., velocity, pressure, temperature) from upstream disturbances and an inverse model to recover upstream disturbances from downstream wall-pressure data.
• Incorporate data augmentation through phase and frequency variation, and apply weighted loss to balance amplitudes across decaying downstream modes.

Experimental results

Research questions

• RQ1 Can a DeepONet learn the operator that maps upstream instability disturbances to downstream perturbation fields in a high-speed boundary layer?
• RQ2 How accurately can DeepONet predict downstream fields for unseen disturbance frequencies and phases?
• RQ3 Can DeepONet solve the inverse problem of inferring upstream disturbances from downstream wall-pressure measurements?
• RQ4 How does DeepONet performance compare when predicting different fields (velocity, pressure, temperature) derived from the same governing equations?
• RQ5 What is the data assimilation workflow combining forward and inverse DeepONets for wall-pressure–based disturbance reconstruction?

Key findings

• DeepONets can reproduce downstream perturbation fields predicted by the PSE with relative errors below 5% on validation modes.
• Forward cases show accurate recovery of wall-normal profiles, wavelengths, phases, and amplitudes for unseen frequencies.
• Inverse cases successfully reconstruct downstream input disturbances from wall-pressure measurements, enabling a data assimilation loop.
• Training requires substantial data generation but yields extremely fast evaluations compared to solving the PSE for new inputs.
• Feature expansion and loss weighting help mitigate training challenges due to amplitude disparities across modes, improving convergence and accuracy.
• Evaluation time per DeepONet evaluation is orders of magnitude faster than running new PSE simulations, enabling rapid many-query assessments.