[Paper Review] Deep Dynamical Modeling and Control of Unsteady Fluid Flows

Jeremy Morton, Freddie Witherden+2 more|arXiv (Cornell University)|May 18, 2018

Model Reduction and Neural Networks40 references73 citations

TL;DR

The paper develops a Deep Koopman framework to learn forced and unforced fluid dynamics from CFD data, enabling stable long-horizon prediction and MPC-based control to suppress cylinder wake vortex shedding with an interpretable control law.

ABSTRACT

The design of flow control systems remains a challenge due to the nonlinear nature of the equations that govern fluid flow. However, recent advances in computational fluid dynamics (CFD) have enabled the simulation of complex fluid flows with high accuracy, opening the possibility of using learning-based approaches to facilitate controller design. We present a method for learning the forced and unforced dynamics of airflow over a cylinder directly from CFD data. The proposed approach, grounded in Koopman theory, is shown to produce stable dynamical models that can predict the time evolution of the cylinder system over extended time horizons. Finally, by performing model predictive control with the learned dynamical models, we are able to find a straightforward, interpretable control law for suppressing vortex shedding in the wake of the cylinder.

Motivation & Objective

• Motivate flow control for nonlinear fluid dynamics and reduce reliance on costly CFD-based controller design.
• Learn a Koopman-invariant subspace through a neural encoder to model both forced and unforced cylinder flow dynamics.
• Demonstrate long-horizon stable predictions and integrate the learned model into model predictive control for vortex-shedding suppression.
• Show that the derived control law is interpretable and aligns with simple, implementable feedback strategies.

Proposed method

• Represent the nonlinear fluid dynamics in a finite-dimensional space via a neural encoder g(x) that outputs state mappings spanning a Koopman invariant subspace.
• Compute a linear dynamics A in the Koopman subspace by least-squares fitting on encoded snapshots.
• Train end-to-end by enforcing accurate reconstruction and forward-prediction of time-evolved states, with recursive prediction during training.
• Extend the model to forced dynamics by incorporating control influence with a known B matrix and learn a global B while estimating A from data.
• Apply the learned (A,B) in a model predictive control (MPC) framework to minimize a quadratic cost over a horizon, using CVXPY for optimization.

Experimental results

Research questions

• RQ1 Can a Deep Koopman representation learn invariant subspace mappings that accurately predict unforced CFD dynamics from cylinder flow data?
• RQ2 How does the Deep Koopman model compare to baseline Deep Variational Bayes Filters and Takeishi et al.’s approach in short- and long-horizon prediction?
• RQ3 Can the learned Koopman-based model be extended to forced (control-influenced) dynamics with interpretable control behavior?
• RQ4 Is MPC leveraging the learned model capable of suppressing vortex shedding and producing an interpretable control law?

Key findings

• The Deep Koopman model achieves stable multi-step predictions, with mean error remaining below 0.2% over a horizon of 128 time steps (≈ eight vortex-shedding periods).
• Compared to the Deep Variational Bayes Filter, the Deep Koopman model remains stable under long-horizon prediction, whereas VBF degrades beyond ~32 steps.
• When applied to forced dynamics, the model can incorporate control inputs through a linear-in-the-coordinates form g(x_{t+1}) = A g(x_t) + B u_t, enabling MPC to suppress vortex shedding in the cylinder wake.
• The MPC-based control with the learned model yields a monotonic decrease in flow residuals toward a laminar state, and reveals an interpretable control pattern resembling a proportional controller based on y-velocity measurements.
• The resulting control law is easy to implement and shows strong alignment between the optimal MPC actions and a single-hotspot feedback location in the wake (d*), supporting practical applicability.