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Deep Dynamical Modeling and Control of Unsteady Fluid Flows
Jeremy Morton, Freddie D. Witherden, Antony Jameson, Mykel J. Kochenderfer
TL;DR
Flow control remains difficult because nonlinear fluid dynamics limit conventional control methods and accurate model-free learning can be costly. The paper learns forced and unforced cylinder-flow dynamics using Koopman-based models, then embeds them in model predictive control. The models remain stable over long horizons and yield an interpretable control law that suppresses vortex shedding.
Problem
Nonlinear fluid dynamics challenge conventional flow control, while model-free methods may require too many costly simulation interactions and model-based methods require accurate dynamical models.
Method
The paper learns forced and unforced cylinder-flow dynamics with Koopman-based Deep Koopman models and incorporates them into model predictive control.
Results
Stable Deep Koopman predictions maintain a mean error below 0.2% over 128 time steps, and MPC produces an interpretable control law that suppresses vortex shedding.
Takeaways & Limitations
The learned models support sample-efficient flow control from approximately 4000 training examples, with a control law aligned with established flow-control approaches.
Takeaways & Limitations
The controlled model treats B as known when estimating A, making known actuation dynamics an assumption of the modeling procedure.
Abstract
from arXiv · showhide
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.
1 Introduction
Flow control is difficult because fluid dynamics are nonlinear, yet CFD and data-driven modeling create a route toward practical controllers. The cylinder wake provides a canonical test case where model-based learning can connect complex flow dynamics to interpretable control.
- Motivation: Nonlinear Navier–Stokes dynamics make many control techniques developed for linear systems ineffective for fluid flows.Controller effectiveness is also sensitive to flow conditions, measurement configuration, and feedback gains.
- Motivation: CFD can simulate previously intractable complex flows but produces computationally expensive simulations and vast quantities of data.Reduced-order modeling seeks efficient dynamical models learned from those data.
- Learning-based control: Model-free control may require too many environment interactions for computationally expensive flow applications, whereas model-based control depends on accurate dynamical models.This trade-off motivates learning accurate models before controller design.
- Contribution: The paper learns forced and unforced cylinder-flow dynamics with Koopman theory and uses the learned models in model predictive control to suppress vortex shedding.The authors also examine how the resulting actions reveal a simple, interpretable control law.
2 Modeling unforced dynamics
The paper learns nonlinear cylinder-flow dynamics by mapping observations into a Koopman-inspired latent space where linear evolution can be fitted and recursively applied. On the Reynolds-number-50 test case, Deep Koopman produces substantially more stable long-horizon predictions than the compared baselines.
- 2 Modeling unforced dynamics: A = Y X† estimates linear dynamics from paired snapshot matrices, but this approximation is generally accurate only for linear systems.The matrices collect consecutive snapshots, and the estimate approaches the underlying linear dynamics as the sequence length increases.
- 2.1 The Koopman operator: For unforced flow, the method seeks a finite-dimensional mapping whose latent dynamics can be approximated by linear least-squares evolution.Koopman theory represents nonlinear dynamics through linear evolution of observable functions; the learned mapping approximately spans an invariant subspace.
- 2.2 Deep Koopman dynamical model: Deep Koopman encodes snapshot sequences, fits a linear A-matrix in the encoded space, recursively propagates it, and decodes the results to reconstruct and predict flow states.Training minimizes reconstruction and prediction errors, encouraging an invertible mapping and accurate time evolution during deployment-like recursive simulation.
- 2.3.1 Test case: The evaluation uses two-dimensional cylinder flow at Reynolds number 50, where periodic von Kármán vortex shedding produces strong transverse forces and unsteady lift.CFD snapshots are formatted as 128 × 256 inputs with four physical channels: density, x-momentum, y-momentum, and energy.
- 2.3.3 Results: Over 128 time steps, Deep Koopman’s mean prediction error remains below 0.2%, whereas the Variational Bayes Filter rapidly accumulates error beyond its 32-step reconstruction horizon.Over 32 steps, Deep Koopman and VBF perform comparably; the longer-horizon difference is evaluated using relative L1 error across test sequences.
3 Modeling forced dynamics
The paper extends Deep Koopman modeling to learn forced fluid dynamics by incorporating control inputs into a linear latent-space model. Training uses controlled cylinder simulations with varied inputs, while optimizing a global control matrix alongside the neural-network parameters.
- Forced-dynamics model: The extended model represents forced dynamics as g(x_t+1) = Ag(x_t) + Bu_t, with control inputs entering through matrix B.The target is a linear dynamical model in the learned state mapping, with u_t as the control input.
- Parameter estimation: The method defines Γ from the input sequence and estimates A by linear least squares while treating B as known during that fit.The resulting A matrix is generated through the stated least-squares equation.
- Parameter estimation: Although B is fixed when estimating A, a global B-matrix is optimized by gradient descent together with the neural-network parameters during training.This accounts for the fact that B must ultimately be learned rather than truly known.
- Controlled experiment: The controlled cylinder simulation permits scalar actuation by prescribing the cylinder’s angular velocity through a spatially varying wall velocity.The spatially varying velocity keeps the computational grid static.
- Controlled experiment: The training data contains 4238 snapshots arranged into 1600 staggered 32-step sequences, with inputs altered every 1500 solver steps.The snapshots come from simulations with time-varying angular velocity.
- Training inputs: Linearly chirped sinusoidal inputs, interspersed with no-control periods, expose the model to forced dynamics and unforced dynamics from different initial conditions.The input design is analogous to frequency sweeps used in system identification.
4 Model predictive control
The learned Koopman models are embedded in MPC to drive cylinder flow toward a steady laminar target. The resulting controller reduces residuals and yields an interpretable proportional-control law based on a specific wake measurement location.
- MPC formulation: MPC selects finite-horizon input sequences using the learned dynamical models to suppress vortex shedding.At each time step, the controller solves an optimization problem over candidate inputs.
- MPC formulation: The MPC objective penalizes deviations between the observable state and a goal observable, while also penalizing nonzero control inputs.Q weights state deviation and R weights control effort; input and dynamical constraints are imposed in the resulting quadratic program.
- MPC target: The goal state is a steady flow snapshot at Reynolds number 45, where vortex shedding does not occur, despite the controlled system operating at Reynolds number 50.The authors report that this formulation provides a reliable cost estimate even though the two flows are qualitatively different.
- MPC implementation: The controller uses a 16-step MPC horizon, solves for an optimal sequence, and applies only the first input to the CFD solver.The horizon aligns with the model’s prediction procedure, while Q is the identity and R is set near 10^5 to discourage extreme actions.
- Performance measure: Residual norms for x- and y-momentum measure closeness to the desired steady flow because steady solutions have zero time derivatives and residuals.The desired outcome is a steady laminar flow without vortex shedding.
- MPC results: The wake changes from a vortex-shedding profile to a more laminar profile, while residuals decrease monotonically and correspond strongly with the MPC cost.This correspondence supports using the chosen cost measure for this control problem.
- MPC results: After an initial 16-step delay, MPC inputs vary sinusoidally with decreasing amplitude and overlap with y-velocity at wake location d* when scaled by 0.4.The resulting law is functionally similar to proportional feedback from y-velocity with gain 0.4.
- Proportional control: Proportional control using y-velocity at d* suppresses vortex shedding, whereas the same gain at two other locations fails to move the system closer to steady laminar flow.The experiments show that effectiveness depends strongly on measurement location.
5 Related work
Related work connects Koopman-based reduced-order modeling with DMD, nonlinear invariant-subspace approximations, and learned latent state-space models. The paper is positioned alongside approaches seeking useful low-dimensional dynamics for control.
- Koopman and DMD: DMD approximates the Koopman operator under an invariant-subspace assumption that generally fails when the underlying dynamics are nonlinear.Extended DMD approaches address this setting by building on the Koopman perspective.
- Deep dynamical models: Recent machine-learning studies learn deep dynamical models and low-dimensional state representations to model high-dimensional systems and support sample-efficient controller learning.The paper identifies state representation learning as its closest related line of work.
6 Conclusions
The method trains Deep Koopman models that stably simulate cylinder airflow and supports interpretable model-predictive control for vortex-shedding suppression.
- Deep Koopman models stably simulated airflow over a cylinder for significant prediction horizons.
- The models were modified to incorporate control inputs and suppress vortex shedding through flow control.
- Approximately 4000 training examples made the method sample efficient for CFD-based modeling.The conclusion identifies sample efficiency as important because CFD simulations are computationally expensive.
- Embedding the Deep Koopman model in an MPC framework produced an interpretable, sensible control law aligned with established flow-control approaches.
- Future work will test scalability at higher Reynolds numbers and applicability to other flow-control problems.