Deep Dynamical Modeling and Control of Unsteady Fluid Flows
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.
Update: after author response. I think that the authors have done good job in addressing my concerns in their response. Prior to receiving the feedback, my main concern could be summarized as: the paper proposes a modification to an existing method (optimize multi-step prediction error instead of one-step prediction error) in an effort to improve accuracy and stability of the model during open-loop simulations. However, there was no direct comparison to the existing method that was modified, and so in the absence of any theoretical or empirical evidence, it is difficult to say whether the proposed approach is effective. In their response, the authors provided a comparison with Takeishi, which suggests that the modification does indeed (at least in this particular experiment) have the desired effect of improving multi-step prediction accuracy. In light of these results (and with the cost function specified more clearly) I'm happy to revise my 'score' to a 6.   Summary ------- This paper concerns data-driven modeling and control of fluid flows. The dynamics describing such flows are nonlinear, making modeling and control challenging. The approach adopted in this paper is based on spectral analysis of the Koopman operator (a popular approach that has received considerable attention in the literature), which (roughly speaking) models the finite-dimensional nonlinear dynamics with infinite-dimensional linear dynamics. For tractable analysis it is desirable to find a finite-dimensional approximation of these infinite-dimensional dynamics (i.e., a Koopman invariant subspace). The strategy presented in this paper is dynamic mode decomposition (DMD), another popular approach that has received much attention in the literature. The key challenge in DMD is coming up with a basis for the Koopman invariant subspace (i.e., a collection of nonlinear functions referred to as ‘observables’). Here lies the main contribution of the paper: a data-driven way to learn appropriate observables, where each observable function is modeled by a deep neural network (encoder). The resulting finite-dimensional linear model is then used for control design (standard MPC for linear systems). It is observed that the MPC strategy coincides with proportional feedback (i.e. control action proportional to the velocity), *as long as* the velocity used for feedback is measured at a specific point (spatially). This proportional control then constitutes a ‘simple, interpretable’ policy.   Originality ----------- The idea of learning a basis for the Koopman invariant subspace (using deep neural networks) seems very promising. Such an approach was proposed in Takeishi (NIPS ’17, ref [19]). As the authors themselves state, the approach in this paper is a ‘modified form of the training algorithm proposed by Takeishi’, c.f., pg. 2 - this raises the question: how significant are the modifications? In my view, the modifications are rather minor. in the authors’ own words:  ‘One main difference between our algorithm and that proposed by Takeishi et al. is that we force the model to simulate the time evolution of the system during training in a manner that mirrors how the model will be deployed at test time.’ Specifically, the error term for the decoder includes mismatch for both the measured states and the predicted states (where the predicted states are obtained by simulating the linear model forward in time, recursively). While this does not directly effect the estimation of A (which is estimated by least squares) the argument is (presumably) that this indirectly (via the nonlinear function g) leads to better prediction performance (specifically, longer-term stability of predictions, c.f. e.g., Figure 3).  A few comments: i) while it is plausible that this could improve predictive performance, it is not obvious; e.g., this does not directly influence the estimation of A (which is still fit by least squares, i.e., minimization of one-step prediction error), and it is A that affects stability of the long-term predictions. ii) in my view, this is a rather minor modification of the algorithm proposed in Takeishi. iii) further, this appears to be the only difference? (i.e., ‘one main difference’ could more accurately be described as ‘the only difference’?). The other difference is that Takeishi has an additional step to estimate a linear delay embedding (because Takeishi do not assume that the states are observable, unlike the present paper, which makes the stronger assumption that the states are observable).  The material on MPC is entirely standard, given the finite-dimensional linear model.  Quality ------- There are no theoretical claims made in the paper, so quality can only be assessed via the numerical illustrations. The method is evaluated against a 'deep variational Bayes filter'. A natural question is: why not compare performance to the method proposed by Takeishi? After all, the proposed method is a modification of Takeishi, would it not make sense to see if the modification had the intended effect, namely, better long-term predictive performance and stability?   Clarity ------- The paper is fairly well written, however, the main contribution (namely, a method for training deep Koopman dynamical models) is only really presented via Figure 1. It might be desirable to describe the method with more precision, e.g., specifically state the cost functions that are minimized during training (particularly for the encoder/decoder; in fairness, the least squares cost is stated), along with any regularization penalties.   Significance  ------------ I think the paper tackles a highly relevant/significant problem (i.e., learning basis functions for DMD in spectral analysis of the Koopman operator), however, the differences compared to previous work are unclear at best. The modifications to existing algorithms (that of Takeishi) appear minor, and the numerical studies do not evaluate against the method of Takeishi, making it very difficult to say whether the modification is effective or not.  Further comments ---------------- I think numerical comparison with Takeishi is quite important/necessary.