42 research outputs found
Koopman operator-based model reduction for switched-system control of PDEs
We present a new framework for optimal and feedback control of PDEs using
Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a
linear but infinite-dimensional operator which describes the dynamics of
observables. A numerical approximation of the Koopman operator therefore yields
a linear system for the observation of an autonomous dynamical system. In our
approach, by introducing a finite number of constant controls, the dynamic
control system is transformed into a set of autonomous systems and the
corresponding optimal control problem into a switching time optimization
problem. This allows us to replace each of these systems by a K-ROM which can
be solved orders of magnitude faster. By this approach, a nonlinear
infinite-dimensional control problem is transformed into a low-dimensional
linear problem. In situations where the Koopman operator can be computed
exactly using Extended Dynamic Mode Decomposition (EDMD), the proposed approach
yields optimal control inputs. Furthermore, a recent convergence result for
EDMD suggests that the approach can be applied to more complex dynamics as
well. To illustrate the results, we consider the 1D Burgers equation and the 2D
Navier--Stokes equations. The numerical experiments show remarkable performance
concerning both solution times and accuracy.Comment: arXiv admin note: text overlap with arXiv:1801.0641
Tensor-based dynamic mode decomposition
Dynamic mode decomposition (DMD) is a recently developed tool for the
analysis of the behavior of complex dynamical systems. In this paper, we will
propose an extension of DMD that exploits low-rank tensor decompositions of
potentially high-dimensional data sets to compute the corresponding DMD modes
and eigenvalues. The goal is to reduce the computational complexity and also
the amount of memory required to store the data in order to mitigate the curse
of dimensionality. The efficiency of these tensor-based methods will be
illustrated with the aid of several different fluid dynamics problems such as
the von K\'arm\'an vortex street and the simulation of two merging vortices
Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients
In this article, we build on previous work to present an optimization
algorithm for nonlinearly constrained multi-objective optimization problems.
The algorithm combines a surrogate-assisted derivative-free trust-region
approach with the filter method known from single-objective optimization.
Instead of the true objective and constraint functions, so-called fully linear
models are employed, and we show how to deal with the gradient inexactness in
the composite step setting, adapted from single-objective optimization as well.
Under standard assumptions, we prove convergence of a subset of iterates to a
quasi-stationary point and if constraint qualifications hold, then the limit
point is also a KKT-point of the multi-objective problem
Learning a model is paramount for sample efficiency in reinforcement learning control of PDEs
The goal of this paper is to make a strong point for the usage of dynamical
models when using reinforcement learning (RL) for feedback control of dynamical
systems governed by partial differential equations (PDEs). To breach the gap
between the immense promises we see in RL and the applicability in complex
engineering systems, the main challenges are the massive requirements in terms
of the training data, as well as the lack of performance guarantees. We present
a solution for the first issue using a data-driven surrogate model in the form
of a convolutional LSTM with actuation. We demonstrate that learning an
actuated model in parallel to training the RL agent significantly reduces the
total amount of required data sampled from the real system. Furthermore, we
show that iteratively updating the model is of major importance to avoid biases
in the RL training. Detailed ablation studies reveal the most important
ingredients of the modeling process. We use the chaotic Kuramoto-Sivashinsky
equation do demonstarte our findings