Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories

Data-driven models for nonlinear dynamical systems based on approximating the underlying Koopman operator or generator have proven to be successful tools for forecasting, feature learning, state estimation, and control. It has become well known that the Koopman generators for control-affine systems also have affine dependence on the input, leading to convenient finite-dimensional bilinear approximations of the dynamics. Yet there are still two main obstacles that limit the scope of current approaches for approximating the Koopman generators of systems with actuation. First, the performance of existing methods depends heavily on the choice of basis functions over which the Koopman generator is to be approximated; and there is currently no universal way to choose them for systems that are not measure preserving. Secondly, if we do not observe the full state, then it becomes necessary to account for the dependence of the output time series on the sequence of supplied inputs when constructing observables to approximate Koopman operators. To address these issues, we write the dynamics of observables governed by the Koopman generator as a bilinear hidden Markov model, and determine the model parameters using the expectation-maximization (EM) algorithm. The E-step involves a standard Kalman filter and smoother, while the M-step resembles control-affine dynamic mode decomposition for the generator. We demonstrate the performance of this method on three examples, including recovery of a finite-dimensional Koopman-invariant subspace for an actuated system with a slow manifold; estimation of Koopman eigenfunctions for the unforced Duffing equation; and model-predictive control of a fluidic pinball system based only on noisy observations of lift and drag

Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariance

Data-driven reduced-order models often fail to make accurate forecasts of high-dimensional nonlinear dynamical systems that are sensitive along coordinates with low-variance because such coordinates are often truncated, e.g., by proper orthogonal decomposition, kernel principal component analysis, and autoencoders. Such systems are encountered frequently in shear-dominated fluid flows where non-normality plays a significant role in the growth of disturbances. In order to address these issues, we employ ideas from active subspaces to find low-dimensional systems of coordinates for model reduction that balance adjoint-based information about the system's sensitivity with the variance of states along trajectories. The resulting method, which we refer to as covariance balancing reduction using adjoint snapshots (CoBRAS), is analogous to balanced truncation with state and adjoint-based gradient covariance matrices replacing the system Gramians and obeying the same key transformation laws. Here, the extracted coordinates are associated with an oblique projection that can be used to construct Petrov-Galerkin reduced-order models. We provide an efficient snapshot-based computational method analogous to balanced proper orthogonal decomposition. This also leads to the observation that the reduced coordinates can be computed relying on inner products of state and gradient samples alone, allowing us to find rich nonlinear coordinates by replacing the inner product with a kernel function. In these coordinates, reduced-order models can be learned using regression. We demonstrate these techniques and compare to a variety of other methods on a simple, yet challenging three-dimensional system and a nonlinear axisymmetric jet flow simulation with $10^5$ state variables

Analysis of amplification mechanisms and cross-frequency interactions in nonlinear flows via the harmonic resolvent

We propose a framework that elucidates the input-output characteristics of flows with complex dynamics arising from nonlinear interactions between different time scales. More specifically, we consider a periodically time-varying base flow, and perform a frequency-domain analysis of periodic perturbations about this base flow; the response of these perturbations is governed by the harmonic resolvent, which is a linear operator similar to the harmonic transfer function introduced by Wereley (1991). This approach makes it possible to explicitly capture the triadic interactions that are responsible for the energy transfer between different time scales in the flow. For instance, perturbations at frequency $\alpha$ are coupled with perturbations at frequency $\omega$ through the base flow at frequency $\omega-\alpha$. We draw a connection with resolvent analsyis, which is a special case of the harmonic resolvent when evaluated about a steady base flow. We show that the left and right singular vectors of the harmonic resolvent are the optimal response and forcing modes, which can be understood as full spatio-temporal signals that reveal space-time amplification characteristics of the flow. We illustrate the method on examples, including a three-dimensional system of ordinary differential equations and the flow over an airfoil at near-stall angle of attack

Fitting high-energy Littlest Seesaw parameters using low-energy neutrino data and leptogenesis

We show that the four high-energy Littlest Seesaw parameters in the flavour basis,namely two real Yukawa couplings plus the two right-handed neutrino masses, can be determined by an excellent fit to the seven currently constrained observables of low-energy neutrino data and leptogenesis. Taking into account renormalisation group corrections, we estimate $\chi^2 \simeq 1.5-2.6$ for the three d.o.f., depending on the high-energy scale and the type of non supersymmetric Littlest Seesaw model. We extract allowed ranges of neutrino parameters from our fit data, including the approximate mu-tau symmetric predictions $\theta_{23}=45^o\pm 1^o$ and $\delta = -90^o \pm 5^o$, which, together with a normal mass ordering with $m_1=0$, will enable Littlest Seesaw models to be tested in future neutrino experiments.Comment: Typos corrected, references added. 25 pages, 20 figure

Learning Nonlinear Projections for Reduced-Order Modeling of Dynamical Systems using Constrained Autoencoders

Recently developed reduced-order modeling techniques aim to approximate nonlinear dynamical systems on low-dimensional manifolds learned from data. This is an effective approach for modeling dynamics in a post-transient regime where the effects of initial conditions and other disturbances have decayed. However, modeling transient dynamics near an underlying manifold, as needed for real-time control and forecasting applications, is complicated by the effects of fast dynamics and nonnormal sensitivity mechanisms. To begin to address these issues, we introduce a parametric class of nonlinear projections described by constrained autoencoder neural networks in which both the manifold and the projection fibers are learned from data. Our architecture uses invertible activation functions and biorthogonal weight matrices to ensure that the encoder is a left inverse of the decoder. We also introduce new dynamics-aware cost functions that promote learning of oblique projection fibers that account for fast dynamics and nonnormality. To demonstrate these methods and the specific challenges they address, we provide a detailed case study of a three-state model of vortex shedding in the wake of a bluff body immersed in a fluid, which has a two-dimensional slow manifold that can be computed analytically. In anticipation of future applications to high-dimensional systems, we also propose several techniques for constructing computationally efficient reduced-order models using our proposed nonlinear projection framework. This includes a novel sparsity-promoting penalty for the encoder that avoids detrimental weight matrix shrinkage via computation on the Grassmann manifold

Modelling buildings during flood inundation using TELEMAC-2D

Global climate change significantly increases flood risk. Inundation around buildings in both urban and rural areas can pose significant risk to life and property. Accurate and precise modelling is the key to a better understanding and quantification of flood risk. This study applies three different methods for modelling buildings within TELEMAC-2D from a dyke breach scenario: a) buildings excluded from the mesh; b) buildings modelled as elevated bathymetry; and c) buildings modelled as vegetation friction. The maximum flood hazard rating for each method is then calculated from the hydrodynamics generated by the model and compared. The results show that using vegetation friction to represent the buildings in the model is the most effective and accurate approach in evaluating the flood risk

Development of novel catalytic solutions applied for the hydrogen evolution and oxygen reduction reactions

This thesis reports the utilisation of 2D nanomaterials, namely molybdenum disulphide (2D-MoS2) and molybdenum diselenide (2D-MoSe2), as cheap, earth abundant and effective catalytic alternatives to platinum (Pt) for hydrogen production (via the hydrogen evolution reaction (HER)) within electrolysers and energy generation (via the oxygen reduction reaction (ORR)) within proton exchange membrane fuel cells (PEMFC). Chapter 1 introduces the chemical reactions associated with electrolysers and PEMFCs, then gives an overview of the relevant fundamental electrochemical concepts utilised throughout this thesis. Subsequent to this, Chapter 2 specifically describes the equipment and fabrication techniques implemented herein, in addition to providing the full physicochemical characterisation of the 2D-MoS2 and 2D-MoSe2 utilised in later chapters. Chapter 3 demonstrates that a commonly employed surfactant (sodium cholate) used in the liquid exfoliation of 2D-MoS2 has a profound effect upon its electrocatalytic activity. It is shown that the surfactant has a negative effect upon the observed HER signal output (decreasing the current density and increasing the electronegativity of the HER onset potential) of the 2D-MoS2 compared to “pristine” 2D-MoS2 (produced without a surfactant present). This suggests that future studies utilising 2D nanomaterials should carefully consider their use of a surfactant as well as perform the necessary control experiments. Chapters 4 and 5 reveal that, in specific conditions, 2D-MoS2 nanosheets are effective at reducing the electronegativity of the HER and ORR onset potentials, increasing their achievable current density and allowing the ORR reaction mechanism to occur via the desirable 4 electron process (product: H2O). This electrocatalytic effect is reported herein for the first time. Research was undertaken by electrically wiring the 2D-MoS2 to four commonly employed commercially available carbon based electrode support materials, namely edge plane pyrolytic graphite (EPPG), glassy carbon (GC), boron-doped diamond (BDD) and screen-printed graphite electrodes (SPE). The reduction in the electronegativity of the HER and ORR onset potential is shown to be associated with each supporting electrode's individual electron transfer kinetics/properties and is thus distinct from the literature, which predominately uses just GC as a supporting electrode material. It is revealed that the ability to catalyse the HER and ORR is dependent on the mass deposited until a critical coverage of 2D-MoS2 nanosheets is achieved, after which its electrocatalytic benefits and/or surface stability curtail. In Chapter 6, 2D-MoS2 screen-printed electrodes (2D-MoS2-SPEs) are designed, fabricated and their performance is evaluated towards the electrochemical HER and ORR within acidic aqueous media. A screen-printable ink is developed, which allows for the tailoring of the 2D-MoS2 content/mass used in the fabrication of the 2D-MoS2-SPEs. The 2D-MoS2-SPEs are shown to exhibit an electrocatalytic behaviour towards the ORR, which is found, critically, to be reliant upon the percentage mass incorporation of 2D-MoS2 in the 2D-MoS2-SPEs. Chapter 7 utilises the exact methodology for electrocatalytic ink production as Chapter 6, however it incorporates 2D-MoSe2 and explores the fabricated 2D-MoSe2-SPEs towards the HER where beneficial electrochemistry is observed. Both the 2D-MoS2-SPEs and 2D-MoSe2-SPEs display remarkable stability with no degradation in their respective performances over the course of 1000 repeat scans. The electrocatalytic inks produced in these chapters and the resultant mass producible electrodes mitigate the need to post hoc modify an electrode via the drop-casting technique that has been shown to result in poor stability. This thesis reports that novel 2D nanomaterials can be implemented as beneficial electrode materials towards enhancing “green” energy generation technologies. Specifically, 2D-MoS2 is shown to be effective at lowering the onset potential and increasing the achievable current density for the HER and ORR, giving rise to further benefits when 2D-MoS2 (and 2D-MoSe2 towards the HER) are incorporated into SPEs. These novel electrodes exhibit the inherent unique electrochemical behaviour of the 2D nanomaterials incorporated and benefit from the remarkable stability attributed to the intrinsic properties of a SPE. Consequently, the findings of this thesis are highly applicable to industrial electrolyser/fuel cell applications