Introduction to Stochastic Dynamics and Transition Operators
- Markov Property and Transition Kernels
- Perron-Frobenius and Koopman Operator
- Examples (ODEs, Brownian Motion)
- Stationarity, Reversibility
Introduction to traditional (intrusive, projection-based) model reduction
- POD/PCA, greedy
- Galerkin ROMs
- DEIM
Introduction to data-driven modeling
- DMD
- Koopman
- Time-delay embeddings (HAVOK)
We propose a supervised learning methodology for use of the random feature model as a data-driven surrogate for operators mapping between spaces of functions. Although our methodology is quite general, we consider operators defined by partial differential equations (PDEs); here, the inputs and outputs are themselves functions, with the input parameters being functions required to specify a...
Partial differential equations (PDEs) are commonly used to model complex systems in applied sciences. Methods for estimating PDE parameters require repeatedly solving PDEs numerically under thousands of candidate parameter values, and thus the computational load is high and expensive. To make these problems tractable we use reduced-order models (ROMs) to reduce the computational cost of PDE...
The formation and oscillation of bubbles is important in cavitation related to turbomachinery, and in biomedical applications, such as contrast-enhanced ultrasound imaging and drug delivery for cancer treatment. There is an extensive literature on the modeling and analysis of bubble oscillations in these settings, allowing for detailed simulations from first principles. However, there are...
Harmful algal blooms (HABs) are a growing public health concern both nation and worldwide. Last year there were 25 major sites of HABs in the state of Utah alone. These blooms are caused in part by excess nutrients (nitrogen and phosphorus) being discharged from wastewater treatment plants (WWTPs). To combat the growing prevalence of HABs the state of Utah is imposing new nitrogen and...
We are developing a pneumatic Hybrid-Fluidic Elastomer Actuator (H-FEA) by combining an additively manufactured internal structure and silicone elastomer. It is evident that in many soft robotic applications, there is a need to be able to sense shape of the robot and collision with the environment. To address these needs, we are developing an analytical model of the nonlinear kinematics of the...
Tendon-Driven Continuum Manipulators (TD-CMs) have gained increasing popularity in various minimally invasive surgical robotic applications. However, the adverse effects of tendon-sheath friction along the transmission path may result in significant non-uniform cable tension and subsequently motion losses, which affects the deformation behavior of a TD-CM. Most of the current approaches for...
The Dynamic Mode Decomposition (DMD) algorithm was first introduced in the fluid mechanics community for analyzing the behavior of nonlinear systems. DMD processes empirical data and produces approximations of eigenvalues and eigenvectors (“DMD modes”) of the linear Koopman operator that represents the nonlinear dynamics. In fluid dynamics, this approach has been used to both analyze...
Within each animal cell is a complex infrastructure of microtubules and motor proteins that translate energy from ATP cycles into a complex fluid flow. Although this process is vital for intracellular transport of nutrients, a quantitative mathematical model for this system remains elusive. Recent experimental work has produced high-resolution video of this system and made possible attempts to...
The mass of a nucleus is its fundamental quantity. It dictates the stability of a particular nucleus, the type of decays and nuclear reactions it can undergo, and much more. Yet after decades of experimental efforts, we are unable to experimentally measure the masses of thousands of exotic isotopes. They cannot be produced in the laboratory so we have to rely on theoretical models. However,...
Cognitive impairment is one of the most prominent symptoms of age-related diseases such as Alzhei-mer’s disease or Lewy body disease. Therefore, it is not surprising that cognitive impairment is one of the variables that is usually measured in longitudinal studies of Alzheimer’s disease. However, if we look naively at the progression of cognitive impairment in a patient, we cannot obtain...
We present a new neural-network architecture, called the Cholesky-factored symmetric positive definite neural network (SPD-NN), for modeling constitutive relations in computational mechanics. Instead of directly predicting the stress of the material, the SPD-NN trains a neural network to predict the Cholesky factor of the tangent stiffness matrix, based on which the stress is calculated in the...
The ability for sparse symbolic machine learning techniques to discover governing equations from data [1], [2] has opened up many opportunities in fluid mechanics. The equations solved in fluid mechanics are conservation of mass, momentum, and energy as well as the closure models. Closure models arise from averaging the conservation equations. Averaging introduces additional terms, which...
In the context of multi-material lightweight assemblies, structural joints such as adhesives and bolts should be taken into account in the FE models for a reliable representation of the reality. The goal of this research work is to identify the parameters of the joints models exploiting the potential of the Virtual Sensing techniques.
Parameters identification can be achieved via the...
The need to solve discrete ill-posed problems arises in many areas of science and engineering. Solutions of these problems, if they exist, are very sensitive to perturbations in available data. Regularization replaces the original problem by a nearby regularized problem, whose solution is less sensitive to the error in the data. The
regularized problem contains a fidelity term and a...
Recent years have seen a massive explosion of datasets across all areas of science, engineering, technology, medicine, and the social sciences. The central questions are: How do we optimally learn from data through the lens of models? And how do we do so taking into account uncertainty in both data and models? These questions can be mathematically framed as Bayesian inverse problems. While...
Due to the notable potentials of additive manufacturing (AM), the interest in AM has risen significantly across several industries during the past decade. One of the key factors governing the mechanical properties of an additively-manufactured part is the solidification microstructure. However, the spatial and temporal resolution required for the simulation of the solidification process is...
Intense lasers have the ability to accelerate ions to high energies over very short distances, but the beam quality generated through these methods is not yet ready for many applications. We developed a framework using evolutionary algorithms to automatically run thousands of one-dimensional (1D) particle-in-cell simulations to optimize the conversion from laser energy to ion energy. The...
Recently, the advent of deep learning has spurred interest in the development of physics-informed neural networks (PINN) for efficiently solving partial differential equations (PDEs), particularly in a parametric setting. Among all different classes of deep neural networks, the convolutional neural network (CNN) has attracted increasing attention in the scientific machine learning community,...
Phase field models, in particular, the Allen-Cahn type and Cahn-Hilliard type equations, have been widely used to investigate interfacial dynamic problems. Designing accurate, efficient, and stable numerical algorithms for solving the phase field models has been an active field for decades. We focus on using the deep neural network to design an automatic numerical solver for the Allen-Cahn and...
System identification from noisy data is challenging in many science and engineering fields. In current work, we present an approach of system identification by sparse Bayesian learning methods. The key idea is to determine the sparse relevant weights from a constructed library by learning from noisy data. The sparse promoting prior is used to regularize the learning process. Furthermore, to...
We present a weak formulation and discretization of the system discovery problem from noisy measurement data. This method of learning differential equations from data replaces point-wise derivative approximations with local integration and improves on the standard SINDy algorithm by orders of magnitude. Linear transformations associated with local integration are used to construct covariance...
Stochastic Calculus
- Stochastic Integral and SDEs
- Ito’s Formula
- The Generator of an SDE
- Examples
- Intrusive ROMs: Error bounds: residual-based; offline-online splitting
- Recovering reduced models from data with operator inference (re-projection) and relation to DMD
Introduction to DMD/Koopman for ROMs data-driven modeling
- DMD integration into Galerkin ROMs
- Koopman/DMD models for ROMs
(E)DMD for Stochastic Dynamics
- Basic EDMD
- Variational Principle for Reversible Dynamics
- Generator EDMD
- Operator inference (non-Markovian terms for partial observations)
- Probabilistic generalization bounds in limited situations for operator inference
Introduction to neural networks for DMD & Koopman approximations
- NN for learning coordinate transformations
- Koopman reductions more linear ROMs
Metabolism plays a key role in a multitude of different biological processes ranging from food production and biofuel production to human health. Predicting the metabolism of a living organism, however, can be a challenging task. Genome-scale models (GEMs) can provide this predictive power by accounting for all metabolic reactions in an organism's genome. So far, GEMs have been used to model...
Interpolatory methods offer a powerful framework for generating reduced‑order models for non‑parametric or parametric systems with time‑varying inputs. Choosing the interpolation points adaptively remains an area of active interest. A greedy framework has been introduced in [1, 2] to choose interpolation points automatically using a posteriori error estimators....
Artificial neural network for bifurcating phenomena modelled by nonlinear parametrized PDEs
The aim of this work is to show the applicability of the Reduced Basis (RB) model reduction and Artificial Neural Network (ANN) dealing with parametrized Partial Differential Equations (PDEs) in nonlinear systems undergoing bifurcations.
Bifurcation analysis, i.e., following the different...
In the context of industrial applications involving machine learning techniques, a challenging problem is represented by object detection, as can be seen in [1]. A particular application of it inside a leading company in the field of professional appliances, such as Electrolux Professional, is represented by the recognition and localization of different types of objects.
A possible approach...
Identifying dynamical systems from measured data is an important step towards accurate modeling and control. Model order reduction (MOR) constitutes a class of methods that can be used to replace large, complex models of dynamical processes with simpler, smaller models. The reduced-order models (ROMs) can be then used for further tasks such as control, design, and simulation. One typical...
Computer simulations of natural and physical systems are subject to various sources of uncertainty necessitating the facilitation of uncertainty quantification and sensitivity analysis methods in the development of mathematical models. As complexity of mathematical models grows, non-intrusive methods draw the attention for identification and characterisation of uncertainties in model outputs....
The scope of this contribution is to present some recent results on how interpolation-based data-driven methods such as
- The Loewner framework [Mayo/Antoulas '07];
- The AAA algorithm [Nakatsukasa/Sete/Trefethen '18];
can handle noisy data sets. More precisely, it will be assumed that the input-output measurements used in these methods, i.e., transfer function evaluations, are...
Multibody systems are the state-of-the-art tool to model complex mechanical mechanisms. However, they typically include redundant coordinates plus constraints, leading to differential algebraic equations for the dynamics which require dedicated integration schemes and control/estimation algorithms.
In my work, autoencoder neural networks are combined with the multibody physics information. In...
In this work, we investigate the capabilities of deep neural networks for solving hyperbolic conservation laws with non-convex flux functions. The behavior of the solution of these problems depends on the underlying small scale regularization. In many applications concerning phase transition phenomena, the regularization terms consist of diffusion and dispersion which are kept in balance in...
Scanning quantum dot microscopy is a technique for imaging electrostatic surface potentials with atomic resolution. To this end it uses a sensor molecule, the so-called quantum dot (QD), which is bonded to the tip of a frequency modulated non-contact atomic force microscope. The QD is moved in the vicinity of the surface atoms so that it experiences the surface potential. By superimposing an...
The need to devise model order reduction methods is strictly related to the finite nature of the available resources, including the computational budget, the amount of memory at disposal and the limited time, which may range from a life-time to real-time queries. Parametric studies, from optimization tasks to the design of response surfaces, suffer particularly from the curse of dimensionality...
Data-driven methods are a promising approach for optimizing traffic control systems. Today’s vehicle technology allows to collect an increasing amount of data to improve the vehicles’ performance, reliability and safety. Concerning mobility infrastructure and communication technology, larger and larger datasets can be transmitted faster every year. Our goal is to use (real-time) data,...
Until now, only classical approaches for the parameter identification of gradient-enhanced damage models combined with e.g. finite plasticity or rate-dependent phenomena are used in order to characterize the damage evolution in metal forming processes. In the future, the models will be extended to simulate hot forming processes. Considering the increasingly complex material models with...
Physical phenomena like chemically reacting flows are computationally expensive to simulate due to the interaction between different physics at a wide range of time and length scales. Chemically reacting flows can be described by systems of hyperbolic partial differential equations with stiff source terms. The governing equations can be simplified by assuming chemical equilibrium and then it...
Human mortality patterns and trajectories in closely related subpopulation are likely linked together and share similarities. It is always desirable to model them simultaneously while taking their heterogeneity into account. This poster introduces two new models for jointly mortality modelling and forecasting of multiple subpopulations in adaptations of the multivariate functional principal...
The poster will give insights into my PhD research. I combine time-series prediction and heuristic optimization algorithms to cope with time-varying optimization problems. A frequent task in dynamic optimization is to track the moving optimum as accurately as possible. Originally designed for static optimization, nature-inspired algorithms on dynamic problems suffer from premature convergence....
‘Virtual Acoustics’ is the field of science that deals with simulating and synthesizing sound in virtual domains. The areas of application are widespread, e.g., building design, virtual entertainment and hearing research. The problem is extremely challenging because it involves simulating time-dependent wave propagation over a broad frequency spectrum in large and complex domains – ideally...
Modelling data assimilation allows to fill the gap between numerical simulations and experimental data. Optimal control problems governed by parametrized partial differential equations is suited for this kind of application, where you want to track problem solutions towards known quantities, given by data collections or previous knowledge. Still, the computational effort increases when one has...
Mathematical models of physical processes often depend on parameters, such as material properties or source terms, that are known only with some uncertainty. Measurement data can help estimate these parameters and thereby improve the meaningfulness of the model. As experiments can be costly, it is important to choose sensor positions carefully to obtain informative data on the unknown...
Deep learning approaches are widely used for many tasks and applications, spanning from object detection, to classification and control. Certifying or enforcing performance and stability guarantees for controllers based on deep learning is, however, challenging. This work considers the use of so called non-autonomous input-output stable deep neural networks for the control of dynamical...
Dynamic Mode Decomposition (DMD) has emerged as a prominent data-driven technique to identify the spatio-temporal coherent structures in dynamical systems, owing to its strong relation with the Koopman operator. For dynamical systems with inputs (external forcing) and outputs (measurement), the input-output DMD (ioDMD) provides a natural extension to DMD so that the learned model approximates...
In my PhD work, I am combining established numerical methods with machine learning techniques to build adaptive and highly accurate numerical schemes for fluid mechanics. Currently, I am interested in how neural networks can enhance the flux reconstruction process in finite-volume schemes. Most recently, I have submitted the journal paper “A data-driven physics-informed finite-volume scheme...
Mathematical models for physical phenomena typically show certain structures if formulated correctly. Hamiltonian systems are an example for such structured systems. They rely on the so-called symplectic structure, which is responsible for the characteristic property to preserve the Hamiltonian function over time. In numerical mathematics, preservation of these structures shows great...
The optimization of vibro-acoustic systems in terms of vibration or sound radiation requires many system evaluations for varying parameters. Often, material or geometric uncertainties have to be considered. Vibro-acoustic systems are typically large and numerically expensive to solve, so it is desirable to use an efficient parametrized surrogate model for optimization tasks. Classic reduced...
We are concerned with optimal control strategies subject to uncertain demands. In many real-world situations, taking uncertainty into account gains in importance. Supply chain management and the energy transition are just two examples where control strategies coping with uncertainties are of high practical importance. A compensation of deviations from the actual demand might be very costly and...
The poster presents a novel approach to diagnose rotordynamic faults like unbalance and coupling misalignment from measured vibration. For that purpose, a large database of virtual hydropower rotors and their vibration has been calculated. The goal is to create a data-driven diagnosis system from this database, that will be applicable to a variety of real hydropower rotors. In a first step, a...
We are interested in real-time capable simulation of soil and soil-tool interaction forces. In previous work, we have successfully implemented a solution of precomputing data using the Discrete Element Method (DEM) and efficiently processing and saving it in a lookup table. Within the respective online phase, the data is accessed in an efficient way [1,2].
We also perform measurements at a...
My research's topic focuses on developing and investigating computational data-driven methods in order to model the material laws from observed data. The methodology is expected to deliver the governing mathematical model of the observed problem in the form of a set of symbolic equations that potentially enable new discoveries in data-rich fields of continuous physical problems. Artificial...
A non-intrusive data-driven model order reduction method is introduced that learns low-dimensional dynamical models for a parametrized non-traditional shallow water equation (NTSWE). The reduced-order model is learnt by setting an appropriate least-squares optimization problem in a low-dimen-sional subspace. Computational challenges that particularly arise from the optimization problem, such...
Piezoelectric energy harvesters (PEHs) are a potential alternative to batteries in large-scale sensor networks and implanted health trackers, but the low output power and the narrow work range has been a bottleneck for its practical application.
To alleviate this problem, the present research will develop a data-driven reduced-order model for flow-induced PEHs based on the dataset obtained...
Kernel methods provide a mathematically rigorous way of learning, however they usually lack efficiency on large amounts of data due to a bad scaling in the number of data points. Furthermore, they are flat models, in the sense that they consist only of one linear combination of non-linear functions. Another drawback is that they do not allow for end-to-end learning, since the model learning is...
In this paper, we intend to use a deep-learning based approach for the construction of locally conservative flux fields with heterogeneous and high-contrast media in the context of flow models. In previous work, the problem is solved through a variation of the Generalized Multiscale Finite Element Method(GMsFEM), which is computationally expensive. The key ingredients of GMsFEM include...
We aim to utilize machine learning methods to learn superstructures in turbulent flow to obtain a data-driven reduced model for turbulent convection. The underlying data will stem from both numerical simulations and experiments and will be used as training data for various machine learning architectures in order to predict the behavior of the underlying system and to extract hidden structures...
My PhD research concerns mathematical modelling, numerical simulations and applications to electrochemical energy storage devices, in particular Zn-air batteries (ZAB).
Zn-air battery (ZAB) concepts exhibit storage potentialities ranging from low-power portable consumer electronics, to automotive and home applications (see [2]). During recharge, the regeneration of Zn is however daunted by...
Friction brakes can exhibit high-intensity vibrations in the frequency range above 1 kHz, which is typically known as squeal. Those vibrations are self-excited due to the friction-interface between brake pads and disk. Decades of research have been spent on modelling this phenomenon, but even today predictive modelling is out of reach. The root causes, amongst others, are considered to be the...
Modeling and simulations are a pillar in the development of complex technical systems. However, for time-critical applications a conduction of high-fidelity simulations is not always feasible. To mitigate this computational bottleneck model order reduction (MOR) can be applied. For nonlinear models, linear MOR approaches are only practicable to a limited extend. Nonlinear approaches, on the...
Model order reduction for advection dominated problems has always been not effective due to the slow decay of the Kolmogorov $N$-width of the problems. Even very simple problems, such as linear transport equations of sharp gradients, show already this behavior. This difficulty can be overcome with different techniques. What we propose is to change the original solution manifold thanks to a...
This study aims to model transonic airfoil-gust interaction and the gust response on transonic aileron-buzz problems using high fidelity computational fluid dynamics (CFD) and the Long Short-Term Memory (LSTM) based deep learning approach. It first explores the rich physics associated with these interactions, which show strong flow field nonlinearities arising from the complex shock-boundary...
Physics-informed neural networks are applied to incompressible two-phase flow problems. We investigate the forward problem, where the governing equations are solved from initial and boundary conditions, as well as the inverse problem, where continuous velocity and pressure fields are inferred from data on the interface position scattered across time. We employ a volume of fluid approach, i.e....
When conducting measurement on existing structures, e.g. collecting the time-response of a building, and trying to compute the same response by a suitable computational method, one often notices discrepancies between the measurement and model data. These discrepancies are due to a wide range of errors, done in both the measurement and the modelling. The model errors can stem from uncertainty...
With the continuously increasing size of the wind turbine blades, the complexity of the blade casting process and the risk of failures has also increased. The vacuum-assisted resin transfer moulding (VATRM) production process at the Siemens Gamesa Renewable Energy facility in Aalborg, Denmark, does not permit the visual inspection of the process. Hence a sensor system (possibly virtual) for...
In the field of environmental modelling, especially modelling problems in the water resources sector, the acquisition of observation data is usually expensive, and/or the underlying model representations are incredibly complex. The spatially distributed models typically used for water quantity and quality prediction yield significant uncertainties even after being carefully calibrated, and...
As in many engineering fields, Computational Fluid Dynamics (CFD) lives upon modelling reality in a feasible way to come to a desired solution. One good example in fluid dynamics is turbulence, which is mathematical modelled in most simulations, but there are many cases where it is necessary to resolve turbulent eddy’s to take crucial effects into consideration. If this is coupled with a flow...
Introduction to Kernel Methods for Data Driven Modeling
- Kernel-based EDMD and Generator EDMD
- Interpolatory model reduction: transfer function, H2 norm, Loewner
- Learning in the frequency domain: Loewner noise, AAA, vector fitting
Introduction to data-driven learning of physics models
- The SINDy method
- PDE-FIND
- ROMs with SINDy
Coarse Graining for SDEs
- Effective Dynamics
- Reduced Generator
- Parameter Estimation
Multifidelity and using data-fit models together with traditional model for, e.g., uncertainty quantification
Learning coordinates and models
- SINDy autoencoders
- Koopman autoencoders
