arXiv subjects

Silke Glas

Publications and source records attributed to Silke Glas.

16 recordsLinked to original sources

arXiv2026

Spatio-temporal Latent Denoising Diffusion Probabilistic Models for Reduced-order Modeling of Parametrized Dynamical Systems

Many scientific problems require accurate modeling of complex physical phenomena, such as fluid dynamics or climate modeling. These phenomena often result in high-dimensional and thus computationally expensive computational models that can limit their application to real-time and multi-query problems. Model-order reduction (MOR) is an approach that seeks to approximate full-order models (FOMs) using reduced-order models (ROMs), trading a minor reduction in accuracy for a major reduction in computational cost. In this work, we propose a non-intrusive MOR method using generative machine learning by means of denoising diffusion probabilistic models (DDPMs) for generating solutions of the dynamical systems under different instances of their parameters. Unlike conventional DDPMs, which often operate purely in the spatial domain, we aim to generate spatio-temporal solutions to improve quality and temporal coherence. In addition, we embed the DDPM in a latent space obtained by sequentially applying proper orthogonal decomposition and an autoencoder to reduce the data dimensionality and the computational cost of the DDPM. We test our approach on a parametrized 2D fluid flow around an obstacle. The numerical experiments demonstrate that our latent DDPM can (i) produce accurate and temporally coherent solutions, (ii) achieve strong generalization capabilities to scenarios involving unseen parameter values, and (iii) extrapolate in time beyond the training horizon.

math.NA↗

arXiv2026

Deep Invertible Autoencoders for Dimensionality Reduction of Dynamical Systems

Constructing reduced-order models (ROMs) capable of efficiently predicting the evolution of parameter-dependent high-dimensional dynamical systems is crucial in many applications in engineering and applied sciences. A popular class of projection-based ROMs projects the high-dimensional full-order model (FOM) dynamics onto a low-dimensional manifold. These projection-based ROMs approaches often rely on classical model reduction techniques such as proper orthogonal decomposition (POD) or, more recently, on neural network architectures such as autoencoders (AEs). In the case that the ROM is constructed by the POD, one has approximation guaranteed based based on the singular values of the problem at hand. However, POD-based techniques can suffer from slow decay of the singular values in transport- and advection-dominated problems. In contrast to that, AEs allow for better reduction capabilities than the POD, often with the first few modes, but at the price of theoretical considerations. In addition, it is often observed, that AEs exhibits a plateau of the projection error with the increment of the dimension of the trial manifold. In this work, we propose an invertible AE architecture, named inv-AE, that computationally improves upon the stagnation of the reconstruction error typical of traditional AE architectures. Inv-AE is composed of several invertible neural network layers that allows for gradually recovering more information about the FOM solutions the more we increase the dimension of the reduced manifold. Through the application of inv-AE to a 1-dimensional Burgers' equation, a 2-dimensional fluid flow around an obstacle with variable geometry, and a 3-dimensional Korteweg-de Vries, we show that (i) inv-AE mitigates the issue of the characteristic plateau of AEs and (ii) inv-AE can be combined with popular autoencoder-based ROM approaches, e.g., DL-ROM, to improve their accuracy.

cs.LG↗

arXiv2026

Model reduction of port-Hamiltonian systems via neural networks

In this paper, we consider structure-preserving model reduction of port-Hamiltonian (pH) systems which extend classical Hamiltonian systems with dissipation and an input-output port. These pH systems are often used in multi-physics systems, as the interconnection of one or more \pH systems results again in a pH system. If particularly the system matrices associated with the interconnection and/or dissipation of a pH system are state-dependent, then the evaluation of standard reduced-order models (ROMs) may depend on the dimension of the original full-order model, resulting in high computational costs. To circumvent these high costs, we propose to use structure-preserving neural networks. In particular, we perform two steps: (1) we use the generalized manifold Galerkin projection to project the pH system onto the reduced space; then (2) we train a neural network to learn the map from the reduced-order state to the reduced-order interconnection and dissipation system matrices. To ensure that the resulting ROM is again a pH system, the architecture of the neural network is chosen such that the skew-symmetry and positive semi-definiteness of the reduced-order systems matrices are maintained. In a numerical example, we consider a nonlinear mass-spring-damper system with state-dependent system matrices. The numerical results show that the proposed method achieves a significant computational speed-up compared to the original \ROM with comparable accuracy.

math.NA↗

arXiv2026

Structure-Preserving Generalized Manifold Galerkin Reduction for Port-Hamiltonian Systems

This paper considers structure-preserving model order reduction (MOR) techniques for port-Hamiltonian (pH) systems, which are typically derived from energy-based modeling. To keep favorable properties of \pH systems such as passivity in a reduced order model (ROM), we use structure-preserving methods in the reduction process. Although projection-based structure-preserving MOR methods for nonlinear pH systems based on nonlinear approximation ansatzes have recently been proposed, existing approaches typically rely on specific structures of the approximation map and the underlying pH system. To address this limitation, we propose a \MOR framework based on generalized manifold Galerkin (GMG) reduction. The resulting framework can employ general nonlinear approximation maps while preserving the pH structure. We establish sufficient conditions for structure preservation, show that the associated non-degeneracy conditions are generically satisfied. We further present linear and quadratic approximation maps within the proposed framework. Numerical examples for a linear and a nonlinear mass-spring-damper system show that the proposed \MOR methods have lower relative reduction error compared to existing methods.

math.NA↗

arXiv2026

Model order reduction via Lie groups

Lie groups and their actions are ubiquitous in the description of physical systems, and we explore implications in the setting of model order reduction (MOR). We present a novel framework of MOR via Lie groups, called MORLie, in which high-dimensional dynamical systems on manifolds are approximated by low-dimensional dynamical systems on Lie groups. In comparison to other Lie group methods we are able to attack non-equivariant dynamics, which are frequent in practical applications, and we provide new non-intrusive MOR methods based on the presented geometric formulation. We also highlight numerically that MORLie has a lower error bound than the Kolmogorov $N$-width, which limits linear-subspace methods. The method is applied to various examples: 1. MOR of a simplified deforming body modeled by noisy point cloud data following a sheering motion, where MORLie outperforms a naive POD approach in terms of accuracy and dimensionality reduction. 2. Reconstructing liver motion during respiration with data from edge detection in MRI scans, where MORLie reaches performance approaching the state of the art, while reducing the training time from hours on a computing cluster to minutes on a mobile workstation. 3. An analytic example showing that the method of freezing is analytically recovered as a special case, showing the generality of the geometric framework.

math.NA↗

arXiv2026

Fast prediction of plasma instabilities with sparse-grid-accelerated optimized dynamic mode decomposition

Parametric data-driven reduced-order models (ROMs) that embed dependencies in a large number of input parameters are crucial for enabling many-query tasks in large-scale problems. These tasks, including design optimization, control, and uncertainty quantification, are essential for developing digital twins in real-world applications. However, standard grid-based data generation methods are computationally prohibitive due to the curse of dimensionality. This paper investigates efficient training of parametric data-driven ROMs using sparse grid interpolation with (L)-Leja points, specifically targeting scenarios with higher-dimensional input parameter spaces. (L)-Leja points are nested and exhibit slow growth, resulting in sparse grids with low cardinality in low-to-medium dimensional settings, making them ideal for large-scale, computationally expensive problems. Focusing on gyrokinetic simulations of plasma micro-instabilities in fusion experiments as a representative real-world application, we construct parametric ROMs for the full 5D gyrokinetic distribution function via optimized dynamic mode decomposition (optDMD) and sparse grids based on (L)-Leja points. We perform detailed experiments in two scenarios: First, the Cyclone Base Case benchmark assesses optDMD ROM prediction capabilities beyond training time horizons and across variations in the binormal wave number. Second, for a real-world electron-temperature-gradient-driven micro-instability simulation with six input parameters, we demonstrate that a predictive parametric optDMD ROM that is up to three orders of magnitude cheaper to evaluate can be constructed using only 28 high-fidelity gyrokinetic simulations, enabled by the use of sparse grids. In the broader context of fusion research, these results demonstrate the potential of sparse grid-based parametric ROMs to enable otherwise intractable many-query tasks.

physics.comp-ph↗

arXiv2025

Symplectic model order reduction of port-Hamiltonian systems

This work proposes a novel structure-preserving model order reduction (MOR) method for linear, time-invariant port-Hamiltonian (pH) systems. Our goal is to construct a reduced order pH system, which can still be interpreted in the physical domain of the full order model. By this we mean, that if an electrical circuit is the initial high-dimensional pH system, we want the reduced order model to be still interpretable as an electronic circuit. In the case of the well-known mass spring damper (MSD) system, there are MOR methods available, which already guarantee the preservation of this particular structure. Moreover, we show that our new structure-preserving MOR method, which is based on symplectic MOR methods, will recover the known second-order Arnoldi method in the case of MSD systems. However, for the example of an electrical circuit pH model (and more models of similar block structure), our method yields a novel model reduction method. We present numerical results on the aforementioned electronic circuit model, highlighting the advantages of the proposed method.

math.OC↗

arXiv2025

Energy-stable Port-Hamiltonian Systems

We combine energy-stable and port-Hamiltonian (pH) systems to obtain energy-stable port-Hamiltonian (espH) systems. The idea is to extend the known energy-stable systems with an input-output port, which results in a pH formulation. One advantage of the new espH formulation is that it naturally preserves its espH structure throughout discretization (in space and time) and model reduction.

math.NA↗

arXiv2025

Leveraging time and parameters for nonlinear model reduction methods

In this paper, we consider model order reduction (MOR) methods for problems with slowly decaying Kolmogorov $n$-widths as, e.g., certain wave-like or transport-dominated problems. To overcome this Kolmogorov barrier within MOR, nonlinear projections are used, which are often realized numerically using autoencoders. These autoencoders generally consist of a nonlinear encoder and a nonlinear decoder and involve costly training of the hyperparameters to obtain a good approximation quality of the reduced system. To facilitate the training process, we show that extending the to-be-reduced system and its corresponding training data makes it possible to replace the nonlinear encoder with a linear encoder without sacrificing accuracy, thus roughly halving the number of hyperparameters to be trained.

math.NA↗

arXiv2024

How to reveal the rank of a matrix?

We study algorithms called rank-revealers that reveal a matrix's rank structure. Such algorithms form a fundamental component in matrix compression, singular value estimation, and column subset selection problems. While column-pivoted QR has been widely adopted due to its practicality, it is not always a rank-revealer. Conversely, Gaussian elimination (GE) with a pivoting strategy known as global maximum volume pivoting is guaranteed to estimate a matrix's singular values but its exponential complexity limits its interest to theory. We show that the concept of local maximum volume pivoting is a crucial and practical pivoting strategy for rank-revealers based on GE and QR. In particular, we prove that it is both necessary and sufficient; highlighting that all local solutions are nearly as good as the global one. This insight elevates Gu and Eisenstat's rank-revealing QR as an archetypal rank-revealer, and we implement a version that is observed to be at most $2\times$ more computationally expensive than CPQR. We unify the landscape of rank-revealers by considering GE and QR together and prove that the success of any pivoting strategy can be assessed by benchmarking it against a local maximum volume pivot.

math.NA↗

arXiv2024

Model reduction on manifolds: A differential geometric framework

Using nonlinear projections and preserving structure in model order reduction (MOR) are currently active research fields. In this paper, we provide a novel differential geometric framework for model reduction on smooth manifolds, which emphasizes the geometric nature of the objects involved. The crucial ingredient is the construction of an embedding for the low-dimensional submanifold and a compatible reduction map, for which we discuss several options. Our general framework allows capturing and generalizing several existing MOR techniques, such as structure preservation for Lagrangian- or Hamiltonian dynamics, and using nonlinear projections that are, for instance, relevant in transport-dominated problems. The joint abstraction can be used to derive shared theoretical properties for different methods, such as an exact reproduction result. To connect our framework to existing work in the field, we demonstrate that various techniques for data-driven construction of nonlinear projections can be included in our framework.

math.NA↗

arXiv2023

Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds

For projection-based linear-subspace model order reduction (MOR), it is well known that the Kolmogorov n-width describes the best-possible error for a reduced order model (ROM) of size n. In this paper, we provide approximation bounds for ROMs on polynomially mapped manifolds. In particular, we show that the approximation bounds depend on the polynomial degree p of the mapping function as well as on the linear Kolmogorov n-width for the underlying problem. This results in a Kolmogorov (n, p)-width, which describes a lower bound for the best-possible error for a ROM on polynomially mapped manifolds of polynomial degree p and reduced size n.

math.NA↗

arXiv2023

Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds

This work presents two novel approaches for the symplectic model reduction of high-dimensional Hamiltonian systems using data-driven quadratic manifolds. Classical symplectic model reduction approaches employ linear symplectic subspaces for representing the high-dimensional system states in a reduced-dimensional coordinate system. While these approximations respect the symplectic nature of Hamiltonian systems, linear basis approximations can suffer from slowly decaying Kolmogorov $N$-width, especially in wave-type problems, which then requires a large basis size. We propose two different model reduction methods based on recently developed quadratic manifolds, each presenting its own advantages and limitations. The addition of quadratic terms to the state approximation, which sits at the heart of the proposed methodologies, enables us to better represent intrinsic low-dimensionality in the problem at hand. Both approaches are effective for issuing predictions in settings well outside the range of their training data while providing more accurate solutions than the linear symplectic reduced-order models.

math.NA↗

arXiv2021

Symplectic Model Reduction of Hamiltonian Systems on Nonlinear Manifolds

Classical model reduction techniques project the governing equations onto linear subspaces of the high-dimensional state-space. For problems with slowly decaying Kolmogorov-n-widths such as certain transport-dominated problems, however, classical linear-subspace reduced-order models (ROMs) of low dimension might yield inaccurate results. Thus, the concept of classical linear-subspace ROMs has to be extended to more general concepts, like Model Order Reduction (MOR) on manifolds. Moreover, as we are dealing with Hamiltonian systems, it is crucial that the underlying symplectic structure is preserved in the reduced model, as otherwise it could become unphysical in the sense that the energy is not conserved or stability properties are lost. To the best of our knowledge, existing literature addresses either MOR on manifolds or symplectic model reduction for Hamiltonian systems, but not their combination. In this work, we bridge the two aforementioned approaches by providing a novel projection technique called symplectic manifold Galerkin (SMG), which projects the Hamiltonian system onto a nonlinear symplectic trial manifold such that the reduced model is again a Hamiltonian system. We derive analytical results such as stability, energy-preservation and a rigorous a-posteriori error bound. Moreover, we construct a weakly symplectic deep convolutional autoencoder as a computationally practical approach to approximate a nonlinear symplectic trial manifold. Finally, we numerically demonstrate the ability of the method to outperform (non-)structure-preserving linear-subspace ROMs and non-structure-preserving MOR on manifold techniques.

math.NA↗

arXiv2021

Global Stochastic Optimization of Stellarator Coil Configurations

In the construction of a stellarator, the manufacturing and assembling of the coil system is a dominant cost. These coils need to satisfy strict engineering tolerances, and if those are not met the project could be canceled as in the case of the National Compact Stellarator Experiment (NCSX) project [25]. Therefore, our goal is to find coil configurations that increase construction tolerances without compromising the performance of the magnetic field. In this paper, we develop a gradient-based stochastic optimization model which seeks robust stellarator coil configurations in high dimensions. In particular, we design a two-step method: first, we perform an approximate global search by a sample efficient trust-region Bayesian optimization; second, we refine the minima found in step one with a stochastic local optimizer. To this end, we introduce two stochastic local optimizers: BFGS applied to the Sample Average Approximation and Adam, equipped with a control variate for variance reduction. Numerical experiments performed on a W7-X-like coil configuration demonstrate that our global optimization approach finds a variety of promising local solutions at less than 0.1% of the cost of previous work, which considered solely local stochastic optimization.

physics.plasm-ph↗

arXiv2019

The Oracle of DLphi

We present a novel technique based on deep learning and set theory which yields exceptional classification and prediction results. Having access to a sufficiently large amount of labelled training data, our methodology is capable of predicting the labels of the test data almost always even if the training data is entirely unrelated to the test data. In other words, we prove in a specific setting that as long as one has access to enough data points, the quality of the data is irrelevant.

cs.LG↗