arXiv subjects

Patrick Buchfink

Publications and source records attributed to Patrick Buchfink.

16 recordsLinked to original sources

arXiv2025

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↗

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↗

arXiv2024

Data-driven identification of latent port-Hamiltonian systems

Conventional physics-based modeling techniques involve high effort, e.g., time and expert knowledge, while data-driven methods often lack interpretability, structure, and sometimes reliability. To mitigate this, we present a data-driven system identification framework that derives models in the port-Hamiltonian (pH) formulation. This formulation is suitable for multi-physical systems while guaranteeing the useful system theoretical properties of passivity and stability. Our framework combines linear and nonlinear reduction with structured, physics-motivated system identification. In this process, high-dimensional state data obtained from possibly nonlinear systems serves as input for an autoencoder, which then performs two tasks: (i) nonlinearly transforming and (ii) reducing this data onto a low-dimensional latent space. In this space, a linear pH system, that satisfies the pH properties per construction, is parameterized by the weights of a neural network. The mathematical requirements are met by defining the pH matrices through Cholesky factorizations. The neural networks that define the coordinate transformation and the pH system are identified in a joint optimization process to match the dynamics observed in the data while defining a linear pH system in the latent space. The learned, low-dimensional pH system can describe even nonlinear systems and is rapidly computable due to its small size. The method is exemplified by a parametric mass-spring-damper and a nonlinear pendulum example, as well as the high-dimensional model of a disc brake with linear thermoelastic behavior.

math.DS↗

arXiv2024

Greedy Kernel Methods for Approximating Breakthrough Curves for Reactive Flow from 3D Porous Geometry Data

We address the challenging application of 3D pore scale reactive flow under varying geometry parameters. The task is to predict time-dependent integral quantities, i.e., breakthrough curves, from the given geometries. As the 3D reactive flow simulation is highly complex and computationally expensive, we are interested in data-based surrogates that can give a rapid prediction of the target quantities of interest. This setting is an example of an application with scarce data, i.e., only having available few data samples, while the input and output dimensions are high. In this scarce data setting, standard machine learning methods are likely to ail. Therefore, we resort to greedy kernel approximation schemes that have shown to be efficient meshless approximation techniques for multivariate functions. We demonstrate that such methods can efficiently be used in the high-dimensional input/output case under scarce data. Especially, we show that the vectorial kernel orthogonal greedy approximation (VKOGA) procedure with a data-adapted two-layer kernel yields excellent predictors for learning from 3D geometry voxel data via both morphological descriptors or principal component analysis.

math.NA↗

arXiv2024

Error Analysis of Randomized Symplectic Model Order Reduction for Hamiltonian systems

Solving high-dimensional dynamical systems in multi-query or real-time applications requires efficient surrogate modelling techniques, as e.g., achieved via model order reduction (MOR). If these systems are Hamiltonian systems their physical structure should be preserved during the reduction, which can be ensured by applying symplectic basis generation techniques such as the complex SVD (cSVD). Recently, randomized symplectic methods such as the randomized complex singular value decomposition (rcSVD) have been developed for a more efficient computation of symplectic bases that preserve the Hamiltonian structure during MOR. In the current paper, we present two error bounds for the rcSVD basis depending on the choice of hyperparameters and show that with a proper choice of hyperparameters, the projection error of rcSVD is at most a constant factor worse than the projection error of cSVD. We provide numerical experiments that demonstrate the efficiency of randomized symplectic basis generation and compare the bounds numerically.

math.NA↗

arXiv2023

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↗

arXiv2023

Dictionary-based Online-adaptive Structure-preserving Model Order Reduction for Parametric Hamiltonian Systems

Classical model order reduction (MOR) for parametric problems may become computationally inefficient due to large sizes of the required projection bases, especially for problems with slowly decaying Kolmogorov n-widths. Additionally, Hamiltonian structure of dynamical systems may be available and should be preserved during the reduction. In the current presentation, we address these two aspects by proposing a corresponding dictionary-based, online-adaptive MOR approach. The method requires dictionaries for the state-variable, non-linearities and discrete empirical interpolation (DEIM) points. During the online simulation, local basis extensions/simplifications are performed in an online-efficient way, i.e. the runtime complexity of basis modifications and online simulation of the reduced models do not depend on the full state dimension. Experiments on a linear wave equation and a non-linear Sine-Gordon example demonstrate the efficiency of the approach.

math.NA↗

arXiv2023

Improved a posteriori Error Bounds for Reduced port-Hamiltonian Systems

Projection-based model order reduction of dynamical systems usually introduces an error between the high-fidelity model and its counterpart of lower dimension. This unknown error can be bounded by residual-based methods, which are typically known to be highly pessimistic in the sense of largely overestimating the true error. This work applies two improved error bounding techniques, namely (a) a hierarchical error bound and (b) an error bound based on an auxiliary linear problem, to the case of port-Hamiltonian systems. The approaches rely on a second approximation of (a) the dynamical system and (b) the error system. In this paper, these methods are for the first time adapted to port-Hamiltonian systems by exploiting their structure. The mathematical relationship between the two methods is discussed both, theoretically and numerically. The effectiveness of the described methods is demonstrated using a challenging three-dimensional port-Hamiltonian model of a classical guitar with fluid-structure interaction.

math.NA↗

arXiv2023

Randomized Symplectic Model Order Reduction for Hamiltonian Systems

Simulations of large scale dynamical systems in multi-query or real-time contexts require efficient surrogate modelling techniques, as e.g. achieved via Model Order Reduction (MOR). Recently, symplectic methods like the complex singular value decomposition (cSVD) or the SVD-like decomposition have been developed for preserving Hamiltonian structure during MOR. In the current contribution, we show how symplectic structure preserving basis generation can be made more efficient with randomized matrix factorizations. We present a randomized complex SVD (rcSVD) algorithm and a randomized SVD-like (rSVD-like) decomposition. We demonstrate the efficiency of the approaches with numerical experiments on high dimensional systems.

math.NA↗

arXiv2022

Port-Hamiltonian Fluid-Structure Interaction Modeling and Structure-Preserving Model Order Reduction of a Classical Guitar

A fluid-structure interaction model in a port-Hamiltonian representation is derived for a classical guitar. We combine the laws of continuum mechanics for solids and fluids within a unified port-Hamiltonian (pH) modeling approach by adapting the discretized equations on second-order level in order to obtain a damped multi-physics model. The high-dimensionality of the resulting system is reduced by model order reduction. The article focuses on pH-systems in different state transformations, a variety of basis generation techniques as well as structure-preserving model order reduction approaches that are independent from the projection basis. As main contribution a thorough comparison of these method combinations is conducted. In contrast to typical frequency-based simulations in acoustics, transient time simulations of the system are presented. The approach is embedded into a straightforward workflow of sophisticated commercial software modeling and flexible in-house software for multi-physics coupling and model order reduction.

math.DS↗

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

Surrogate-data-enriched Physics-Aware Neural Networks

Neural networks can be used as surrogates for PDE models. They can be made physics-aware by penalizing underlying equations or the conservation of physical properties in the loss function during training. Current approaches allow to additionally respect data from numerical simulations or experiments in the training process. However, this data is frequently expensive to obtain and thus only scarcely available for complex models. In this work, we investigate how physics-aware models can be enriched with computationally cheaper, but inexact, data from other surrogate models like Reduced-Order Models (ROMs). In order to avoid trusting too-low-fidelity surrogate solutions, we develop an approach that is sensitive to the error in inexact data. As a proof of concept, we consider the one-dimensional wave equation and show that the training accuracy is increased by two orders of magnitude when inexact data from ROMs is incorporated.

cs.LG↗

arXiv2021

Reduced Basis Methods for Efficient Simulation of a Rigid Robot Hand Interacting with Soft Tissue

We present efficient reduced basis (RB) methods for the simulation of the coupled problem consisting of a rigid robot hand interacting with soft tissue material which is modeled by the linear elasticity equation and discretized with the Finite Element Method. We look at two different scenarios: (i) the forward simulation and (ii) a feedback control formulation of the model. In both cases, large-scale systems of equations appear, which need to be solved in real-time. This is essential in practice for the implementation in a real robot. For the feedback-scenario, in the context of the linear quadratic regulator, we encounter a high-dimensional Algebraic Riccati Equation (ARE). To overcome the real-time constraint by significantly reducing the computational complexity, we use several structure-preserving and non-structure-preserving reduction methods. These include proper orthogonal decomposition-based reduced basis techniques. For the ARE, instead of solving a full dimensional problem we compute a low-rank-factor and hence a low-dimensional ARE is solved. Numerical examples for both cases are provided. These illustrate the approximation quality of the reduced solution and speedup factors of the different reduction approaches.

math.NA↗

arXiv2019

Symplectic Model Order Reduction with Non-Orthonormal Bases

Parametric high-fidelity simulations are of interest for a wide range of applications. But the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g. structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such a ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction.

math.NA↗