arXiv subjects

Peter Benner

Publications and source records attributed to Peter Benner.

At least 55 records · Page 3Linked to original sources

arXiv2024

Interpolatory Necessary Optimality Conditions for Reduced-order Modeling of Parametric Linear Time-invariant Systems

Interpolatory necessary optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of non-parametric linear time-invariant (LTI) systems are known and well-investigated. In this work, using the general framework of $\mathcal{L}_2$-optimal reduced-order modeling of parametric stationary problems, we derive interpolatory $\mathcal{H}_2 \otimes \mathcal{L}_2$-optimality conditions for parametric LTI systems with a general pole-residue form. We then specialize this result to recover known conditions for systems with parameter-independent poles and develop new conditions for a certain class of systems with parameter-dependent poles.

math.OC↗

arXiv2024

Discrete empirical interpolation in the tensor t-product framework

The discrete empirical interpolation method (DEIM) is a well-established approach, widely used for state reconstruction using sparse sensor/measurement data, nonlinear model reduction, and interpretable feature selection. We introduce the tensor t-product Q-DEIM (t-Q-DEIM), an extension of the DEIM framework for dealing with tensor-valued data. The proposed approach seeks to overcome one of the key drawbacks of DEIM, viz., the need for matricizing the data, which can distort any structural and/or geometric information. Our method leverages the recently developed tensor t-product algebra to avoid reshaping the data. In analogy with the standard DEIM, we formulate and solve a tensor-valued least-squares problem, whose solution is achieved through an interpolatory projection. We develop a rigorous, computable upper bound for the error resulting from the t-Q-DEIM approximation. Using five different tensor-valued datasets, we numerically illustrate the better approximation properties of t-Q-DEIM and the significant computational cost reduction it offers.

math.NA↗

arXiv2024

Data-Augmented Predictive Deep Neural Network: Enhancing the extrapolation capabilities of non-intrusive surrogate models

Numerically solving a large parametric nonlinear dynamical system is challenging due to its high complexity and the high computational costs. In recent years, machine-learning-aided surrogates are being actively researched. However, many methods fail in accurately generalizing in the entire time interval $[0, T]$, when the training data is available only in a training time interval $[0, T_0]$, with $T_0<T$. To improve the extrapolation capabilities of the surrogate models in the entire time domain, we propose a new deep learning framework, where kernel dynamic mode decomposition (KDMD) is employed to evolve the dynamics of the latent space generated by the encoder part of a convolutional autoencoder (CAE). After adding the KDMD-decoder-extrapolated data into the original data set, we train the CAE along with a feed-forward deep neural network using the augmented data. The trained network can predict future states outside the training time interval at any out-of-training parameter samples. The proposed method is tested on two numerical examples: a FitzHugh-Nagumo model and a model of incompressible flow past a cylinder. Numerical results show accurate and fast prediction performance in both the time and the parameter domain.

cs.LG↗

arXiv2024

Interpolatory $\mathcal{H}_2$-optimality Conditions for Structured Linear Time-invariant Systems

Interpolatory necessary optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of unstructured linear time-invariant (LTI) systems are well-known. Based on previous work on $\mathcal{L}_2$-optimal reduced-order modeling of stationary parametric problems, in this paper we develop and investigate optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of structured LTI systems, in particular, for second-order, port-Hamiltonian, and time-delay systems. Under certain diagonalizability assumptions, we show that across all these different structured settings, bitangential Hermite interpolation is the common form for optimality, thus proving a unifying optimality framework for structured reduced-order modeling.

math.NA↗

arXiv2024

Structure-preserving learning for multi-symplectic PDEs

This paper presents an energy-preserving machine learning method for inferring reduced-order models (ROMs) by exploiting the multi-symplectic form of partial differential equations (PDEs). The vast majority of energy-preserving reduced-order methods use symplectic Galerkin projection to construct reduced-order Hamiltonian models by projecting the full models onto a symplectic subspace. However, symplectic projection requires the existence of fully discrete operators, and in many cases, such as black-box PDE solvers, these operators are inaccessible. In this work, we propose an energy-preserving machine learning method that can infer the dynamics of the given PDE using data only, so that the proposed framework does not depend on the fully discrete operators. In this context, the proposed method is non-intrusive. The proposed method is grey box in the sense that it requires only some basic knowledge of the multi-symplectic model at the partial differential equation level. We prove that the proposed method satisfies spatially discrete local energy conservation and preserves the multi-symplectic conservation laws. We test our method on the linear wave equation, the Korteweg-de Vries equation, and the Zakharov-Kuznetsov equation. We test the generalization of our learned models by testing them far outside the training time interval.

cs.LG↗

arXiv2024

Active Sampling of Interpolation Points to Identify Dominant Subspaces for Model Reduction

Model reduction is an active research field to construct low-dimensional surrogate models of high fidelity to accelerate engineering design cycles. In this work, we investigate model reduction for linear structured systems using dominant reachable and observable subspaces. When the training set $-$ containing all possible interpolation points $-$ is large, then these subspaces can be determined by solving many large-scale linear systems. However, for high-fidelity models, this easily becomes computationally intractable. To circumvent this issue, in this work, we propose an active sampling strategy to sample only a few points from the given training set, which can allow us to estimate those subspaces accurately. To this end, we formulate the identification of the subspaces as the solution of the generalized Sylvester equations, guiding us to select the most relevant samples from the training set to achieve our goals. Consequently, we construct solutions of the matrix equations in low-rank forms, which encode subspace information. We extensively discuss computational aspects and efficient usage of the low-rank factors in the process of obtaining reduced-order models. We illustrate the proposed active sampling scheme to obtain reduced-order models via dominant reachable and observable subspaces and present its comparison with the method where all the points from the training set are taken into account. It is shown that the active sample strategy can provide us $17$x speed-up without sacrificing any noticeable accuracy.

stat.ML↗

arXiv2024

Learning reduced-order Quadratic-Linear models in Process Engineering using Operator Inference

In this work, we address the challenge of efficiently modeling dynamical systems in process engineering. We use reduced-order model learning, specifically operator inference. This is a non-intrusive, data-driven method for learning dynamical systems from time-domain data. The application in our study is carbon dioxide methanation, an important reaction within the Power-to-X framework, to demonstrate its potential. The numerical results show the ability of the reduced-order models constructed with operator inference to provide a reduced yet accurate surrogate solution. This represents an important milestone towards the implementation of fast and reliable digital twin architectures.

math.NA↗

arXiv2024

Estimates of the Kolmogorov n-width for nonlinear transformations with application to distributed-parameter control systems

This paper aims at characterizing the approximability of bounded sets in the range of nonlinear operators in Banach spaces by finite-dimensional linear varieties. In particular, the class of operators we consider includes the endpoint maps of nonlinear distributed-parameter control systems. We describe the relationship between the Kolmogorov n-width of a bounded subset and the width of its image under an essentially nonlinear transformation. We propose explicit estimates of the n-width in the space of images in terms of the affine part of the corresponding operator and the width of its nonlinear perturbation. These $n$-width estimates enable us to describe the reachable sets for infinite-dimensional bilinear control systems, with applications to controlling the Euler-Bernoulli beam using a contraction force and to a single-input Schrödinger equation.

math.OC↗

arXiv2024

Modelling Gas Networks with Compressors: A port-Hamiltonian Approach

Transient gas network simulations can significantly assist in design and operational aspects of gas networks. Models used in these simulations require a detailed framework integrating various models of the network constituents - pipes and compressor stations among others. In this context, the port-Hamiltonian modelling framework provides an energy-based modelling approach with a port-based coupling mechanism. This study investigates developing compressor models in an integrated isothermal port-Hamiltonian model for gas networks. Four different models of compressors are considered and their inclusion in a larger network model is detailed. A numerical implementation for a simple testcase is provided to confirm the validity of the proposed model and to highlight their differences.

math.OC↗

arXiv2024

GS-PINN: Greedy Sampling for Parameter Estimation in Partial Differential Equations

Partial differential equation parameter estimation is a mathematical and computational process used to estimate the unknown parameters in a partial differential equation model from observational data. This paper employs a greedy sampling approach based on the Discrete Empirical Interpolation Method to identify the most informative samples in a dataset associated with a partial differential equation to estimate its parameters. Greedy samples are used to train a physics-informed neural network architecture which maps the nonlinear relation between spatio-temporal data and the measured values. To prove the impact of greedy samples on the training of the physics-informed neural network for parameter estimation of a partial differential equation, their performance is compared with random samples taken from the given dataset. Our simulation results show that for all considered partial differential equations, greedy samples outperform random samples, i.e., we can estimate parameters with a significantly lower number of samples while simultaneously reducing the relative estimation error. A Python package is also prepared to support different phases of the proposed algorithm, including data prepossessing, greedy sampling, neural network training, and comparison.

math.DS↗

arXiv2024

Roadmap on Data-Centric Materials Science

Science is and always has been based on data, but the terms "data-centric" and the "4th paradigm of" materials research indicate a radical change in how information is retrieved, handled and research is performed. It signifies a transformative shift towards managing vast data collections, digital repositories, and innovative data analytics methods. The integration of Artificial Intelligence (AI) and its subset Machine Learning (ML), has become pivotal in addressing all these challenges. This Roadmap on Data-Centric Materials Science explores fundamental concepts and methodologies, illustrating diverse applications in electronic-structure theory, soft matter theory, microstructure research, and experimental techniques like photoemission, atom probe tomography, and electron microscopy. While the roadmap delves into specific areas within the broad interdisciplinary field of materials science, the provided examples elucidate key concepts applicable to a wider range of topics. The discussed instances offer insights into addressing the multifaceted challenges encountered in contemporary materials research.

cond-mat.mtrl-sci↗

arXiv2024

Stability-Certified Learning of Control Systems with Quadratic Nonlinearities

This work primarily focuses on an operator inference methodology aimed at constructing low-dimensional dynamical models based on a priori hypotheses about their structure, often informed by established physics or expert insights. Stability is a fundamental attribute of dynamical systems, yet it is not always assured in models derived through inference. Our main objective is to develop a method that facilitates the inference of quadratic control dynamical systems with inherent stability guarantees. To this aim, we investigate the stability characteristics of control systems with energy-preserving nonlinearities, thereby identifying conditions under which such systems are bounded-input bounded-state stable. These insights are subsequently applied to the learning process, yielding inferred models that are inherently stable by design. The efficacy of our proposed framework is demonstrated through a couple of numerical examples.

cs.LG↗

arXiv2024

MaRDIFlow: A CSE workflow framework for abstracting meta-data from FAIR computational experiments

Numerical algorithms and computational tools are instrumental in navigating and addressing complex simulation and data processing tasks. The exponential growth of metadata and parameter-driven simulations has led to an increasing demand for automated workflows that can replicate computational experiments across platforms. In general, a computational workflow is defined as a sequential description for accomplishing a scientific objective, often described by tasks and their associated data dependencies. If characterized through input-output relation, workflow components can be structured to allow interchangeable utilization of individual tasks and their accompanying metadata. In the present work, we develop a novel computational framework, namely, MaRDIFlow, that focuses on the automation of abstracting meta-data embedded in an ontology of mathematical objects. This framework also effectively addresses the inherent execution and environmental dependencies by incorporating them into multi-layered descriptions. Additionally, we demonstrate a working prototype with example use cases and methodically integrate them into our workflow tool and data provenance framework. Furthermore, we show how to best apply the FAIR principles to computational workflows, such that abstracted components are Findable, Accessible, Interoperable, and Reusable in nature.

cs.DC↗

arXiv2024

Balanced Truncation of Descriptor Systems with a Quadratic Output

This work discusses model reduction for differential-algebraic systems with quadratic output equations. Under mild conditions, these systems can be transformed into a Weierstraß canonical form and, thus, be decoupled into differential equations and algebraic equations. The corresponding decoupled states are referred to as proper and improper states. Due to the quadratic function of the state as an output, the proper and improper states are coupled in the output equation, which imposes a challenge from a model reduction viewpoint. Keeping the coupling in mind, our goal in this work is to find important subspaces of the proper and improper states and to reduce the system accordingly. To that end, we first propose the system's matrices, the so-called Gramians, to characterize the system's dominant subspaces. We pay particular attention to the computation of the observability Gramians that take into account the nonlinear coupling between the proper and the improper states. We furthermore show that the proposed Gramians are related to certain kernel functions, which are used to identify important subspaces. This allows us to propose a reduction algorithm to obtain reduced-order systems by removing the subspaces that are difficult to reach, as well as, difficult to observe. Moreover, we quantify the error between the full-order and reduced-order models and demonstrate the proposed methodology using three numerical experiments.

math.DS↗

arXiv2024

Data-Driven Identification of Quadratic Representations for Nonlinear Hamiltonian Systems using Weakly Symplectic Liftings

We present a framework for learning Hamiltonian systems using data. This work is based on a lifting hypothesis, which posits that nonlinear Hamiltonian systems can be written as nonlinear systems with cubic Hamiltonians. By leveraging this, we obtain quadratic dynamics that are Hamiltonian in a transformed coordinate system. To that end, for given generalized position and momentum data, we propose a methodology to learn quadratic dynamical systems, enforcing the Hamiltonian structure in combination with a weakly-enforced symplectic auto-encoder. The obtained Hamiltonian structure exhibits long-term stability of the system, while the cubic Hamiltonian function provides relatively low model complexity. For low-dimensional data, we determine a higher-dimensional transformed coordinate system, whereas for high-dimensional data, we find a lower-dimensional coordinate system with the desired properties. We demonstrate the proposed methodology by means of both low-dimensional and high-dimensional nonlinear Hamiltonian systems.

cs.LG↗

arXiv2024

Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inference

Scientific machine learning for inferring dynamical systems combines data-driven modeling, physics-based modeling, and empirical knowledge. It plays an essential role in engineering design and digital twinning. In this work, we primarily focus on an operator inference methodology that builds dynamical models, preferably in low-dimension, with a prior hypothesis on the model structure, often determined by known physics or given by experts. Then, for inference, we aim to learn the operators of a model by setting up an appropriate optimization problem. One of the critical properties of dynamical systems is stability. However, this property is not guaranteed by the inferred models. In this work, we propose inference formulations to learn quadratic models, which are stable by design. Precisely, we discuss the parameterization of quadratic systems that are locally and globally stable. Moreover, for quadratic systems with no stable point yet bounded (e.g., chaotic Lorenz model), we discuss how to parameterize such bounded behaviors in the learning process. Using those parameterizations, we set up inference problems, which are then solved using a gradient-based optimization method. Furthermore, to avoid numerical derivatives and still learn continuous systems, we make use of an integral form of differential equations. We present several numerical examples, illustrating the preservation of stability and discussing its comparison with the existing state-of-the-art approach to infer operators. By means of numerical examples, we also demonstrate how the proposed methods are employed to discover governing equations and energy-preserving models.

cs.LG↗

arXiv2024

Generalized Quadratic Embeddings for Nonlinear Dynamics using Deep Learning

The engineering design process often relies on mathematical modeling that can describe the underlying dynamic behavior. In this work, we present a data-driven methodology for modeling the dynamics of nonlinear systems. To simplify this task, we aim to identify a coordinate transformation that allows us to represent the dynamics of nonlinear systems using a common, simple model structure. The advantage of a common simple model is that customized design tools developed for it can be applied to study a large variety of nonlinear systems. The simplest common model -- one can think of -- is linear, but linear systems often fall short in accurately capturing the complex dynamics of nonlinear systems. In this work, we propose using quadratic systems as the common structure, inspired by the lifting principle. According to this principle, smooth nonlinear systems can be expressed as quadratic systems in suitable coordinates without approximation errors. However, finding these coordinates solely from data is challenging. Here, we leverage deep learning to identify such lifted coordinates using only data, enabling a quadratic dynamical system to describe the system's dynamics. Additionally, we discuss the asymptotic stability of these quadratic dynamical systems. We illustrate the approach using data collected from various numerical examples, demonstrating its superior performance with the existing well-known techniques.

math.DS↗

arXiv2023

Fast and Reliable Reduced-Order Models for Cardiac Electrophysiology

Mathematical models of the human heart are increasingly playing a vital role in understanding the working mechanisms of the heart, both under healthy functioning and during disease. The aim is to aid medical practitioners diagnose and treat the many ailments affecting the heart. Towards this, modelling cardiac electrophysiology is crucial as the heart's electrical activity underlies the contraction mechanism and the resulting pumping action. The governing equations and the constitutive laws describing the electrical activity in the heart are coupled, nonlinear, and involve a fast moving wave front, which is generally solved by the finite element method. The simulation of this complex system as part of a virtual heart model is challenging due to the necessity of fine spatial and temporal resolution of the domain. Therefore, efficient surrogate models are needed to predict the dynamics under varying parameters and inputs. In this work, we develop an adaptive, projection-based surrogate model for cardiac electrophysiology. We introduce an a posteriori error estimator that can accurately and efficiently quantify the accuracy of the surrogate model. Using the error estimator, we systematically update our surrogate model through a greedy search of the parameter space. Furthermore, using the error estimator, the parameter search space is dynamically updated such that the most relevant samples get chosen at every iteration. The proposed adaptive surrogate modelling technique is tested on three benchmark models to illustrate its efficiency, accuracy, and ability of generalization.

math.NA↗