arXiv subjects

Igor Pontes Duff

Publications and source records attributed to Igor Pontes Duff.

At least 19 recordsLinked to original sources

arXiv2025

Reduced-Order Inference with Structure-Preserving Parametrization for Bending and Rotating Systems

Mechanical systems are often characterized only by their response to certain loads known from experiments or simulations. The obtained data can be used for various purposes: system analysis, design of mathematical models, or construction of reduced-order models for further simulations under different loading conditions. The use of data for reduced-order modeling is an important developing research direction, especially when the high-dimensional system operators are unknown and their low-dimensional approximation is required for accurate but fast simulations. Our goal is to obtain the low-dimensional surrogate model from the available input signal and deformation trajectory data, capturing the correct system behavior for the basic deformation cases, namely bending and rotation, which are present in almost every complex mechanical system. In this work, we propose a methodology to infer the system operators by solving a nonlinear unconstrained optimization problem. The methodology is based on the operator inference approach for second-order systems and includes a parametrization of the unknown operators that preserves their symmetric positive definite or skew-symmetric structure. We demonstrate the performance of the novel approach for three numerical examples that are used to simulate basic bending and rotating.

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arXiv2025

Data-driven balanced truncation for linear systems with quadratic outputs

We develop the framework for a non-intrusive, quadrature-based method for approximate balanced truncation (QuadBT) of linear systems with quadratic outputs, thus extending the applicability of QuadBT, which was originally designed for data-driven balanced truncation of standard linear systems with linear outputs only. The new approach makes use of the time-domain and frequency-domain quadrature-based representation of the system's infinite Gramians, only implicitly. We show that by sampling solely the extended impulse responses of the original system and their derivatives (or the corresponding transfer functions), we construct a reduced-order model that mimics the approximation quality of the intrusive (projection-based) balanced truncation. We validate the proposed framework on a numerical example.

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arXiv2025

Application of operator inference to reduced-order modeling of constrained mechanical systems

Constrained mechanical systems occur in many applications, such as modeling of robots and other multibody systems. In this case, the motion is governed by a system of differential-algebraic equations (DAE), often with large and sparse system matrices. The problem dimension strongly influences the effectiveness of simulations for system analysis, optimization, and control, given limited computational resources. Therefore, we aim to obtain a simplified surrogate model with a few degrees of freedom that is able to accurately represent the motion and other important properties of the original high-dimensional DAE model. Classical model reduction methods intrusively exploit the system matrices to construct the projection of the high-fidelity model onto a low-dimensional subspace. In practice, the dynamical equations are frequently an inaccessible part of proprietary software products. In this work, we show an application of the non-intrusive operator inference (OpInf) method to DAE systems of index 2 and 3. Considering the fact that for proper DAEs there exists an ODE realization on the so-called hidden manifold, the OpInf optimization problem directly provides the underlying ODE representation of the given DAE system in the reduced subspace. A significant advantage is that only the DAE solution snapshots in a compressed form are required for identification of the reduced system matrices. Stability and interpretability of the reduced-order model is guaranteed by enforcing the symmetric positive definite structure of the system operators using semidefinite programming. The numerical results demonstrate the implementation of the proposed methodology for different examples of constrained mechanical systems, tested for various loading conditions.

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arXiv2025

Inference of Substructured Reduced-Order Models for Dynamic Contact from Contact-free Simulations

In this paper, we propose an operator-inference-based reduction approach for contact problems, leveraging snapshots from simulations without active contact. Contact problems are solved using adjoint methods, by switching to the dual system, where the corresponding Lagrange multipliers represent the contact pressure. The Craig-Bampton-like substructuring method is incorporated into the inference process to provide the reduced system matrices and the coupling of the contact and interior nodes. The maximum possible set of contact nodes must be known a priori. Characteristic properties of the inferred matrices, such as symmetry and positive definiteness, are enforced by appending additional constraints to the underlying least-squares problem. The resulting dual system, which forms a linear complementarity problem, is well-defined and can be effectively solved using methods such as Lemke's algorithm. The performance of the proposed method is validated on three-dimensional finite element models.

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arXiv2025

$\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems by multivariate rational interpolation

This paper addresses the $\mathcal{H}_2$-optimal approximation of linear dynamical systems with quadratic-output functions, also known as linear quadratic-output systems. Our major contributions are threefold. First, we derive interpolatory first-order optimality conditions for the linear quadratic-output $\mathcal{H}_2$ minimization problem. These conditions correspond to the mixed-multipoint tangential interpolation of the full-order linear- and quadratic-output transfer functions, and generalize the Meier-Luenberger optimality framework for the $\mathcal{H}_2$-optimal model reduction of linear time-invariant systems. Second, given the optimal interpolation data, we show how to enforce the interpolatory optimality conditions explicitly by Petrov-Galerkin projection of the full-order model. Third, to find the optimal interpolation data, we build on this projection framework and propose a generalization of the iterative rational Krylov algorithm for the $\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems, called LQO-IRKA. Upon convergence, LQO-IRKA produces reduced linear quadratic-output systems that satisfy the interpolatory optimality conditions. The method only requires solving shifted linear systems and matrix-vector products, thus making it suitable for large-scale problems. Numerical examples are included to illustrate the effectiveness of the proposed method.

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arXiv2025

On the representation of energy-preserving quadratic operators with application to Operator Inference

In this work, we investigate a skew-symmetric parameterization for energy-preserving quadratic operators. Earlier, [Goyal et al., 2023] proposed this parameterization to enforce energy-preservation for quadratic terms in the context of dynamical system data-driven inference. We here prove that every energy-preserving quadratic term can be equivalently formulated using a parameterization of the corresponding operator via skew-symmetric matrix blocks. Based on this main finding, we develop an algorithm to compute an equivalent quadratic operator with skew-symmetric sub-matrices, given an arbitrary energy-preserving operator. Consequently, we employ the skew-symmetric sub-matrix representation in the framework of non-intrusive reduced-order modeling (ROM) via Operator Inference (OpInf) for systems with an energy-preserving nonlinearity. To this end, we propose a sequential, linear least-squares (LS) problems formulation for the inference task, to ensure energy-preservation of the data-driven quadratic operator. The potential of this approach is indicated by the numerical results for a 2D Burgers' equation benchmark, compared to classical OpInf. The inferred system dynamics are accurate, while the corresponding operators are faithful to the underlying physical properties of the system.

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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.

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arXiv2024

$H_2$ optimal model reduction of linear systems with multiple quadratic outputs

In this work, we consider the $H_2$ optimal model reduction of dynamical systems that are linear in the state equation and up to quadratic nonlinearity in the output equation. As our primary theoretical contributions, we derive gradients of the squared $H_2$ system error with respect to the reduced model quantities and, from the stationary points of these gradients, introduce Gramian-based first-order necessary conditions for the $H_2$ optimal approximation of a linear quadratic output (LQO) system. The resulting $H_2$ optimality framework neatly generalizes the analogous Gramian-based optimality framework for purely linear systems. Computationally, we show how to enforce the necessary optimality conditions using Petrov-Galerkin projection; the corresponding projection matrices are obtained from a pair of Sylvester equations. Based on this result, we propose an iteratively corrected algorithm for the $H_2$ model reduction of LQO systems, which we refer to as LQO-TSIA (linear quadratic output two-sided iteration algorithm). Numerical examples are included to illustrate the effectiveness of the proposed computational method against other existing approaches.

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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.

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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.

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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.

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arXiv2023

Semi-active damping optimization of vibrational systems using the reduced basis method

In this article, we consider vibrational systems with semi-active damping that are described by a second-order model. In order to minimize the influence of external inputs to the system response, we are optimizing some damping values. As minimization criterion, we evaluate the energy response, that is the $\cH_2$-norm of the corresponding transfer function of the system. Computing the energy response includes solving Lyapunov equations for different damping parameters. Hence, the minimization process leads to high computational costs if the system is of large dimension. We present two techniques that reduce the optimization problem by applying the reduced basis method to the corresponding parametric Lyapunov equations. In the first method, we determine a reduced solution space on which the Lyapunov equations and hence the resulting energy response values are computed approximately in a reasonable time. The second method includes the reduced basis method in the minimization process. To evaluate the quality of the approximations, we introduce error estimators that evaluate the error in the controllability Gramians and the energy response. Finally, we illustrate the advantages of our methods by applying them to two different examples.

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arXiv2023

Inference of Continuous Linear Systems from Data with Guaranteed Stability

Machine-learning technologies for learning dynamical systems from data play an important role in engineering design. This research focuses on learning continuous linear models from data. Stability, a key feature of dynamic systems, is especially important in design tasks such as prediction and control. Thus, there is a need to develop methodologies that provide stability guarantees. To that end, we leverage the parameterization of stable matrices proposed in [Gillis/Sharma, Automatica, 2017] to realize the desired models. Furthermore, to avoid the estimation of derivative information to learn continuous systems, we formulate the inference problem in an integral form. We also discuss a few extensions, including those related to control systems. Numerical experiments show that the combination of a stable matrix parameterization and an integral form of differential equations allows us to learn stable systems without requiring derivative information, which can be challenging to obtain in situations with noisy or limited data.

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arXiv2023

Dominant Subspaces of High-Fidelity Nonlinear Structured Parametric Dynamical Systems and Model Reduction

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.

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arXiv2022

An Operator Inference Oriented Approach for Mechanical Systems

Model-order reduction techniques allow the construction of low-dimensional surrogate models that can accelerate engineering design processes. Often, these techniques are intrusive, meaning that they require direct access to underlying high-fidelity models. Accessing these models is laborious or may not even be possible in some cases. Therefore, there is an interest in developing non-intrusive model reduction techniques to construct low-dimensional models directly from simulated or experimental data. In this work, we focus on a recent data-driven methodology, namely operator inference, that aims at inferring the reduced operators using only trajectories of high-fidelity models. We present an extension of operator inference for mechanical systems, preserving the second-order structure. We also study a particular case in which complete information about the external forces is available. In this formulation, the reduced operators having certain properties inspired by the original system matrices are enforced by adding constraints to the optimization problem. We illustrate the presented methodology using three numerical examples.

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arXiv2022

Model reduction for second-order systems with inhomogeneous initial conditions

In this paper, we consider the problem of finding surrogate models for large-scale second-order linear time-invariant systems with inhomogeneous initial conditions. For this class of systems, the superposition principle allows us to decompose the system behavior into three independent components. The first behavior corresponds to the transfer between the input and output having zero initial conditions. In contrast, the other two correspond to the transfer between the initial position and the initial velocity conditions having zero input, respectively. Based on this superposition of systems, our goal is to propose model reduction schemes allowing to preserve the second-order structure in the surrogate models. To this aim, we introduce tailored second-order Gramians for each system component and compute them numerically, solving Lyapunov equations. As a consequence, two methodologies are proposed. The first one consists in reducing each of the components independently using a suitable balanced truncation procedure. The sum of these reduced systems provides an approximation of the original system. This methodology allows flexibility on the order of the reduced-order model. The second proposed methodology consists in extracting the dominant subspaces from the sum of Gramians to build the projection matrices leading to a surrogate model. Additionally, we discuss error bounds for the overall output approximation. Finally, the proposed methods are illustrated by means of benchmark problems.

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arXiv2022

Model order reduction for bilinear systems with non-zero initial states -- different approaches with error bounds

In this paper, we consider model order reduction for bilinear systems with non-zero initial conditions. We discuss choices of Gramians for both the homogeneous and the inhomogeneous parts of the system individually and prove how these Gramians characterize the respective dominant subspaces of each of the two subsystems. Proposing different, not necessarily structure preserving, reduced order methods for each subsystem, we establish several strategies to reduce the dimension of the full system. For all these approaches, error bounds are shown depending on the truncated Hankel singular values of the subsystems. Besides the error analysis, stability is discussed. In particular, a focus is on a new criterion for the homogeneous subsystem guaranteeing the existence of the associated Gramians and an asymptotically stable realization of the system.

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arXiv2021

Full state approximation by Galerkin projection reduced order models for stochastic and bilinear systems

In this paper, the problem of full state approximation by model reduction is studied for stochastic and bilinear systems. Our proposed approach relies on identifying the dominant subspaces based on the reachability Gramian of a system. Once the desired subspace is computed, the reduced order model is then obtained by a Galerkin projection. We prove that, in the stochastic case, this approach either preserves mean square asymptotic stability or leads to reduced models whose minimal realization is mean square asymptotically stable. This stability preservation guarantees the existence of the reduced system reachability Gramian which is the basis for the full state error bounds that we derive. This error bound depends on the neglected eigenvalues of the reachability Gramian and hence shows that these values are a good indicator for the expected error in the dimension reduction procedure. Subsequently, we establish the stability preservation result and the error bound for a full state approximation to bilinear systems in a similar manner. These latter results are based on a recently proved link between stochastic and bilinear systems. We conclude the paper by numerical experiments using a benchmark problem. We compare this approach with balanced truncation and show that it performs well in reproducing the full state of the system. \end{abstract}

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