arXiv subjects

Maciej Balajewicz

Publications and source records attributed to Maciej Balajewicz.

13 recordsLinked to original sources

arXiv2024

Efficient Transonic Aeroelastic Model Reduction Using Optimized Sparse Multi-Input Polynomial Functionals

Nonlinear aeroelastic reduced-order models (ROMs) based on machine learning or artificial intelligence algorithms can be complex and computationally demanding to train, meaning that for practical aeroelastic applications, the conservative nature of linearization is often favored. Therefore, there is a requirement for novel nonlinear aeroelastic model reduction approaches that are accurate, simple and, most importantly, efficient to generate. This paper proposes a novel formulation for the identification of a compact multi-input Volterra series, where Orthogonal Matching Pursuit is used to obtain a set of optimally sparse nonlinear multi-input ROM coefficients from unsteady aerodynamic training data. The framework is exemplified using the Benchmark Supercritical Wing, considering; forced response, flutter and limit cycle oscillation. The simple and efficient Optimal Sparsity Multi-Input ROM (OSM-ROM) framework performs with high accuracy compared to the full-order aeroelastic model, requiring only a fraction of the tens-of-thousands of possible multi-input terms to be identified and allowing a 96% reduction in the number of training samples.

physics.flu-dyn↗

arXiv2024

Optimal Sparsity in Nonlinear Non-Parametric Reduced Order Models for Transonic Aeroelastic Systems

Machine learning and artificial intelligence algorithms typically require large amount of data for training. This means that for nonlinear aeroelastic applications, where small training budgets are driven by the high computational burden associated with generating data, usability of such methods has been limited to highly simplified aeroelastic systems. This paper presents a novel approach for the identification of optimized sparse higher-order polynomial-based aeroelastic reduced order models (ROM) to significantly reduce the amount of training data needed without sacrificing fidelity. Several sparsity promoting algorithms are considered, including; rigid sparsity, LASSO regression, and Orthogonal Matching Pursuit (OMP). The study demonstrates that through OMP, it is possible to efficiently identify optimized s-sparse nonlinear aerodynamic ROMs using only aerodynamic response information. This approach is exemplified in a three-dimensional aeroelastic stabilator model experiencing high amplitude freeplay-induced limit cycles. The comparison shows excellent agreement between the ROM and the full-order aeroelastic response, including the ability to generalize to new freeplay and velocity index values, with online computational savings of several orders of magnitude. The development of an Optimally Sparse ROM (OS-ROM) extends previous higher-order polynomial-based ROM approaches for feasible application to complex three-dimensional nonlinear aeroelastic problems, without incurring significant computational burdens or loss of accuracy.

physics.flu-dyn↗

arXiv2022

Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks

Physics-informed neural networks (PINNs) leverage neural-networks to find the solutions of partial differential equation (PDE)-constrained optimization problems with initial conditions and boundary conditions as soft constraints. These soft constraints are often considered to be the sources of the complexity in the training phase of PINNs. Here, we demonstrate that the challenge of training (i) persists even when the boundary conditions are strictly enforced, and (ii) is closely related to the Kolmogorov n-width associated with problems demonstrating transport, convection, traveling waves, or moving fronts. Given this realization, we describe the mechanism underlying the training schemes such as those used in eXtended PINNs (XPINN), curriculum regularization, and sequence-to-sequence learning. For an important category of PDEs, i.e., governed by non-linear convection-diffusion equation, we propose reformulating PINNs on a Lagrangian frame of reference, i.e., LPINNs, as a PDE-informed solution. A parallel architecture with two branches is proposed. One branch solves for the state variables on the characteristics, and the second branch solves for the low-dimensional characteristics curves. The proposed architecture conforms to the causality innate to the convection, and leverages the direction of travel of the information in the domain. Finally, we demonstrate that the loss landscapes of LPINNs are less sensitive to the so-called "complexity" of the problems, compared to those in the traditional PINNs in the Eulerian framework.

cs.LG↗

arXiv2020

Physics-aware registration based auto-encoder for convection dominated PDEs

We design a physics-aware auto-encoder to specifically reduce the dimensionality of solutions arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by a large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Moreover, the realized latent variables are often hard to interpret. Therefore, many of these methods are often dismissed in the reduced order modeling of dynamical systems governed by the partial differential equations (PDEs). Accordingly, we propose an auto-encoder type nonlinear dimensionality reduction algorithm. The unsupervised learning problem trains a diffeomorphic spatio-temporal grid, that registers the output sequence of the PDEs on a non-uniform parameter/time-varying grid, such that the Kolmogorov n-width of the mapped data on the learned grid is minimized. We demonstrate the efficacy and interpretability of our approach to separate convection/advection from diffusion/scaling on various manufactured and physical systems.

math.DS↗

arXiv2019

Transported snapshot model order reduction approach for parametric, steady-state fluid flows containing parameter dependent shocks

A new model order reduction approach is proposed for parametric steady-state nonlinear fluid flows characterized by shocks and discontinuities whose spatial locations and orientations are strongly parameter dependent. In this method, solutions in the predictive regime are approximated using a linear superposition of parameter dependent basis. The sought after parametric reduced-basis are obtained by transporting the snapshots in a spatially and parametrically dependent transport field. Key to the proposed approach is the observation that the transport fields are typically smooth and continuous, despite the solution themselves not being so. As a result, the transport fields can be accurately expressed using a low-order polynomial expansion. Similar to traditional projection-based model order reduction approaches, the proposed method is formulated mathematically as a residual minimization problem for the generalized coordinates. The proposed approach is also integrated with well-known hyper-reduction strategies to obtain significant computational speed-ups. The method is successfully applied to the reduction of a parametric 1-D flow in a converging-diverging nozzle, a parametric 2-D supersonic flow over a forward facing step and a parametric 2-D jet diffusion flame in a combustor.

physics.flu-dyn↗

arXiv2018

Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems

Model reduction of high-dimensional dynamical systems alleviates computational burdens faced in various tasks from design optimization to model predictive control. One popular model reduction approach is based on projecting the governing equations onto a subspace spanned by basis functions obtained from the compression of a dataset of solution snapshots. However, this method is intrusive since the projection requires access to the system operators. Further, some systems may require special treatment of nonlinearities to ensure computational efficiency or additional modeling to preserve stability. In this work we propose a deep learning-based strategy for nonlinear model reduction that is inspired by projection-based model reduction where the idea is to identify some optimal low-dimensional representation and evolve it in time. Our approach constructs a modular model consisting of a deep convolutional autoencoder and a modified LSTM network. The deep convolutional autoencoder returns a low-dimensional representation in terms of coordinates on some expressive nonlinear data-supporting manifold. The dynamics on this manifold are then modeled by the modified LSTM network in a computationally efficient manner. An offline unsupervised training strategy that exploits the model modularity is also developed. We demonstrate our model on three illustrative examples each highlighting the model's performance in prediction tasks for fluid systems with large parameter-variations and its stability in long-term prediction.

math.DS↗

arXiv2017

Lagrangian basis method for dimensionality reduction of convection dominated nonlinear flows

Foundations of a new projection-based model reduction approach for convection dominated nonlinear fluid flows are summarized. In this method the evolution of the flow is approximated in the Lagrangian frame of reference. Global basis functions are used to approximate both the state and the position of the Lagrangian computational domain. It is demonstrated that in this framework, certain wave-like solutions exhibit low-rank structure and thus, can be efficiently compressed using relatively few global basis. The proposed approach is successfully demonstrated for the reduction of several simple but representative problems.

physics.flu-dyn↗

arXiv2016

Reduced Order Models for Pricing European and American Options under Stochastic Volatility and Jump-Diffusion Models

European options can be priced by solving parabolic partial(-integro) differential equations under stochastic volatility and jump-diffusion models like Heston, Merton, and Bates models. American option prices can be obtained by solving linear complementary problems (LCPs) with the same operators. A finite difference discretization leads to a so-called full order model (FOM). Reduced order models (ROMs) are derived employing proper orthogonal decomposition (POD). The early exercise constraint of American options is enforced by a penalty on subset of grid points. The presented numerical experiments demonstrate that pricing with ROMs can be orders of magnitude faster within a given model parameter variation range.

cs.CE↗

arXiv2015

Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier-Stokes equations

For a projection-based reduced order model (ROM) of a fluid flow to be stable and accurate, the dynamics of the truncated subspace must be taken into account. This paper proposes an approach for stabilizing and enhancing projection-based fluid ROMs in which truncated modes are accounted for a priori via a minimal rotation of the projection subspace. Attention is focused on the full non-linear compressible Navier-Stokes equations in specific volume form as a step toward a more general formulation for problems with generic non-linearities. Unlike traditional approaches, no empirical turbulence modeling terms are required, and consistency between the ROM and the full order model from which the ROM is derived is maintained. Mathematically, the approach is formulated as a trace minimization problem on the Stiefel manifold. The reproductive as well as predictive capabilities of the method are evaluated on several compressible flow problems, including a problem involving laminar flow over an airfoil with a high angle of attack, and a channel-driven cavity flow problem.

physics.comp-ph↗

arXiv2015

Projection-based model reduction for contact problems

To be feasible for computationally intensive applications such as parametric studies, optimization and control design, large-scale finite element analysis requires model order reduction. This is particularly true in nonlinear settings that tend to dramatically increase computational complexity. Although significant progress has been achieved in the development of computational approaches for the reduction of nonlinear computational mechanics models, addressing the issue of contact remains a major hurdle. To this effect, this paper introduces a projection-based model reduction approach for both static and dynamic contact problems. It features the application of a non-negative matrix factorization scheme to the construction of a positive reduced-order basis for the contact forces, and a greedy sampling algorithm coupled with an error indicator for achieving robustness with respect to model parameter variations. The proposed approach is successfully demonstrated for the reduction of several two-dimensional, simple, but representative contact and self contact computational models.

physics.comp-ph↗

arXiv2014

Reduction of Nonlinear Embedded Boundary Models for Problems with Evolving Interfaces

Embedded boundary methods alleviate many computational challenges, including those associated with meshing complex geometries and solving problems with evolving domains and interfaces. Developing model reduction methods for computational frameworks based on such methods seems however to be challenging. Indeed, most popular model reduction techniques are projection-based, and rely on basis functions obtained from the compression of simulation snapshots. In a traditional interface-fitted computational framework, the computation of such basis functions is straightforward, primarily because the computational domain does not contain in this case a fictitious region. This is not the case however for an embedded computational framework because the computational domain typically contains in this case both real and ghost regions whose definitions complicate the collection and compression of simulation snapshots. The problem is exacerbated when the interface separating both regions evolves in time. This paper addresses this issue by formulating the snapshot compression problem as a weighted low-rank approximation problem where the binary weighting identifies the evolving component of the individual simulation snapshots. The proposed approach is application independent and therefore comprehensive. It is successfully demonstrated for the model reduction of several two-dimensional, vortex-dominated, fluid-structure interaction problems.

physics.comp-ph↗

arXiv2013

Lyapunov stable Galerkin models of post-transient incompressible flows

A method for deriving provably stable low-dimensional Galerkin models of post-transient incompressible flows is introduced. The proposed approach involves an iterative procedure for expansion modes that satisfy Lyapunov stability in the neighborhood of a fixed point. The approach is demonstrated using two prototypical flow configurations: a two-dimensional mixing layer, and two-dimensional flow inside a lid-driven cavity. This new methodology can be a building block in an effort to develop more accurate and more robust low-dimensional models of incompressible flows.

physics.flu-dyn↗

arXiv2012

A Novel Model Order Reduction Approach for Navier-Stokes Equations at High Reynolds Number

A new approach to model order reduction of the Navier-Stokes equations at high Reynolds number is proposed. Unlike traditional approaches, this method does not rely on empirical turbulence modeling or modification of the Navier-Stokes equations. It provides spatial basis functions different from the usual proper orthogonal decomposition basis function in that, in addition to optimally representing the training data set, the new basis functions also provide stable and accurate reduced-order models. The proposed approach is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer.

physics.flu-dyn↗