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Organizing Committee
University of Massachusetts, Dartmouth
University of Massachusetts Dartmouth
Virginia Tech
Tufts University
Rensselaer Polytechnic Institute
University of Utah
Abstract
Reduced order modeling (ROM) has become an important tool in computational science for accelerating model-based simulations, including those governed by parametrized differential equations. Through the approximation of high-dimensional features with low-dimensional representations, ROM consists of proven strategies that build accurate emulators for the field or response of computationally expensive high-fidelity models using only a fraction of the simulation cost. In forward prediction or outer loop design and optimization, ROM has the potential to substantially improve the efficiency of current simulation-based techniques.
While ROM has seen considerable success in numerous applications, it continues to attract active research and development. This workshop showcases emerging frontiers in ROM by bringing together researchers whose core interests lie in model reduction and approximation theory, but who have also explored and developed novel methods that utilize various aspects of statistical learning and data science. Topics of the workshop will include: 1) new mathematical and computational nonlinear ROM formulations required for prediction of transport phenomena, 2) new model reduction and meta-learning opportunities necessitated by the prevalence of large neural network models, 3) new paradigms for ROM capable of attacking parameter inference and ill-posed inverse problems, 4) new frontiers in the automated learning of latent representations due to the availability of computationally feasible optimization in statistical and machine learning, 5) an appropriate computational balancing of observational or experimental data with simulation-based models and ROM that would lead to the usage of ROM in digital twin-scale applications, and 6) challenges and new developments in ROM that aims to preserve inherent physical structure of the underlying dynamics.
While ROM has seen considerable success in numerous applications, it continues to attract active research and development. This workshop showcases emerging frontiers in ROM by bringing together researchers whose core interests lie in model reduction and approximation theory, but who have also explored and developed novel methods that utilize various aspects of statistical learning and data science. Topics of the workshop will include: 1) new mathematical and computational nonlinear ROM formulations required for prediction of transport phenomena, 2) new model reduction and meta-learning opportunities necessitated by the prevalence of large neural network models, 3) new paradigms for ROM capable of attacking parameter inference and ill-posed inverse problems, 4) new frontiers in the automated learning of latent representations due to the availability of computationally feasible optimization in statistical and machine learning, 5) an appropriate computational balancing of observational or experimental data with simulation-based models and ROM that would lead to the usage of ROM in digital twin-scale applications, and 6) challenges and new developments in ROM that aims to preserve inherent physical structure of the underlying dynamics.
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Workshop Schedule
Monday, January 06, 2025
8:50 - 9:00 AM EST
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:45 AM EST
Large Eddy Simulation Reduced Order Models (LES-ROMs) for Turbulent Flows
11th Floor Lecture Hall
Speaker
Traian Iliescu, Virginia Tech
Session Chair
Yanlai Chen, University of Massachusetts, Dartmouth
Abstract
In this talk, I will survey our group's work to bridge two distinct research fields, large eddy simulation (LES) and reduced order models (ROMs), for efficient and accurate numerical simulations of engineering and geophysical turbulent flows. The models that we have developed, called LES-ROMs, leverage ideas from both LES and ROMs, as well as modern data-driven modeling strategies. The main tool used to construct the new LES-ROMs is explicit spatial filtering, which is employed to model the large spatial structures in the underlying turbulent flow. We have used the new LES-ROMs in the numerical simulation of the turbulent channel flow, which is an extremely challenging test problem for ROMs.
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EST
Parametric model reduction of stochastic systems via population dynamics
11th Floor Lecture Hall
Virtual Speaker
Benjamin Peherstorfer, Courant Institute, New York University
Session Chair
Yanlai Chen, University of Massachusetts, Dartmouth
Abstract
The aim of this work is to learn models of population dynamics of physical systems that feature stochastic and mean-field effects and that depend on physics parameters. The learned models can act as surrogates of classical numerical models to efficiently predict the system behavior over the physics parameters. Building on the Benamou-Brenier formula from optimal transport and action matching, we use a variational problem to infer parameter- and time-dependent gradient fields that represent approximations of the population dynamics. The inferred gradient fields can then be used to rapidly generate sample trajectories that mimic the dynamics of the physical system on a population level over varying physics parameters. We show that combining Monte Carlo sampling with higher-order quadrature rules is critical for accurately estimating the training objective from sample data and for stabilizing the training process. We demonstrate on Vlasov-Poisson instabilities as well as on high-dimensional particle and chaotic systems that our approach accurately predicts population dynamics over a wide range of parameters and outperforms state-of-the-art diffusion-based and flow-based modeling that simply condition on time and physics parameters.
11:30 AM - 12:15 PM EST
Digital Twins and Optimization
11th Floor Lecture Hall
Speaker
Harbir Antil, George Mason University
Session Chair
Yanlai Chen, University of Massachusetts, Dartmouth
Abstract
"With recent advancements in computing resources and interdisciplinary collaborations, a new research field called Digital Twins (DTs) is starting to emerge. Data from sensors located on a physical system is fed into its DT, the DT in-turn help make decisions about the physical system. This cycle then continues for the life-time of the physical system. A typical example is a bridge or a human heart.
In many cases, these problems can be cast as optimization problems with finite or infinite dimensional (partial differential equations) constraints. This talk will provide an introduction to this topic. Special attention will be given to: 1) Optimization algorithms that are adaptive and can handle inexactness, e.g., Trust-Regions and ALESQP; 2) Optimization under uncertainty and tensor train decomposition to overcome the curse of dimensionality; 3) Reduced order modeling for dynamic optimization using randomized compression.
Additionally, the DT framework may require coupling mutiphysics / systems / data with very different time scales. Keeping this in mind, a newly introduced notion of barely coupled problems will be discussed.
Realistic examples of DTs to identify weakness in structures such as bridges, wind turbines, electric motors, and neuromorphic imaging will be considered."
12:30 - 2:30 PM EST
Lunch/Free Time
2:30 - 3:15 PM EST
Operator Learning via an Auto-Encoder-Based Nonlinear Model Reduction and its Error Analysis
11th Floor Lecture Hall
Speaker
Rongjie Lai, Purdue University
Session Chair
Misha Kilmer, Tufts University
Abstract
Many physical processes in science and engineering are naturally represented by operators between infinite-dimensional function spaces. The problem of operator learning, in this context, seeks to extract these physical processes from empirical data, which is challenging due to the infinite or high dimensionality of data. An integral component in addressing this challenge is model reduction, which reduces both the data dimensionality and problem size. In this talk, I will discuss our recent work on utilizing low-dimensional nonlinear structures in model reduction by investigating Auto-Encoder-based Neural Network (AENet). The proposed AENet first learns the latent variables of the input data and then learns the transformation from these latent variables to corresponding output data. Our numerical experiments validate the ability of AENet to accurately learn the solution operator of nonlinear partial differential equations. Furthermore, we establish a mathematical and statistical estimation theory that analyzes the generalization error of AENet. Our theoretical framework shows that the sample complexity of training AENet is intricately tied to the intrinsic dimension of the modeled process, while also demonstrating the robustness of AENet to noise.
3:30 - 4:00 PM EST
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EST
Advantages of Weak-Form Equation Learning in Reduced-Order Models for Parametric PDEs
11th Floor Lecture Hall
Speaker
Chi Tran, University of Colorado at Boulder
Session Chair
Misha Kilmer, Tufts University
Abstract
"Recent advancements in data-driven modeling highlight that adopting a weak formulation of model equations significantly improves noise robustness across various computational methods. This presentation explores how the weak form enhances the LaSDI (Latent Space Dynamics
Identification) algorithm, a recently developed data-driven
reduced order modeling technique. We introduce a weak form-based extension of LaSDI, WLaSDI - Weak-form Latent Space
Dynamics Identification. WLaSDI first compresses data,
then projects onto the test functions and learns the local
latent space models. Notably, WLaSDI demonstrates significantly enhanced robustness to noise. WLaSDI obtains
the local latent space using weak-form equation learning.
Compared to the standard SINDy used in LaSDI, the variance reduction of the weak form guarantees a robust and precise latent space recovery, hence allowing for a fast, robust, and
accurate simulation."
5:00 - 6:30 PM EST
Reception
11th Floor Collaborative Space
Tuesday, January 07, 2025
9:00 - 9:45 AM EST
From Data to Structure: Learning Mechanical Systems Via Rational Functions
11th Floor Lecture Hall
Speaker
Steffen W. R. Werner, Virginia Tech
Session Chair
Fengyan Li, Rensselaer Polytechnic Institute
Abstract
Data-driven reduced-order modeling is an essential tool in constructing high-fidelity compact models to approximate physical phenomena when explicit models, such as state-space formulations with access to internal variables, are not available yet abundant input/output data are. When deriving models from theoretical principles, the resulting models typically contain differential structures that lead to certain system properties. A common example are dynamical systems with second-order time derivatives arising in the modeling of mechanical or electro-mechanical processes. In this case, data are often available in the frequency domain, where the systems' input-to-output behavior is described by rational functions rather than differential equations. Classical frequency domain approaches like the Loewner framework, vector fitting and AAA are available and can be used to learn unstructured (first-order) models from data. However, these models in their original formulation do not reflect the structure-inherited properties. The key element in the derivation of frequency domain approaches is the barycentric form of rational functions. In this work, we present structured extensions of the barycentric form for the case of mechanical systems. Building on these structured barycentric forms, we develop new algorithms for learning mechanical phenomena in the frequency domain, while enforcing the mechanical system structure in the model description.
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space