Subhayan De

Northern Arizona University

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Alireza Doostan

University of Colorado Boulder

Kurt Maute

University of Colorado Boulder

Malik Hassanaly

National Renewable Energy Laboratory

Ryan King

National Renewable Energy Laboratory

Peter J. Weddle

National Renewable Energy Laboratory

E. A. Johnson

University of Southern California

Steven F. Wojtkiewicz

Clarkson University

Patrick T. Brewick

University of Notre Dame

Eric J. Dufek

Idaho National Laboratory

Corey R. Randall

Colorado School of Mines

Cooperative Institutions

University of Colorado Boulder

National Renewable Energy Laboratory

University of Colorado System

Laboratório Nacional de Computação Científica

Netherlands Aerospace Centre

Universidade Federal do Rio de Janeiro

Colorado School of Mines

Universidade Federal Fluminense

Universidade de Fortaleza

National Energy Research Scientific Computing Center

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Neural Network Training Using 𝓁 1 -Regularization and Bi-fidelity Data.

arXiv (Cornell University) (2021)

Subhayan De Alireza Doostan

With the capability of accurately representing a functional relationship between the inputs of a physical system's model and output quantities of interest, neural networks have become popular for surrogate modeling in scientific applications. However, as these networks are over-parameterized, their training often requires a large amount of data. To prevent overfitting and improve generalization error, regularization based on, e.g., $\ell_1$- and $\ell_2$-norms of the parameters is applied. Similarly, multiple connections of the network may be pruned to increase sparsity in the network parameters. In this paper, we explore the effects of sparsity promoting $\ell_1$-regularization on training neural networks when only a small training dataset from a high-fidelity model is available. As opposed to standard $\ell_1$-regularization that is known to be inadequate, we consider two variants of $\ell_1$-regularization informed by the parameters of an identical network trained using data from lower-fidelity models of the problem at hand. These bi-fidelity strategies are generalizations of transfer learning of neural networks that uses the parameters learned from a large low-fidelity dataset to efficiently train networks for a small high-fidelity dataset. We also compare the bi-fidelity strategies with two $\ell_1$-regularization methods that only use the high-fidelity dataset. Three numerical examples for propagating uncertainty through physical systems are used to show that the proposed bi-fidelity $\ell_1$-regularization strategies produce errors that are one order of magnitude smaller than those of networks trained only using datasets from the high-fidelity models.

Overfitting

Regularization

Training set

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Citations (1)

Efficient optimal design and design‐under‐uncertainty of passive control devices with application to a cable‐stayed bridge

Structural Control and Health Monitoring (2016)

Subhayan De Steven F. Wojtkiewicz E. A. Johnson

Structures today may be equipped with passive structural control devices to achieve some performance criteria. The optimal design of these passive control devices, whether via a formal optimization algorithm or a response surface parameter study, requires multiple solutions of the dynamic response of that structure, incurring a significant computational cost for complex structures. These passive control elements are typically point-located, introducing a local change (possibly nonlinear, possibly uncertain) that affects the global behavior of the rest of the structure. When the structure, other than these localized devices, is linear and deterministic, conventional solvers (e.g., Runge–Kutta, MATLAB's ode45, etc.) ignore the localized nature of the passive control elements. The methodology applied in this paper exploits the locality of the uncertain and/or nonlinear passive control element(s) by exactly converting the form of the dynamics of the high-order structural model to a low-dimensional Volterra integral equation. Design optimization for parameters and placement of linear and nonlinear passive dampers, tuned mass dampers, and their combination, as well as their design-under-uncertainty for a benchmark cable-stayed bridge, is performed using this approach. For the examples considered herein, the proposed method achieves a two-orders-of-magnitude gain in computational efficiency compared with a conventional method of comparable accuracy. Copyright © 2016 John Wiley & Sons, Ltd.

Benchmark (surveying)

Bridge (graph theory)

Tuned mass damper

10.1002/stc.1846

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Citations (19)

Bi-fidelity Variational Auto-encoder for Uncertainty Quantification

arXiv (Cornell University) (2023)

Nuojin Cheng Osman Asif Malik Subhayan De Stephen Becker Alireza Doostan

Quantifying the uncertainty of quantities of interest (QoIs) from physical systems is a primary objective in model validation. However, achieving this goal entails balancing the need for computational efficiency with the requirement for numerical accuracy. To address this trade-off, we propose a novel bi-fidelity formulation of variational auto-encoders (BF-VAE) designed to estimate the uncertainty associated with a QoI from low-fidelity (LF) and high-fidelity (HF) samples of the QoI. This model allows for the approximation of the statistics of the HF QoI by leveraging information derived from its LF counterpart. Specifically, we design a bi-fidelity auto-regressive model in the latent space that is integrated within the VAE's probabilistic encoder-decoder structure. An effective algorithm is proposed to maximize the variational lower bound of the HF log-likelihood in the presence of limited HF data, resulting in the synthesis of HF realizations with a reduced computational cost. Additionally, we introduce the concept of the bi-fidelity information bottleneck (BF-IB) to provide an information-theoretic interpretation of the proposed BF-VAE model. Our numerical results demonstrate that BF-VAE leads to considerably improved accuracy, as compared to a VAE trained using only HF data, when limited HF data is available.

High fidelity

Autoencoder

Information bottleneck method

10.48550/arxiv.2305.16530

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Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantification

arXiv (Cornell University) (2024)

Soban Nasir Lone Subhayan De Rajdip Nayek

We introduce a novel deep operator network (DeepONet) framework that incorporates generalised variational inference (GVI) using R\'enyi's $\alpha$-divergence to learn complex operators while quantifying uncertainty. By incorporating Bayesian neural networks as the building blocks for the branch and trunk networks, our framework endows DeepONet with uncertainty quantification. The use of R\'enyi's $\alpha$-divergence, instead of the Kullback-Leibler divergence (KLD), commonly used in standard variational inference, mitigates issues related to prior misspecification that are prevalent in Variational Bayesian DeepONets. This approach offers enhanced flexibility and robustness. We demonstrate that modifying the variational objective function yields superior results in terms of minimising the mean squared error and improving the negative log-likelihood on the test set. Our framework's efficacy is validated across various mechanical systems, where it outperforms both deterministic and standard KLD-based VI DeepONets in predictive accuracy and uncertainty quantification. The hyperparameter $\alpha$, which controls the degree of robustness, can be tuned to optimise performance for specific problems. We apply this approach to a range of mechanics problems, including gravity pendulum, advection-diffusion, and diffusion-reaction systems. Our findings underscore the potential of $\alpha$-VI DeepONet to advance the field of data-driven operator learning and its applications in engineering and scientific domains.

Uncertainty Quantification

Alpha (finance)

10.48550/arxiv.2408.00681

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Citations (1)

Neural network training using ℓ1-regularization and bi-fidelity data

Journal of Computational Physics (2022)

Subhayan De Alireza Doostan

Overfitting

Regularization

Training set

10.1016/j.jcp.2022.111010

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Citations (16)

Shape and material optimization of problems with dynamically evolving interfaces applied to solid rocket motors

Structural and Multidisciplinary Optimization (2022)

Kurt Maute Subhayan De

Multidisciplinary design optimization

Rocket (weapon)

10.1007/s00158-022-03331-9

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Citations (1)

A Bi-fidelity DeepONet Approach for Modeling Hysteretic Systems under Uncertainty

Applied Mathematical Modelling (2024)

Subhayan De Patrick T. Brewick

Uncertainty Quantification

10.1016/j.apm.2024.06.045

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Topology optimization under uncertainty using a stochastic gradient-based approach

Structural and Multidisciplinary Optimization (2020)

Subhayan De Jerrad Hampton Kurt Maute Alireza Doostan

Topology optimization

Robust Optimization

Stochastic Gradient Descent

Random optimization

Stochastic Approximation

10.1007/s00158-020-02599-z

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Citations (49)

Topology Optimization under Uncertainty using a Stochastic Gradient-based Approach

arXiv (Cornell University) (2019)

Subhayan De Jerrad Hampton Kurt Maute Alireza Doostan

Topology optimization under uncertainty (TOuU) often defines objectives and constraints by statistical moments of geometric and physical quantities of interest. Most traditional TOuU methods use gradient-based optimization algorithms and rely on accurate estimates of the statistical moments and their gradients, e.g., via adjoint calculations. When the number of uncertain inputs is large or the quantities of interest exhibit large variability, a large number of adjoint (and/or forward) solves may be required to ensure the accuracy of these gradients. The optimization procedure itself often requires a large number of iterations, which may render TOuU computationally expensive, if not infeasible. To tackle this difficulty, we here propose an optimization approach that generates a stochastic approximation of the objective, constraints, and their gradients via a small number of adjoint (and/or forward) solves, per iteration. A statistically independent (stochastic) approximation of these quantities is generated at each optimization iteration. The total cost of this approach is only a small factor larger than that of the corresponding deterministic TO problem. We incorporate the stochastic approximation of objective, constraints and their design sensitivities into two classes of optimization algorithms. First, we investigate the stochastic gradient descent (SGD) method and a number of its variants, which have been successfully applied to large-scale optimization problems for machine learning. Second, we study the use of the proposed stochastic approximation approach within conventional nonlinear programming methods, focusing on the Globally Convergent Method of Moving Asymptotes (GCMMA). The performance of these algorithms is investigated with structural design optimization problems utilizing a Solid Isotropic Material with Penalization (SIMP), as well as an explicit level set method.

Stochastic Gradient Descent

Robust Optimization

Stochastic Approximation

Topology optimization

Random optimization

10.48550/arxiv.1902.04562

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Citations (5)

Bi-fidelity variational auto-encoder for uncertainty quantification

Computer Methods in Applied Mechanics and Engineering (2024)

Nuojin Cheng Osman Asif Malik Subhayan De Stephen Becker Alireza Doostan

High fidelity

Autoencoder

Uncertainty Quantification

10.1016/j.cma.2024.116793

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Citations (5)