Topology Optimization under Uncertainty using a Stochastic Gradient-based Approach

De, Subhayan; Hampton, Jerrad; Maute, Kurt; Doostan, Alireza

doi:10.48550/arxiv.1902.04562

Cited by 7 publications

(11 citation statements)

References 72 publications

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“…, N }, and the derivative ∂Ji k ∂p k is calculated using back propagation [2,61]. We, however, use an improved variant of SGD, namely, the Adaptive Moment Estimation (Adam) algorithm [62,63]. Adam leverages past gradient information to retard the descent along large gradients.…”

Section: Training a Neural Networkmentioning

confidence: 99%

On transfer learning of neural networks using bi-fidelity data for uncertainty propagation

Britton

Reynolds

et al. 2020

Preprint

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Due to their high degree of expressiveness, neural networks have recently been used as surrogate models for mapping inputs of an engineering system to outputs of interest. Once trained, neural networks are computationally inexpensive to evaluate and remove the need for repeated evaluations of computationally expensive models in uncertainty quantification applications. However, given the highly parameterized construction of neural networks, especially deep neural networks, accurate training often requires large amounts of simulation data that may not be available in the case of computationally expensive systems. In this paper, to alleviate this issue for uncertainty propagation, we explore the application of transfer learning techniques using training data generated from both high-and low-fidelity models. We explore two strategies for coupling these two datasets during the training procedure, namely, the standard transfer learning and the bi-fidelity weighted learning. In the former approach, a neural network model mapping the inputs to the outputs of interest is trained based on the low-fidelity data. The high-fidelity data is then used to adapt the parameters of the upper layer(s) of the low-fidelity network, or train a simpler neural network to map the output of the low-fidelity network to that of the high-fidelity model. In the latter approach, the entire low-fidelity network parameters are updated using data generated via a Gaussian process model trained with a small high-fidelity dataset. The parameter updates are performed via a variant of stochastic gradient descent with learning rates given by the Gaussian process model. Using three numerical examples, we illustrate the utility of these bi-fidelity transfer learning methods where we focus on accuracy improvement achieved by transfer learning over standard training approaches.

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Section: Training a Neural Networkmentioning

confidence: 99%

On transfer learning of neural networks using bi-fidelity data for uncertainty propagation

Britton

Reynolds

et al. 2020

Preprint

Self Cite

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“…where ûq and vq denote the incremental state and incremental adjoint, respectively. By taking the variation of (31) with respect to the adjoint v q and using (2), we obtain the incremental state problem: find ûq ∈ U such that ṽ, ∂ vu rq ûq = − ṽ, ∂ vm rq mq , ∀ṽ ∈ V,…”

Section: Computation Of the Gradient And Hessian Actionmentioning

confidence: 99%

“…where the derivatives ∂ vu rq : U → V and ∂ vm rq : M → V are linear operators. The incremental adjoint problem, obtained by taking variation of (31) with respect to the state u and using (2), reads:…”

Section: Computation Of the Gradient And Hessian Actionmentioning

confidence: 99%

“…where ∂ uv rq : V → U is the adjoint of ∂ vu rq : U → V in the sense of (2). The Hessian of q at m acting in direction mq can then be computed by taking variation of (31) with respect to m and using (2) as…”

Section: Computation Of the Gradient And Hessian Actionmentioning

confidence: 99%

“…It is critical to incorporate this uncertainty in the optimization problem to make the optimal solution more reliable and robust. Optimization under uncertainty has become an important research area and received increasing attentions in recent years [74,10,45,41,70,76,49,26,78,19,52,63,20,51,48,8,3,5,84,85,42,69,53,50,55,33,59,47,79,80,81,27,18,86,82,35,54,61,6,38,39,31,37,36]. To account for the uncertainty in the optimization problem, different statistical measures of the objective function have been studied, e.g., mean, variance, conditional value-at-risk, worst case scenario, etc., [70,48,85,3,…”

Section: Introductionmentioning

confidence: 99%

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Taylor approximation for chance constrained optimization problems governed by partial differential equations with high-dimensional random parameters

Chen¹,

Ghattas²

2020

Preprint

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We propose a fast and scalable optimization method to solve chance or probabilistic constrained optimization problems governed by partial differential equations (PDEs) with high-dimensional random parameters. To address the critical computational challenges of expensive PDE solution and high-dimensional uncertainty, we construct surrogates of the constraint function by Taylor approximation, which relies on efficient computation of the derivatives, low rank approximation of the Hessian, and a randomized algorithm for eigenvalue decomposition. To tackle the difficulty of the non-differentiability of the inequality chance constraint, we use a smooth approximation of the discontinuous indicator function involved in the chance constraint, and apply a penalty method to transform the inequality constrained optimization problem to an unconstrained one. Moreover, we design a gradientbased optimization scheme that gradually increases smoothing and penalty parameters to achieve convergence, for which we present an efficient computation of the gradient of the approximate cost functional by the Taylor approximation. Based on numerical experiments for a problem in optimal groundwater management, we demonstrate the accuracy of the Taylor approximation, its ability to greatly accelerate constraint evaluations, the convergence of the continuation optimization scheme, and the scalability of the proposed method in terms of the number of PDE solves with increasing random parameter dimension from one thousand to hundreds of thousands.

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Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

et al. 2019

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Ill-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral properties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems in parallel, at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines.

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Topology Optimization under Uncertainty using a Stochastic Gradient-based Approach

Cited by 7 publications

References 72 publications

On transfer learning of neural networks using bi-fidelity data for uncertainty propagation

On transfer learning of neural networks using bi-fidelity data for uncertainty propagation

Taylor approximation for chance constrained optimization problems governed by partial differential equations with high-dimensional random parameters

Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

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