Iterative surrogate model optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks

Lye, Kjetil Olsen; Mishra, Siddhartha; Ray, Deep; Chandrashekar, Praveen

doi:10.1016/j.cma.2020.113575

Cited by 52 publications

(38 citation statements)

References 20 publications

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“…In addition, the shape functions can be computed in parallel, which boosts their applications in analysis of super-large problems, although their achievement still depends on the availability of appropriate computational facilities. Developing a surrogate model by using techniques on model reduction [42,35] or deep learning [43,44] is expected to help resolve the existing limitations and should be explored in future studies.…”

Section: Discussionmentioning

confidence: 99%

Analysis of Heterogeneous Structures of Non-separated Scales on Curved Bridge Nodes

Li,

2021

Preprint

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Numerically predicting the performance of heterogenous structures without scale separation represents a significant challenge to meet the critical requirements on computational scalability and efficiency -adopting a mesh fine enough to fully account for the small-scale heterogeneities leads to prohibitive computational costs while simply ignoring these fine heterogeneities tends to drastically over-stiffen the structure's rigidity.This study proposes an approach to construct new material-aware shape (basis) functions per element on a coarse discretization of the structure with respect to each curved bridge nodes (CBNs) defined along the elements' boundaries. Instead of formulating their derivation by solving a nonlinear optimization problem, the shape functions are constructed by building a map from the CBNs to the interior nodes and are ultimately presented in an explicit matrix form as a product of a Bézier interpolation transformation and a boundary-interior transformation. The CBN shape function accomodates more flexibility in closely capturing the coarse element's heterogeneity, overcomes the important and challenging issues of inter-element stiffness and displacement discontinuity across interface between coarse elements, and improves the analysis accuracy by orders of magnitude; they also meet the basic geometric properties of shape functions that avoid aphysical analysis results. Extensive numerical examples, including a 3D industrial example of billions of degrees of freedom, are also tested to demonstrate the approach's performance in comparison with results obtained from classical approaches.

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Section: Discussionmentioning

confidence: 99%

Analysis of Heterogeneous Structures of Non-separated Scales on Curved Bridge Nodes

Li,

2021

Preprint

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“…In their active learning, they chose the design point with the largest uncertainty quantified by the network model. Lye et al [14] proposed active learning for surrogate modeling of PDE solutions that chooses the next query point minimizing the cost function with the sequentially updated DNN model. In order to ensure the feasibility, they confined the explorable settings with the feasible region known a priori.…”

Section: A Active Learning For Engineering Systemsmentioning

confidence: 99%

Failure-averse Active Learning for Physics-constrained Systems

Lee¹,

Wang²,

Wu³

et al. 2021

Preprint

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Active learning is a subfield of machine learning that is devised for design and modeling of systems with highly expensive sampling costs. Industrial and engineering systems are generally subject to physics constraints that may induce fatal failures when they are violated, while such constraints are frequently underestimated in active learning. In this paper, we develop a novel active learning method that avoids failures considering implicit physics constraints that govern the system. The proposed approach is driven by two tasks: the safe variance reduction explores the safe region to reduce the variance of the target model, and the safe region expansion aims to extend the explorable region exploiting the probabilistic model of constraints. The global acquisition function is devised to judiciously optimize acquisition functions of two tasks, and its theoretical properties are provided. The proposed method is applied to the composite fuselage assembly process with consideration of material failure using the Tsai-wu criterion, and it is able to achieve zero-failure without the knowledge of explicit failure regions.

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“…Moreover, deep neural networks are being increasingly used successfully in scientific computing, particular in simulating physical and engineering systems modeled by partial differential equations (PDEs). Examples include the use of physics informed neural networks [29,30,27,28] for solving forward and inverse problems for PDEs and supervised learning algorithms for high-dimensional parabolic PDEs [11] and parametric elliptic [14,32] and hyperbolic [22,23] PDEs, among others.…”

Section: Introductionmentioning

confidence: 99%

On universal approximation and error bounds for Fourier Neural Operators

Kovachki¹,

Lanthaler²,

Mishra³

2021

Preprint

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Fourier neural operators (FNOs) have recently been proposed as an effective framework for learning operators that map between infinitedimensional spaces. We prove that FNOs are universal, in the sense that they can approximate any continuous operator to desired accuracy. Moreover, we suggest a mechanism by which FNOs can approximate operators associated with PDEs efficiently. Explicit error bounds are derived to show that the size of the FNO, approximating operators associated with a Darcy type elliptic PDE and with the incompressible Navier-Stokes equations of fluid dynamics, only increases sub (log)-linearly in terms of the reciprocal of the error. Thus, FNOs are shown to efficiently approximate operators arising in a large class of PDEs.

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Iterative surrogate model optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks

Cited by 52 publications

References 20 publications

Analysis of Heterogeneous Structures of Non-separated Scales on Curved Bridge Nodes

Analysis of Heterogeneous Structures of Non-separated Scales on Curved Bridge Nodes

Failure-averse Active Learning for Physics-constrained Systems

On universal approximation and error bounds for Fourier Neural Operators

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