A learning-based multiscale method and its application to inelastic impact problems

Liu, Burigede; Kovachki, Nikola B.; Li, Zongyi; Azizzadenesheli, Kamyar; Anandkumar, Anima; Stuart, Andrew M.; Bhattacharya, Kaushik

doi:10.1016/j.jmps.2021.104668

Cited by 70 publications

(99 citation statements)

References 79 publications

(103 reference statements)

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“…The computational performance and inference efficiency of our hybrid solvers can be further optimized by training the neural network itself using bfloat16 [26], applying convolutions in the Fourier domain [40], and applying model quantization techniques [41]. The GPU memory savings obtained from the reduced numerical precision can enable simulations with a greater number of mesh elements than would otherwise be possible.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Deep Neural Networks to Correct Sub-Precision Errors in CFD

Haridas¹,

Vadlamani²,

Minamoto³

2022

Preprint

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Loss of information in numerical simulations can arise from various sources while solving discretized partial differential equations. In particular, precisionrelated errors can accumulate in the quantities of interest when the simulations are performed using low-precision 16-bit floating-point arithmetic compared to an equivalent 64-bit simulation. Here, low-precision computation requires much lower resources than high-precision computation. Several machine learning (ML) techniques proposed recently have been successful in correcting the errors arising from spatial discretization. In this work, we extend these techniques to improve Computational Fluid Dynamics (CFD) simulations performed using low numerical precision. We first quantify the precision related errors accumulated in a Kolmogorov forced turbulence test case. Subsequently, we employ a Convolutional Neural Network together with a fully differentiable numerical solver performing 16-bit arithmetic to learn a tightly-coupled ML-CFD hybrid solver 1 . Compared to the 16-bit

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Section: Conclusion and Discussionmentioning

confidence: 99%

Deep Neural Networks to Correct Sub-Precision Errors in CFD

Haridas¹,

Vadlamani²,

Minamoto³

2022

Preprint

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“…Or we can model the microscopic problem but this requires additional information thereby defeating the purpose of multiscale modeling. So we seek to construct a surrogate Φ app by learning the full solution operator of the microscale problem with no further empirical or expert input following [56].…”

Section: Machine-learning Materials Behaviormentioning

confidence: 99%

“…We have demonstrated the approach to the problems of impact of polycrystalline magnesium in Figure 7 [56]. High fidelity crystal plasticity unit cell calculations were used to create a machine learned surrogate that accurately predicts the material response against various strain histories (e.g.…”

Section: Machine-learning Materials Behaviormentioning

confidence: 99%

“…All of this leads to the question: Can we use use techniques from machine learning, such as Gaussian processes, deep neural networks or other supervised learning techniques to link scales? Section 3 reviews the recent work of Liu et al [56] that studies the two scale problem of the response of polycrystalline magnesium to impact loading. The key idea is to view the lower-scale model (a polycrystalline ensemble described by crystal plasticity augmented to include twinning in this case) as a map from a strain history to a stress history, and to approximate this map using a combination of model reduction and machine learning [13].…”

Section: Introductionmentioning

confidence: 99%

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Multiscale modeling of materials: Computing, data science,uncertainty and goal-oriented optimization

Kovachki¹,

Liu²,

Sun³

et al. 2021

Preprint

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The recent decades have seen various attempts at accelerating the process of developing materials targeted towards specific applications. The performance required for a particular application leads to the choice of a particular material system whose properties are optimized by manipulating its underlying microstructure through processing. The specific configuration of the structure is then designed by characterizing the material in detail, and using this characterization along with physical principles in system level simulations and optimization. These have been advanced by multiscale modeling of materials, high-throughput experimentations, materials data-bases, topology optimization and other ideas. Still, developing materials for extreme applications involving large deformation, high strain rates and high temperatures remains a challenge. This article reviews a number of recent methods that advance the goal of designing materials targeted by specific applications.

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“…Strategies to approximate solutions of partial differential equations (PDEs) based on neural networks can be traced back to Dissanayake and Phan-Thien (1994) and Lagaris et al (1998). Their approaches are currently being revived and extended thanks to increased computational power and ease of implementation in frameworks like Tensorflow and PyTorch, see Sirignano and Spiliopoulos (2018); E and Yu (2018); Raissi and Karniadakis (2018); Li et al (2020); Abadi et al (2016); Paszke et al (2019).…”

Section: Introductionmentioning

confidence: 99%

Robin Pre-Training for the Deep Ritz Method

Courte¹,

Zeinhofer²

2021

Preprint

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We compare different training strategies for the Deep Ritz Method for elliptic equations with Dirichlet boundary conditions and highlight the problems arising from the boundary values. We distinguish between an exact resolution of the boundary values by introducing a distance function and the approximation through a Robin Boundary Value problem. However, distance functions are difficult to obtain for complex domains. Therefore, it is more feasible to solve a Robin Boundary Value problem which approximates the solution to the Dirichlet Boundary Value problem, yet the naïve approach to this problem becomes unstable for large penalizations. A novel method to compensate this problem is proposed using a small penalization strength to pre-train the model before the main training on the target penalization strength is conducted. We present numerical and theoretical evidence that the proposed method is beneficial.

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A learning-based multiscale method and its application to inelastic impact problems

Cited by 70 publications

References 79 publications

Deep Neural Networks to Correct Sub-Precision Errors in CFD

Deep Neural Networks to Correct Sub-Precision Errors in CFD

Multiscale modeling of materials: Computing, data science,uncertainty and goal-oriented optimization

Robin Pre-Training for the Deep Ritz Method

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