Automatic Differentiation Applied for Optimization of Dynamical Systems

Enciu, Petre; Gerbaud, Laurent; Würtz, Fréderic

doi:10.1109/tmag.2010.2044770

Cited by 20 publications

(10 citation statements)

References 8 publications

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“…Theorem 2 Let initial data y n and dy n d p be given. The algorithmic derivative of a single step of the scheme ( 14), (15) with step size h applied to (8) yields the same value dy n+1 d p as an application of the same integration step to the combined system ( 8) and ( 9) as long as dh d p = 0 and as long as the derivatives of equation solves are recovered according to the implicit function theorem. In terms of automatic differentiation, it is sufficient if h does not carry derivative values and equation solves are treated as elementary operations.…”

Section: Rosenbrock and Runge-kutta Schemes With Adaptive Step Sizementioning

confidence: 99%

“…In [62], fixed step size explicit Runge-Kutta methods are applied for the discretization of optimal control problems and it is shown that the sensitivities obtained by blackbox forward AD are consistent with the corresponding tangent linear model. [15] reports on an application of AD to explicit Runge-Kutta methods with adaptive step size control. Further research on AD of ODE integration schemes was conducted in an optimal control context with a focus on the reverse mode of AD [48,49], including adaptive step sizes [2] and reverse mode specifics such as interpolation strategies for the forward solution [1].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

AutoMat: automatic differentiation for generalized standard materials on GPUs

2021

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We propose a universal method for the evaluation of generalized standard materials that greatly simplifies the material law implementation process. By means of automatic differentiation and a numerical integration scheme, AutoMat reduces the implementation effort to two potential functions. By moving AutoMat to the GPU, we close the performance gap to conventional evaluation routines and demonstrate in detail that the expression level reverse mode of automatic differentiation as well as its extension to second order derivatives can be applied inside CUDA kernels. We underline the effectiveness and the applicability of AutoMat by integrating it into the FFT-based homogenization scheme of Moulinec and Suquet and discuss the benefits of using AutoMat with respect to runtime and solution accuracy for an elasto-viscoplastic example.

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Section: Rosenbrock and Runge-kutta Schemes With Adaptive Step Sizementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

AutoMat: automatic differentiation for generalized standard materials on GPUs

2021

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“…In the second case, automatic derivation is used [6]. In CADES, the analytical models are translated into software components that allow the computation of their outputs and their jacobian according to their inputs.…”

Section: Modelling Approachmentioning

confidence: 99%

“…-using Newton-Raphson algorithm (NR) that gives the set of solving parameters at each iteration of the optimization. The derivative of such algorithm is done using the implicit function theorem [6].…”

Section: Magnetic Model R(t)i² Thermal Model (Joule Losses)mentioning

confidence: 99%

Multiphysical approach including equivalent circuit models for the sizing by optimization

Baraston¹,

Gerbaud²,

Reinbold³

et al. 2016

COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering

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Purpose Multiphysical models are often useful for the design of electrical devices such as electrical machines. In this way, the modeling of thermal, magnetic and electrical phenomena by using an equivalent circuit approach is often used in sizing problems. The coupling of such models with other models is difficult to take into account, partly because it adds complexity to the process. The paper proposes an automatic modelling of thermal and magnetic aspects from an equivalent circuit approach, with its computation of gradients, using selectivity on the variables. Then, it discusses the coupling of various physical models, for the sizing by optimization algorithms. Sensibility analyses are discussed and the multiphysical approach is applied on a permanent magnet synchronous machine. Design/methodology/approach The paper allows one to describe thermal and magnetic models by equivalent circuits. Magnetic aspects are represented by reluctance networks and thermal aspects by thermal equivalent circuits. From circuit modelling and analytical equations, models are generated, coupled and translated into computational codes (Java, C), including the computation of their jacobians. To do so, model generators are used: CADES, Reluctool, Thermotool. The paper illustrates the modelling and automatic programming aspects with Thermotool. The generated codes are directly available for optimization algorithms. Then, the formulation of the coupling with other models is studied in the case of a multiphysical sizing by optimization of the Toyota PRIUS electrical motor. Findings A main specificity of the approach is the ability to easily deal with the selectivity of the inputs and outputs of the generated model according to the problem specifications, thus reducing drastically the size of the jacobian matrix and the computational complexity. Another specificity is the coupling of the models using analytical equations, possibly implicit equations. Research limitations/implications At the present time, the multiphysical modeling is considered only for static phenomena. However, this limit is not important for numerous sizing applications. Originality/value The analytical approach with the selectivity gives fast models, well-adapted for optimization. The use of model generators allows robust programming of the models and their jacobians. The automatic calculation of the gradients allows the use of determinist algorithms, such as SQP, well adapted to deal with numerous constraints.

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“…Over the past several decades, automatic differentiation has been explored across a number of contexts [12][13][14][15][16][17] . However, in recent years the growing interest in machine learning, and particularly gradient-based model training, has driven the development modern automatic differentiation libraries.…”

Section: Introductionmentioning

confidence: 99%

Inverse design of photonic crystals through automatic differentiation

Minkov,

Williamson,

Andreani

et al. 2020

Preprint

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Gradient-based inverse design in photonics has already achieved remarkable results in designing small-footprint, high-performance optical devices. The adjoint variable method, which allows for the efficient computation of gradients, has played a major role in this success. However, gradientbased optimization has not yet been applied to the mode-expansion methods that are the most common approach to studying periodic optical structures like photonic crystals. This is because, in such simulations, the adjoint variable method cannot be defined as explicitly as in standard finitedifference or finite-element time-or frequency-domain methods. Here, we overcome this through the use of automatic differentiation, which is a generalization of the adjoint variable method to arbitrary computational graphs. We implement the plane-wave expansion and the guided-mode expansion methods using an automatic differentiation library, and show that the gradient of any simulation output can be computed efficiently and in parallel with respect to all input parameters. We then use this implementation to optimize the dispersion of a photonic crystal waveguide, and the quality factor of an ultra-small cavity in a lithium niobate slab. This extends photonic inverse design to a whole new class of simulations, and more broadly highlights the importance that automatic differentiation could play in the future for tracking and optimizing complicated physical models.

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Automatic Differentiation Applied for Optimization of Dynamical Systems

Cited by 20 publications

References 8 publications

AutoMat: automatic differentiation for generalized standard materials on GPUs

AutoMat: automatic differentiation for generalized standard materials on GPUs

Multiphysical approach including equivalent circuit models for the sizing by optimization

Inverse design of photonic crystals through automatic differentiation

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