• High Power Laser Science and Engineering
  • Vol. 11, Issue 5, 05000e55 (2023)
Andreas Döpp1、2、*, Christoph Eberle1, Sunny Howard1、2, Faran Irshad1, Jinpu Lin1, and Matthew Streeter3
Author Affiliations
  • 1Ludwig-Maximilians-Universität München, Garching, Germany
  • 2Department of Physics, Clarendon Laboratory, University of Oxford, Oxford, UK
  • 3School for Mathematics and Physics, Queen’s University Belfast, Belfast, UK
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    DOI: 10.1017/hpl.2023.47 Cite this Article Set citation alerts
    Andreas Döpp, Christoph Eberle, Sunny Howard, Faran Irshad, Jinpu Lin, Matthew Streeter. Data-driven science and machine learning methods in laser–plasma physics[J]. High Power Laser Science and Engineering, 2023, 11(5): 05000e55 Copy Citation Text show less

    Abstract

    Laser-plasma physics has developed rapidly over the past few decades as lasers have become both more powerful and more widely available. Early experimental and numerical research in this field was dominated by single-shot experiments with limited parameter exploration. However, recent technological improvements make it possible to gather data for hundreds or thousands of different settings in both experiments and simulations. This has sparked interest in using advanced techniques from mathematics, statistics and computer science to deal with, and benefit from, big data. At the same time, sophisticated modeling techniques also provide new ways for researchers to deal effectively with situation where still only sparse data are available. This paper aims to present an overview of relevant machine learning methods with focus on applicability to laser-plasma physics and its important sub-fields of laser-plasma acceleration and inertial confinement fusion.
    $$\begin{align}\log p\left(y|\theta \right) = \sum \limits_{i = 1}^n\log p\left({y}_i|\theta \right).\end{align}$$ ((1))

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    $$\begin{align}{\widehat{\theta}}_{\mathrm{MLE}} = \underset{\theta }{\arg \max}\left\{p\left(y|\theta \right)\right\} = \underset{\theta }{\arg \min}\left\{\log p\left(y|\theta \right)\right\}.\end{align}$$ ((2))

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    $$\begin{align}p\left(\theta |y\right) = \frac{p\left(y|\theta \right)p\left(\theta \right)}{p(y)},\end{align}$$ ((3))

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    $$\begin{align}{\widehat{\theta}}_{\mathrm{MAP}} = \underset{\theta }{\arg \max}\left\{p\left(\theta |y\right)\right\}.\end{align}$$ ((4))

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    $$\begin{align}f(x)\sim \mathcal{GP}\left(\mu (x),\sigma \left(x,{x}^{\prime}\right)\right),\end{align}$$ ((5))

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    $$\begin{align}\sigma \left(x,{x}^{\prime}\right) = \exp \left(-\frac{{\left(x-{x}^{\prime}\right)}^2}{2{\mathrm{\ell}}^2}\right),\end{align}$$ ((6))

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    $$\begin{align}\sigma \left(x,{x}^{\prime}\right) = \exp \left(-\frac{2\sin^2\left(\pi d\left(x,{x}^{\prime}\right)/\lambda \right)}{{\mathrm{\ell}}^2}\right),\end{align}$$ ((7))

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    $$\begin{align}\mathrm{ReLU}(x) = \left\{\!\!\begin{array}{ll}x,& \mathrm{if}\;x\ge 0,\\ {}0,& \mathrm{otherwise}.\end{array}\right.\end{align}$$ ((8))

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    $$\begin{align}{y}_t = \sum \limits_{i = 1}^p{\varphi}_i{y}_{t-i}+{\varepsilon}_t,\end{align}$$ ((9))

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    $$\begin{align}\left\{{\widehat{\varphi}}_1,\dots, {\widehat{\varphi}}_p\right\} = \mathop{\mathrm{argmin}}_{{\varphi_1,\dots, {\varphi}_p}}{\left({y}_t-\sum \limits_{i = 1}^p{\varphi}_i{y}_{t-i}\right)}^2.\end{align}$$ ((10))

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    $$\begin{align}{y}_t = \mu +\sum \limits_{i = 1}^q{\vartheta}_i{\varepsilon}_{t-i}+{\varepsilon}_t,\end{align}$$ ((11))

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    $$\begin{align}{y}_t = \mu +\sum \limits_{i = 1}^p{\varphi}_i{y}_{t-i}+\sum \limits_{j = 1}^q{\vartheta}_j{\varepsilon}_{t-j}+{\varepsilon}_t.\end{align}$$ ((12))

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    $$\begin{align}{z}_t = \sum \limits_{i = 1}^p{\varphi}_i{z}_{t-i}+\sum \limits_{j = 1}^q{\vartheta}_j{\varepsilon}_{t-j}+{\varepsilon}_t.\end{align}$$ ((13))

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    $$\begin{align}{x}_t &= {A}_t{x}_{t-1}+{B}_t{u}_t+{C}_t{\varepsilon}_t,\nonumber\\ {}{y}_t &= {D}_t{x}_t+{E}_t{\eta}_t,\end{align}$$ ((14))

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    $$\begin{align}{x}_t = f\left({w}_{\mathrm{rec}}{x}_{t-1}+{b}_{\mathrm{rec}}\right),\end{align}$$ ((15))

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    $$\begin{align}{y}_t = f\left({w}_{\mathrm{out}}{x}_t+{b}_{\mathrm{out}}\right),\end{align}$$ ((16))

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    $$\begin{align}{x}_t = {f}_t\odot {x}_{t-1}+{i}_t\odot {h}_t,\end{align}$$ ((17))

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    $$\begin{align}{\tilde{h}}_t = \sum {\alpha}_{ti}{h}_i, \end{align}$$ ((18))

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    $$\begin{align}Ax = y,\end{align}$$ ((19))

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    $$\begin{align}{\widehat{x}}_{\mathrm{LS}} = \underset{x}{\mathrm{argmin}\kern0.1em }\left\{{\left\Vert Ax-y\right\Vert}^2\right\}.\end{align}$$ ((20))

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    $$\begin{align}{\widehat{x}}_{\mathrm{REG}} = \underset{x}{\mathrm{argmin}\kern0.1em }\left\{{\left\Vert Ax-y\right\Vert}^2+\lambda \mathrm{\mathcal{R}}(x)\right\},\end{align}$$ ((21))

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    $$\begin{align}{\widehat{x}}_{\mathrm{CS}} = \underset{\tilde{x}}{\mathrm{argmin}\kern0.1em }\left\{{\left\Vert A\Psi \tilde{x}-y\right\Vert}^2+\lambda {\left\Vert \tilde{x}\right\Vert}_1\right\}.\end{align}$$ ((22))

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    $$\begin{align}{\widehat{x}}_{\mathcal{A}} = \mathcal{A}y.\end{align}$$ ((23))

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    $$\begin{align}{\mathrm{\ell}}_p\left(x,y\right) = {\left(\sum \parallel x-y{\parallel}^p\right)}^{1/p},\end{align}$$ ((24))

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    $$\begin{align}\mathrm{KL}\left(p|q\right) = \sum p(x)\log \frac{p(x)}{q(x)},\end{align}$$ ((25))

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    $$\begin{align}\mathrm{CE}\left(p,q\right) = -\sum p(x)\log q(x) = \mathrm{KL}\left(p|q\right)-H(p),\end{align}$$ ((26))

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    $$\begin{align}\mathrm{UCB}(x) = \mu (x)+\kappa \sigma (x),\end{align}$$ ((27))

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    $$\begin{align}{r}_{xy} = \frac{\sum_{i = 1}^n\left({x}_i-\overline{x}\right)\left({y}_i-\overline{y}\right)}{\sqrt{\sum_{i = 1}^n{\left({x}_i-\overline{x}\right)}^2}\sqrt{\sum_{i = 1}^n{\left({y}_i-\overline{y}\right)}^2}},\end{align}$$ ((28))

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    Andreas Döpp, Christoph Eberle, Sunny Howard, Faran Irshad, Jinpu Lin, Matthew Streeter. Data-driven science and machine learning methods in laser–plasma physics[J]. High Power Laser Science and Engineering, 2023, 11(5): 05000e55
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