# What Machine Learning Can and Cannot Do for Inertial Confinement Fusion

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## Abstract

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## 1. Introduction

## 2. Machine Learning and Limitations

## 3. Inertial Confinement Fusion

## 4. Tasks Good for Machine Learning

## 5. Physics-Guided Deep Learning

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Mitchell, T. Machine Learning; McGraw Hill: New York, NY, USA, 1997; ISBN 0–07-042807-7. [Google Scholar]
- Bengio, Y.; LeCun, Y.; Hinton, G. Deep Learning. Nature
**2015**, 521, 436–444. [Google Scholar] [CrossRef] - Udrescu, S.-M.; Tegmark, M. AI Feynman: A physics-inspired method for symbolic regression. Sci. Adv.
**2020**, 6, eaay2531. [Google Scholar] [CrossRef] [PubMed] - Ng, A. How Artificial Intelligence Is Transforming the Industry. 2021. Available online: https://www.bosch.com/stories/artificial-intelligence-in-industry/ (accessed on 29 July 2022).
- Hatfield, P.W.; Gaffney, J.A.; Anderson, G.J.; Ali, S.; Antonelli, L.; Başeğmez du Pree, S.; Citrin, J.; Fajardo, M.; Knapp, P.; Kettle, B.; et al. The data-driven future of high-energy-density physics. Nature
**2021**, 593, 351–361. [Google Scholar] [CrossRef] - Humphreys, D.; Kupresanin, A.; Boyer, M.D.; Canik, J.; Chang, C.S.; Cyr, E.C.; Granetz, R.; Hittinger, J.; Kolemen, E.; Lawrence, E.; et al. Advancing Fusion with Machine Learning Research Needs Workshop Report. J. Fusion Energy
**2020**, 39, 123–155. [Google Scholar] [CrossRef] - Hopfield, J.J. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA
**1982**, 79, 2554–2558. [Google Scholar] [CrossRef] [PubMed] - Kim, E.J.; Brunner, R.J. Star–galaxy classification using deep convolutional neural networks. MNRAS
**2017**, 464, 4463–4475. [Google Scholar] [CrossRef] - Iten, R.; Metger, T.; Wilming, H.; del Rio, L.; Renner, R. Discovering Physical Concepts with Neural Networks. Phys. Rev. Lett.
**2020**, 124, 010508. [Google Scholar] [CrossRef] - Keshavan, A.; Yeatman, J.D.; Rokem, A. Combining Citizen Science and Deep Learning to Amplify Expertise in Neuroimaging. Front. Neuroinform.
**2019**, 13, 29. [Google Scholar] [CrossRef] - Beck, M.R.; Scarlata, C.; Fortson, L.F.; Lintott, C.J.; Simmons, B.D.; Galloway, M.A.; Willett, K.W.; Dickinson, H.; Masters, K.L.; Marshall, P.J.; et al. Integrating human and machine intelligence in galaxy morphology classification tasks. MNRAS
**2018**, 476, 5516–5534. [Google Scholar] [CrossRef] - Atzeni, S.; Meyer-ter Vehn, J. 2004 The Physics of Inertial Fusion: BeamPlasma Interaction, Hydrodynamics, Hot Dense Matter International Series of Monographs on Physics; Clarendon Press: Oxford, UK, 2004. [Google Scholar]
- Lindl, J. Inertial Confinement Fusion: The Quest for Ignition and Energy Gain Using Indirect Drive; AIP Press: College Park, MD, USA, 1998. [Google Scholar]
- Mehta, P.; Bukov, M.; Wang, C.-H.; Day, A.G.R.; Richardson, C.; Fisher, C.K.; Schwab, D.J. A high-bias, low-variance introduction to Machine Learning, for physicists. Phys. Rep.
**2019**, 810, 1–124. [Google Scholar] [CrossRef] - Rasmussen, C.E.; Williams, C.K.I. 2006 Gaussian Processes for Machine Learning; MIT Press: Cambridge, MA, USA, 2006. [Google Scholar]
- Pearson, K. On Lines and Planes of Closest Fit to Systems of Points in Space. Philos. Mag.
**1901**, 2, 559–572. [Google Scholar] [CrossRef] - Atwell, J.A.; King, B.B. Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Model.
**2001**, 33, 1–19. [Google Scholar] [CrossRef] - Hinton, G.E.; Salakhutdinov, R.R. Reducing the dimensionality of data with neural networks. Science
**2006**, 313, 504–507. [Google Scholar] [CrossRef] [PubMed] - Kates-Harbeck, J.; Svyatkovskiy, A.; Tang, W. Predicting disruptive instabilities in controlled fusion plasmas through deep learning. Nature
**2019**, 568, 526. [Google Scholar] [CrossRef] - Lee, K.; Carlberg, K. Model Reduction of Dynamical Systems on Nonlinear Manifolds Using Deep Convolutional Autoencoders. arXiv
**2018**, arXiv:1812.08373. [Google Scholar] [CrossRef] - Alibrahim, H.; Ludwig, S.A. Hyperparameter Optimization: Comparing Genetic Algorithm against Grid Search and Bayesian Optimization. In Proceedings of the 2021 IEEE Congress on Evolutionary Computation (CEC), Kraków, Poland, 28 June–1 July 2021. [Google Scholar]
- Andrieu, C.; De Freitas, N.; Doucet, A.; Jordan, M.I. An Introduction to MCMC for Machine Learning. Mach. Learn.
**2003**, 50, 5–43. [Google Scholar] [CrossRef] - Feurer, M.; Hutter, F. Automated Machine Learning: Methods, Systems, Challenges; The Springer Series on Challenges in Machine Learning; Springer: Berlin/Heidelberg, Germany, 2019. [Google Scholar] [CrossRef]
- Moonen, M.; Moor, B.D.; Vandenberghe, L.; Vandewalle, J. On- and Off-Line Identification of Linear State Space Models. Int. J. Control
**1989**, 49, 219–232. [Google Scholar] [CrossRef] - Viberg, M. Subspace-based Methods for the Identification of Linear Time-invariant Systems. Automatica
**1995**, 31, 1835–1851. [Google Scholar] [CrossRef] - Abiodun, O.I.; Jantan, A.; Omolara, A.E.; Dada, K.V.; Mohamed, N.A.; Arshad, H. State-of-the-art in artificial neural network applications: A survey. Heliyon
**2018**, 4, e00938. [Google Scholar] [CrossRef] - Dupond, S. A thorough review on the current advance of neural network structures. Annu. Rev. Control
**2019**, 14, 200–230. [Google Scholar] - Tealab, A. Time series forecasting using artificial neural networks methodologies: A systematic review. Future Comput. Inform. J.
**2018**, 3, 334–340. [Google Scholar] [CrossRef] - Gaffney, J.A.; Brandon, S.T.; Humbird, K.D.; Kruse, M.K.G.; Nora, R.C.; Peterson, J.L.; Spears, B.K. Making inertial confinement fusion models more predictive. Phys. Plasmas
**2019**, 26, 082704. [Google Scholar] [CrossRef] - Spears, B.K.; Brase, J.; Bremer, P.-T.; Chen, B.; Field, J.; Gaffney, J.; Kruse, M.; Langer, S.; Lewis, K.; Nora, R.; et al. Deep learning: A guide for practitioners in the physical sciences. Phys. Plasmas
**2018**, 25, 080901. [Google Scholar] [CrossRef] - Kritcher, A.L.; Young, C.V.; Robey, H.F.; Weber, C.R.; Zylstra, A.B.; Hurricane, O.A.; Callahan, D.A.; Ralph, J.E.; Ross, J.S.; Baker, K.L.; et al. Design of inertial fusion implosions reaching the burning plasma regime. Nat. Phys.
**2022**, 18, 251–258. [Google Scholar] [CrossRef] - Zylstra, A.B.; Hurricane, O.A.; Callahan, D.A.; Kritcher, A.; Ralph, J.E.; Robey, H.F.; Ross, J.S.; Young, C.V.; Baker, K.L.; Casey, D.T.; et al. Burning plasma achieved in inertial fusion. Nature
**2022**, 601, 542–548. [Google Scholar] [CrossRef] - Cheng, B.; Kwan, T.J.T.; Wang, Y.-M.; Merrill, F.E.; Cerjan, C.J.; Batha, S.H. Analysis of NIF experiments with the minimal energy implosion model. Phys. Plasmas
**2015**, 22, 082704. [Google Scholar] [CrossRef] - Cheng, B.; Kwan, T.J.T.; Wang, Y.-M.; Batha, S.H. On Thermonuclear ignition criterion at the National Ignition Facility. Phys. Plasmas
**2014**, 21, 102707. [Google Scholar] [CrossRef] - Cheng, B.; Bradley, P.A.; Finnagan, S.A.; Thomas, C.A. Fundamental factors affecting thermonuclear ignition. Nucl. Fusion
**2020**, 61, 096010. [Google Scholar] [CrossRef] - Cheng, B.; Kwan, T.J.T.; Wang, Y.-M.; Batha, S.H. Scaling laws for ignition at the National Ignition Facility from first principles. Phys. Rev. E
**2013**, 88, 041101. [Google Scholar] [CrossRef] - Cheng, B.; Kwan, T.J.T.; Wang, Y.-M.; Yi, S.A.; Batha, S.H.; Wysocki, F.J. Ignition and pusher adiabat. Phys. Control. Fusion
**2018**, 60, 074011. [Google Scholar] [CrossRef] - Cheng, B.; Kwan, T.J.T.; Wang, Y.-M.; Yi, S.A.; Batha, S.H.; Wysocki, F.J. Effects of preheat and mix on the fuel adiabat of an imploding capsule. Phys. Plasmas
**2016**, 23, 120702. [Google Scholar] [CrossRef] - Cheng, B.; Kwan, T.J.T.; Yi, S.A.; Landen, O.L.; Wang, Y.-M.; Cerjan, C.J.; Batha, S.H.; Wysocki, F.J. Effects of asymmetry and hot-spot shape on ignition capsules. Phys. Rev. E
**2018**, 98, 023203. [Google Scholar] [CrossRef] [PubMed] - Edwards, M.J.; Patel, P.K.; Lindl, J.D.; Atherton, L.J.; Glenzer, S.H.; Haan, S.W.; Kilkenny, J.D.; Landen, O.L.; Moses, E.I.; Nikrooet, A.; et al. Progress towards ignition on the national ignition facility. Phys. Plasmas
**2013**, 20, 070501. [Google Scholar] [CrossRef] - Nakhleh, J.B.; Fernández-Godino, M.G.; Grosskopf, M.J.; Wilson, B.M.; Kline, J.; Srinivasan, G. Exploring Sensitivity of ICF Outputs to Design Parameters in Experiments Using Machine Learning. IEEE Trans. Plasma Sci.
**2021**, 49, 2238–2246. [Google Scholar] [CrossRef] - Vazirani, N.N.; Grosskopf, M.J.; Stark, D.J.; Bradley, P.A.; Haines, B.M.; Loomis, E.; England, S.L.; Scales, W.A. Coupling 1D xRAGE simulations with machine learning for graded inner shell design optimization in double shell capsules. Phys. Plasmas
**2021**, 28, 122709. [Google Scholar] [CrossRef] - Peterson, J.L.; Humbird, K.D.; Field, J.E.; Brandon, S.T.; Langer, S.H.; Nora, R.C.; Spears, B.K.; Springer, P.T. Zonal flow generation in inertial confinement fusion implosions. Phys. Plasmas
**2017**, 24, 032702. [Google Scholar] [CrossRef] - Melvin, J.; Lim, H.; Rana, V.; Cheng, B.; Glimm, J.; Sharp, D.H.; Wilson, D.C. Sensitivity of inertial confinement fusion hot spot properties to the deuterium-tritium fuel adiabat. Phys. Plasmas
**2015**, 22, 022708. [Google Scholar] [CrossRef] - Vander Wal, M.D.; McClarren, R.G.; Humbird, K.D. Transfer learning of hight-fidelity opacity spectra in autoencoders and surrogate models. arXiv
**2022**, arXiv:2203.00853. [Google Scholar] - Michoski, C.; Milosavljevic, M.; Oliver, T.; Hatch, D. Solving Irregular and Data-Enriched Differential Equations Using Deep Neural Networks. arXiv
**2019**, arXiv:1905.04351. [Google Scholar] - Humbird, K.D.; Peterson, J.L.; Salmonson, J.; Spears, B.K. Cognitive simulation models for inertial confinement fusion: Combining simulation and experimental data. Phys. Plasmas
**2021**, 28, 042709. [Google Scholar] [CrossRef] - Gopalaswamy, V.; Betti, R.; Knauer, J.P.; Luciani, N.; Patel, D.; Woo, K.M.; Bose, A.; Igumenshchev, I.V.; Campbell, E.M.; Anderson, K.S.; et al. Tripled yield in direct-drive laser fusion through statistical modelling. Nature
**2019**, 565, 581–586. [Google Scholar] [CrossRef] - Ross, J.S.; Ralph, J.E.; Zylstra, J.E.A.B.; Kritcher, A.L.; Robey, H.F.; Young, C.V.; Hurricane, O.A.; Callahan, D.A.; Baker, K.L.; Casey, D.T.; et al. Experiments conducted in the burning plasma regime with inertial fusion implosions. arXiv
**2021**, arXiv:2111.04640. [Google Scholar] - Abu-Shawared, H.; Acree, R.; Adams, P.; Adams, J.; Addis, B.; Aden, R.; Adrian, P.; Afeyan, B.B.; Aggleton, M.; Indirect Drive ICF Collaboration; et al. Lawson’s criteria for ignition exceeded in an inertial fusion experiment. Phys. Rev. Lett.
**2022**, 129, 075001. [Google Scholar] [CrossRef] [PubMed] - Hsu, A.; Cheng, B.; Bradley, P.A. Analysis of NIF scaling using physics informed machine learning. Phys. Plasmas
**2020**, 27, 012703. [Google Scholar] [CrossRef] - Kramer, O. K-Nearest Neighbors. Dimensionality Reduction with Unsupervised Nearest Neighbors; Springer: Berlin/Heidelberg, Germany, 2013; pp. 13–23. [Google Scholar]
- Liu, W.; Principe, J.C.; Haykin, S.S. Kernel Adaptive Filtering: A Comprehensive Introduction, 1st ed.; Wiley: Hoboken, NJ, USA, 2010. [Google Scholar]
- Vapnik, V.N. The Nature of Statistical Learning Theory; Springer: New York, NY, USA, 2000. [Google Scholar]
- Humbird, K.D.; Peterson, J.L.; Mcclarren, R.G. Deep Neural Network Initialization With Decision Trees. IEEE Trans. Neural Networks Learn. Syst.
**2019**, 30, 1286. [Google Scholar] [CrossRef]

**Figure 1.**A scheme of artificial intelligence, machine learning, deep learning, and physics-guided deep learning, where ANN, CNN, RNN, and DL, respectively, represent artificial neural networks, convolutional neural networks, recurrent neural networks, and deep learning.

Successful areas | Pattern recognition, image classification, cancer diagnosis, and systems with the following features: (a) large digital datasets (inputs, outputs), clear goals, and metrics; (b) not dominated by a long chain of logic and reasoning; (c) no requirement for diverse background knowledge and explanation of decision process; (d) high tolerance for errors and no requirement for provably correct or optimal solutions. |

Inherent limitations | Unable to (a) achieve reasoning; (b) incorporate physics constraints in the framework of machine learning. |

Deep learning features | (a) Input data based on multi-step learning process; (b) Advanced neural network; (c) Able to discover new patterns, requires a new mindset, and can potentially distinguish between causation and correlation; (d) Does not work well for problems with limited data and data with complex hierarchical structures, no mechanism for learning abstractions. |

Specialized methods | (1) Flexible regression method (artificial neural network and Gaussian process regression) for static and low-dimension systems; (2) Principal component analysis, autoencoder, and convolutional neural network methods for high-dimension systems; (3) Hyperparameter-tuning approach for optimization and model accuracy; (4) Linear-star-space system identification method and recurrent neural networks for identifying models. |

Desired tools | Combining physics knowledge with human analysis and deep learning algorithms. |

Required for AI | Cognitive computing algorithms that enable the extraction of information from unstructured data by sorting concepts and relationships into a knowledge base. |

ICF systems | Limited data, requiring a long chain of logical, multi-scale, and multi-dimensional physics; sensitivity to small perturbations; low-error tolerance level. |

Required ML | Physics-informed and human analysis incorporated into deep learning and transfer learning algorithms. |

Suitable problems | (1) Study of sensitivity of outputs to design parameters; (2) Integration of simulations and experimental data into a common framework; (3) Exploration of general correlations among the variables buried in the experimental data and between the measured and simulated data; (4) Optimization of implosion symmetry, pusher mass/thickness/materials, and laser-pulse shape; (5) Advanced neutron image analysis and reconstruction. |

Successful examples | (a) NIF high-yield Hybrid E series ignition target design and optimization guided by the LLNL transfer learning model; (b) OMEGA trip-alpha experiment driven by combining machine learning with human analysis and physics knowledge. |

Future plans | (1) Optimizing energy-coupling coefficients; designing parameter space of implosion (symmetry, pusher mass/thickness/materials, and laser-pulse shape); (2) Minimizing hydrodynamic instabilities using optimized spectrum of perturbations; (3) Quantifying uncertainties for both methods and experimental data; (4) Improving 3D neutron image reconstruction using 2D projection and autocoded features; (5) Combining physics knowledge, human analysis, data, and deep learning algorithms in each step of a design. |

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**MDPI and ACS Style**

Cheng, B.; Bradley, P.A.
What Machine Learning Can and Cannot Do for Inertial Confinement Fusion. *Plasma* **2023**, *6*, 334-344.
https://doi.org/10.3390/plasma6020023

**AMA Style**

Cheng B, Bradley PA.
What Machine Learning Can and Cannot Do for Inertial Confinement Fusion. *Plasma*. 2023; 6(2):334-344.
https://doi.org/10.3390/plasma6020023

**Chicago/Turabian Style**

Cheng, Baolian, and Paul A. Bradley.
2023. "What Machine Learning Can and Cannot Do for Inertial Confinement Fusion" *Plasma* 6, no. 2: 334-344.
https://doi.org/10.3390/plasma6020023