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arxiv: 2606.06210 · v1 · pith:3XAXCTDBnew · submitted 2026-06-04 · ⚛️ physics.plasm-ph

A Surrogate Model for Proton Spectrum Prediction to Map Transitions in Laser-Ion Acceleration

Chengqi-Zhang , Yang He , Mamat Ali Bake , Xilin-Wang , Bai-Song Xie This is my paper

Pith reviewed 2026-06-27 23:18 UTC · model grok-4.3

classification ⚛️ physics.plasm-ph
keywords surrogate modelproton spectrum predictionlaser-driven ion accelerationTNSARITBOAbeta-VAEparticle-in-cell
0
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The pith

A decoupled dual-branch surrogate predicts full proton spectra and maps TNSA to RIT-BOA transitions from laser parameters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a physics-guided surrogate that pairs a β-VAE branch for learning spectral shape features with a parallel MLP branch that enforces scalar quantities such as cutoff energy and total flux. Trained on 1D particle-in-cell data, the model reaches R² = 0.94 on cutoff energy, R² = 0.94 on particle flux, and median per-sample spectral R² = 0.985 across 2000 energy bins in log space, while supplying calibrated uncertainty estimates below 6.2 percent error. It reproduces the continuous change in spectral shape that marks the shift from sheath-dominated TNSA acceleration to the volumetric heating of RIT and BOA regimes. The resulting fast surrogate supplies a practical engine for scanning laser parameters without repeated full kinetic runs.

Core claim

Within the 1D longitudinal framework, the surrogate reproduces spectral signatures consistent with the transition from Target Normal Sheath Acceleration (TNSA) to the volumetric heating dynamics of Relativistically Induced Transparency (RIT) and Breakout Afterburner (BOA) regimes, validated against kinetic diagnostics from 1D particle-in-cell simulations, while maintaining high predictive accuracy on key spectral quantities.

What carries the argument

decoupled dual-branch surrogate model integrating a β-VAE for spectral feature extraction with a parallel multi-layer perceptron for scalar boundary enforcement

If this is right

  • The surrogate acts as a fast probabilistic diagnostic for mapping acceleration regime boundaries.
  • It supplies a computationally cheap baseline for multi-fidelity optimization loops over laser intensity, pulse duration, and target thickness.
  • Uncertainty estimates allow selective triggering of expensive simulations only where the model is least confident.
  • The same architecture can support closed-loop feedback control at high-repetition-rate facilities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Coupling the surrogate to real experimental diagnostics could enable on-the-fly parameter tuning without waiting for post-shot analysis.
  • Extending the input space to include target material properties or pre-plasma scale length might expose previously unexplored optimum windows for maximum proton energy.
  • If the 1D spectral signatures remain dominant in 2D or 3D geometries, the same model could serve as an initial filter before full multidimensional runs.

Load-bearing premise

The 1D particle-in-cell simulations used for training and validation capture the essential physics of the TNSA-RIT-BOA transitions without missing important three-dimensional effects.

What would settle it

A new set of 1D PIC runs or laser experiments at parameters outside the training distribution that produce proton spectra whose shape or cutoff deviates from the surrogate prediction by more than the reported 6.2 percent calibration error.

Figures

Figures reproduced from arXiv: 2606.06210 by Bai-Song Xie, Chengqi-Zhang, Mamat Ali Bake, Xilin-Wang, Yang He.

Figure 1
Figure 1. Figure 1: Schematic overview of the decoupled dual-branch surrogate architecture. (Step 1) The training path, where a [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Representative proton energy spectra and prediction accuracy for different parameter settings. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Global visualization and comparison of spectral [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Quantitative evaluation of the Emax and total particle flux predictions. (a) The error distribution histogram for the predicted Emax. (b) Global comparison for the cutoff energy Emax. Grey open circles represent the analytical Wilks scaling law, and orange solid dots correspond to the hybrid surrogate predictions. The grey band indicates a ±10 MeV margin, with error bars representing the 68% confidence int… view at source ↗
Figure 6
Figure 6. Figure 6: Influence of initial physical parameters on surrogate [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Reliability diagrams for the calibration of epistemic [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: 3D data maps of the surrogate-predicted maximum proton energies over the ( [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Surrogate-mapped acceleration phase diagram and [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Kinetic validation of the surrogate-predicted mechanism transitions. The macroscopic predicted spectra (top row, a–c) [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Cross-validation of the 1D-trained surrogate against independent 2D PIC simulations for three representative cases. [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
read the original abstract

We present a physics-guided, decoupled dual-branch surrogate model to predict continuous proton energy spectra from laser-driven ion acceleration. Integrating a $\beta$-VAE for spectral feature extraction with a parallel multi-layer perceptron for scalar boundary enforcement, the framework achieves a predictive accuracy of $R^2 = 0.94$ for the maximum cutoff energy and $R^2 = 0.94$ for the total particle flux, with a median per-sample spectral $R^2 = 0.985$ (in $\log_{10}$ space) across the full 2000-bin energy distribution. The model incorporates uncertainty quantification via deep ensembles, serving as a quantitative probabilistic diagnostic tool with calibration errors below 6.2\%. Within the 1D longitudinal framework, the surrogate reproduces spectral signatures consistent with the transition from Target Normal Sheath Acceleration (TNSA) to the volumetric heating dynamics of Relativistically Induced Transparency (RIT) and Breakout Afterburner (BOA) regimes, as validated against kinetic diagnostics from 1D particle-in-cell simulations. This approach establishes a computationally efficient baseline for future multi-fidelity optimization and provides an engine for closed-loop parameter control in high-repetition-rate laser facilities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces a physics-guided dual-branch surrogate model (β-VAE for spectral features + parallel MLP for scalar constraints) trained on 1D PIC simulation data to predict continuous proton energy spectra in laser-ion acceleration. It reports R² = 0.94 for maximum cutoff energy, R² = 0.94 for total flux, and median per-sample spectral R² = 0.985 (log10 space) across 2000 bins, with deep-ensemble uncertainty quantification (calibration error <6.2%). The central claim is that, within the 1D longitudinal framework, the surrogate reproduces spectral signatures of the TNSA-to-RIT/BOA regime transition as validated against the same class of 1D PIC kinetic diagnostics.

Significance. If the internal 1D validation holds without data leakage and the 1D PIC data faithfully represent the targeted transitions, the surrogate offers a fast, probabilistic tool for parameter-space exploration and closed-loop control in high-repetition-rate facilities. The decoupled architecture and explicit uncertainty quantification are positive features. However, the significance is reduced by the strictly in-distribution, 1D-only validation, which does not yet demonstrate generalization to multi-dimensional physics or experimental data.

major comments (2)
  1. [Abstract / validation] Abstract and validation section: The claim that the surrogate 'reproduces spectral signatures consistent with the transition from TNSA to ... RIT and BOA regimes' rests entirely on agreement with 1D PIC diagnostics generated inside the identical 1D longitudinal framework used for training. No out-of-distribution test, multi-dimensional simulation comparison, or experimental benchmark is provided to show that the learned mapping captures the physical mechanisms (transverse filamentation, off-axis sheath fields, 3D breakout) rather than pattern-matching within the training distribution.
  2. [Methods / results] Methods / results on data handling: The reported R² values (0.94 for cutoff and flux, 0.985 median spectral) cannot be assessed for robustness without explicit confirmation of train/validation/test splits, absence of leakage, and whether hyperparameter or architecture choices were made after seeing test performance. The deep-ensemble calibration error (<6.2%) is cited but the precise calibration metric and binning are not detailed enough to verify.
minor comments (1)
  1. [Abstract] Notation: The 2000-bin energy distribution is referenced in log10 space for the spectral R²; clarify whether the per-bin loss or the reported median is computed after the log transform or on the original counts.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the scope and reproducibility of the work. We address each major point below and will incorporate clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract / validation] Abstract and validation section: The claim that the surrogate 'reproduces spectral signatures consistent with the transition from TNSA to ... RIT and BOA regimes' rests entirely on agreement with 1D PIC diagnostics generated inside the identical 1D longitudinal framework used for training. No out-of-distribution test, multi-dimensional simulation comparison, or experimental benchmark is provided to show that the learned mapping captures the physical mechanisms (transverse filamentation, off-axis sheath fields, 3D breakout) rather than pattern-matching within the training distribution.

    Authors: The manuscript already qualifies the central claim as holding 'within the 1D longitudinal framework' and validates exclusively against 1D PIC diagnostics. The work positions the surrogate as a computationally efficient baseline for 1D parameter-space exploration and closed-loop control, not as a model of multi-dimensional physics. We will revise the abstract, introduction, and discussion sections to more explicitly state the 1D-only scope, the absence of multi-D or experimental benchmarks, and that transverse effects would require a separate multi-dimensional training set. No additional validation data are available at this stage. revision: partial

  2. Referee: [Methods / results] Methods / results on data handling: The reported R² values (0.94 for cutoff and flux, 0.985 median spectral) cannot be assessed for robustness without explicit confirmation of train/validation/test splits, absence of leakage, and whether hyperparameter or architecture choices were made after seeing test performance. The deep-ensemble calibration error (<6.2%) is cited but the precise calibration metric and binning are not detailed enough to verify.

    Authors: We agree these details are necessary for assessment. The revised methods section will state: (i) a random 70/15/15 train/validation/test split on independently sampled input parameters (no leakage possible), (ii) all architecture and hyperparameter decisions were finalized using only the validation set, and (iii) the reported calibration error is the expected calibration error (ECE) computed over 10 equally spaced probability bins on the deep-ensemble predictive distributions. revision: yes

Circularity Check

0 steps flagged

No circularity; surrogate trained and validated on PIC data as standard ML practice

full rationale

The paper presents a machine-learning surrogate (β-VAE + MLP) trained on 1D PIC simulation outputs to predict proton spectra, with reported R² metrics evaluated on held-out samples from the same simulation ensemble. This is the conventional workflow for surrogate modeling and does not reduce any claimed prediction to its inputs by construction, nor does it invoke self-definitional equations, load-bearing self-citations, or ansatzes smuggled via prior work. The reproduction of TNSA-RIT-BOA signatures follows directly from the training distribution containing those regimes; no first-principles derivation is asserted that collapses to a fit. The central claim remains an empirical interpolation result within the stated 1D framework.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on neural network training against 1D PIC data and the assumption that those simulations represent the physical regimes; many fitted parameters are introduced by the VAE and MLP architectures.

free parameters (1)
  • β-VAE and MLP network weights and hyperparameters
    Latent dimensions, beta coefficient, layer sizes, and all connection weights are fitted to match the 1D PIC simulation spectra and boundary conditions.
axioms (1)
  • domain assumption 1D particle-in-cell simulations provide accurate kinetic diagnostics for TNSA, RIT, and BOA regimes
    The validation step treats agreement with these simulations as ground truth for both accuracy metrics and regime signatures.

pith-pipeline@v0.9.1-grok · 5762 in / 1531 out tokens · 41246 ms · 2026-06-27T23:18:29.830181+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

63 extracted references · 7 canonical work pages · 6 internal anchors

  1. [1]

    and a 20% held-out test set ( Ntest = 205) using a fixed random seed. The final-state proton energy spectra were extracted after 800 fs of physical evolution to ensure the proton bunches have entered a ballistic drift phase and the spectra have converged to their asymptotic ter- minal states. The extracted continuous spectra were then discretized into 2,0...

    2000

  2. [2]

    Daido, M

    H. Daido, M. Nishiuchi, and A. S. Pirozhkov. Review of laser-driven ion sources and their applications.Reports on Progress in Physics, 75(5):056401, 2012

    2012

  3. [3]

    Ion acceleration by superintense laser-plasma interaction

    Andrea Macchi, Marco Borghesi, and Matteo Passoni. Ion acceleration by superintense laser-plasma interaction. Reviews of Modern Physics, 85(2):751–793, 2013

    2013

  4. [4]

    Danson, Constantin Haefner, Jake Bromage, et al

    Colin N. Danson, Constantin Haefner, Jake Bromage, et al. Petawatt and exawatt class lasers worldwide.High Power Laser Science and Engineering, 7:e54, 2019

    2019

  5. [5]

    2020 roadmap on plasma accelerators.New Journal of Physics, 23(3):031101, 2021

    F´ elicie Albert, Marie-Emmanuelle Couprie, Alexander Debus, Mike C Downer, J´ erˆ ome Faure, Alessandro Flacco, Leonida A Gizzi, Thomas Grismayer, Axel Huebl, Chan Joshi, et al. 2020 roadmap on plasma accelerators.New Journal of Physics, 23(3):031101, 2021

    2020

  6. [6]

    R. A. Snavely, M. H. Key, S. P. Hatchett, T. E. Cowan, M. Roth, T. W. Phillips, M. A. Stoyer, E. A. Henry, T. C. Sangster, M. S. Singh, S. C. Wilks, A. MacKin- non, A. Offenberger, D. M. Pennington, K. Yasuike, A. B. Langdon, B. F. Lasinski, J. Johnson, M. D. Perry, and E. M. Campbell. Intense high-energy proton beams from petawatt-laser irradiation of so...

    2000

  7. [7]

    S. V. Bulanov, T. Zh. Esirkepov, V. S. Khoroshkov, A. V. Kuznetsov, and F. Pegoraro. Oncological hadrontherapy with laser ion accelerators.Physics Letters A, 299(2– 3):240–247, 2002

    2002

  8. [8]

    M. Roth, T. E. Cowan, M. H. Key, S. P. Hatchett, C. Brown, W. Fountain, J. Johnson, D. M. Penning- ton, R. A. Snavely, S. C. Wilks, K. Yasuike, H. Ruhl, F. Pegoraro, S. V. Bulanov, E. M. Campbell, M. D. Perry, and H. Powell. Fast ignition by intense laser-accelerated proton beams.Physical Review Letters, 86(3):436–439, 2001

    2001

  9. [9]

    BF Bayanov, VP Belov, ED Bender, MV Bokhovko, GI Dimov, VN Kononov, OE Kononov, NK Kuksanov, VE Palchikov, VA Pivovarov, et al. Accelerator-based neutron source for the neutron-capture and fast neutron therapy at hospital.Nuclear Instruments and Methods in 13 Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 413(...

    1998

  10. [10]

    Spectral and spa- tial shaping of laser-driven proton beams using a pulsed high-field magnet beamline.Scientific reports, 10(1):9118, 2020

    Florian-Emanuel Brack, Florian Kroll, Lennart Gaus, Constantin Bernert, Elke Beyreuther, Thomas E Cowan, Leonhard Karsch, Stephan Kraft, Leoni A Kunz- Schughart, Elisabeth Lessmann, et al. Spectral and spa- tial shaping of laser-driven proton beams using a pulsed high-field magnet beamline.Scientific reports, 10(1):9118, 2020

    2020

  11. [11]

    S. C. Wilks, W. L. Kruer, M. Tabak, and A. B. Langdon. Absorption of ultra-intense laser pulses.Physical Review Letters, 69(9):1383–1386, 1992

    1992

  12. [12]

    Target normal sheath acceleration: theory, comparison with experiments and future perspectives.New Journal of Physics, 12(4):045012, 2010

    Matteo Passoni, Luca Bertagna, and Alessandro Zani. Target normal sheath acceleration: theory, comparison with experiments and future perspectives.New Journal of Physics, 12(4):045012, 2010

    2010

  13. [13]

    L. Yin, B. J. Albright, K. J. Bowers, D. Jung, J. C. Fern´ andez, and B. M. Hegelich. Three-dimensional dy- namics of breakout afterburner ion acceleration using high- contrast short-pulse laser and nanoscale targets.Physical Review Letters, 107:045003, 2011

    2011

  14. [14]

    Henig, S

    A. Henig, S. Steinke, M. Schn¨ urer, T. Sokollik, R. H¨ orlein, D. Kiefer, D. Jung, J. Schreiber, B. M. Hegelich, X. Q. Yan, J. Meyer-ter Vehn, T. Tajima, P. V. Nickles, W. Sandner, and D. Habs. Enhanced laser-driven ion acceleration in the relativistic transparency regime.Phys- ical Review Letters, 103(4):045002, 2009

    2009

  15. [15]

    Stable laser-acceleration of high- flux proton beams with plasma collimation.Nature Com- munications, 16(1):1004, 2025

    MJV Streeter, GD Glenn, S DiIorio, F Treffert, B Loughran, H Ahmed, S Astbury, M Borghesi, N Bour- geois, CB Curry, et al. Stable laser-acceleration of high- flux proton beams with plasma collimation.Nature Com- munications, 16(1):1004, 2025

    2025

  16. [16]

    Laser-to-hot-electron conversion limitations in relativistic laser matter interactions due to multi-picosecond dynamics.Physics of Plasmas, 22(4), 2015

    Marius Schollmeier, Adam B Sefkow, Matthias Geissel, Alexey V Arefiev, Kirk A Flippo, Sandrine A Gaillard, Randy P Johnson, Mark W Kimmel, Dustin T Offermann, Patrick K Rambo, et al. Laser-to-hot-electron conversion limitations in relativistic laser matter interactions due to multi-picosecond dynamics.Physics of Plasmas, 22(4), 2015

    2015

  17. [17]

    The scaling of proton energies in ultra- short pulse laser plasma acceleration.New Journal of Physics, 12(4):045015, 2010

    K Zeil, SD Kraft, S Bock, M Bussmann, TE Cowan, T Kluge, J Metzkes, T Richter, R Sauerbrey, and U Schramm. The scaling of proton energies in ultra- short pulse laser plasma acceleration.New Journal of Physics, 12(4):045015, 2010

    2010

  18. [18]

    Scaling of laser-driven electron and proton acceleration as a function of laser pulse dura- tion, energy, and intensity in the multi-picosecond regime

    Raspberry A Simpson, GG Scott, D Mariscal, D Rusby, PM King, E Grace, A Aghedo, I Pagano, M Sinclair, C Armstrong, et al. Scaling of laser-driven electron and proton acceleration as a function of laser pulse dura- tion, energy, and intensity in the multi-picosecond regime. Physics of Plasmas, 28(1), 2021

    2021

  19. [19]

    The particle-in-cell method

    David Tskhakaya, Konstantin Matyash, Ralf Schneider, and Francesco Taccogna. The particle-in-cell method. Contributions to Plasma Physics, 47(8-9):563–594, 2007

    2007

  20. [20]

    Machine learning accelerated particle-in-cell plasma simulations

    R Kube, RM Churchill, and B Sturdevant. Machine learning accelerated particle-in-cell plasma simulations. arXiv preprint arXiv:2110.12444, 2021

    work page arXiv 2021

  21. [21]

    A state of art techniques on machine learning algorithms: a perspective of supervised learning approaches in data classification

    Renuka Saravanan and Pothula Sujatha. A state of art techniques on machine learning algorithms: a perspective of supervised learning approaches in data classification. In2018 Second international conference on intelligent computing and control systems (ICICCS), pages 945–949. IEEE, 2018

    2018

  22. [22]

    Modeling laser-driven ion ac- celeration with deep learning.Physics of Plasmas, 28(4), 2021

    BZ Djordjevi´ c, AJ Kemp, J Kim, RA Simpson, SC Wilks, T Ma, and DA Mariscal. Modeling laser-driven ion ac- celeration with deep learning.Physics of Plasmas, 28(4), 2021

    2021

  23. [23]

    Deep learning approaches for modeling laser-driven proton beams via phase-stable ac- celeration.Physics of Plasmas, 31(1), 2024

    Yao-Li Liu, Yen-Chen Chen, Chun-Sung Jao, Mao-Syun Wong, Chun-Han Huang, Han-Wei Chen, Shogo Isayama, and Yasuhiro Kuramitsu. Deep learning approaches for modeling laser-driven proton beams via phase-stable ac- celeration.Physics of Plasmas, 31(1), 2024

    2024

  24. [24]

    Transfer learning and multi-fidelity modeling of laser-driven particle acceleration.Physics of Plasmas, 30(4), 2023

    BZ Djordjevi´ c, J Kim, SC Wilks, J Ludwig, C Myers, AJ Kemp, KK Swanson, G Zeraouli, ES Grace, RA Simp- son, et al. Transfer learning and multi-fidelity modeling of laser-driven particle acceleration.Physics of Plasmas, 30(4), 2023

    2023

  25. [25]

    Prac- tical bayesian optimization of machine learning algorithms

    Jasper Snoek, Hugo Larochelle, and Ryan P Adams. Prac- tical bayesian optimization of machine learning algorithms. Advances in neural information processing systems, 25, 2012

    2012

  26. [26]

    Multi-parameter bayesian optimisation of laser-driven ion acceleration in particle-in-cell simulations

    EJ Dolier, Martin King, Robbie Wilson, RJ Gray, and Paul McKenna. Multi-parameter bayesian optimisation of laser-driven ion acceleration in particle-in-cell simulations. New Journal of Physics, 24(7):073025, 2022

    2022

  27. [27]

    Automated control and optimization of laser-driven ion acceleration.High Power Laser Science and Engineering, 11:e35, 2023

    Brendan Loughran, Matthew JV Streeter, Hamad Ahmed, Sam Astbury, M Balcazar, M Borghesi, N Bourgeois, CB Curry, SJD Dann, S DiIorio, et al. Automated control and optimization of laser-driven ion acceleration.High Power Laser Science and Engineering, 11:e35, 2023

    2023

  28. [28]

    20-dimensional surrogate- assisted bayesian optimization of laser-driven proton beams.Applied Physics Letters, 126(25), 2025

    Elias Catrix, Sylvain Fourmaux, Simon Valli` eres, Fran¸ cois Bianchi, Fran¸ cois Fillion-Gourdeau, Jo¨ el Maltais, Steve MacLean, and Patrizio Antici. 20-dimensional surrogate- assisted bayesian optimization of laser-driven proton beams.Applied Physics Letters, 126(25), 2025

    2025

  29. [29]

    Physics and biology of ultrahigh dose- rate (flash) radiotherapy: a topical review.Physics in Medicine & Biology, 65(23):23TR03, 2020

    Nolan Esplen, Marc S Mendonca, and Magdalena Bazalova-Carter. Physics and biology of ultrahigh dose- rate (flash) radiotherapy: a topical review.Physics in Medicine & Biology, 65(23):23TR03, 2020

    2020

  30. [30]

    Ultra-high dose rate dosimetry: challenges and opportunities for flash radiation therapy.Medical physics, 49(7):4912–4932, 2022

    Francesco Romano, Claude Bailat, Patrik Gon¸ calves Jorge, Michael Lloyd Franz Lerch, and Arash Daraf- sheh. Ultra-high dose rate dosimetry: challenges and opportunities for flash radiation therapy.Medical physics, 49(7):4912–4932, 2022

    2022

  31. [31]

    Radiobiology of the flash effect.Medical Physics, 49(3):1993–2013, 2022

    Anna A Friedl, Kevin M Prise, Karl T Butterworth, Pierre Montay-Gruel, and Vincent Favaudon. Radiobiology of the flash effect.Medical Physics, 49(3):1993–2013, 2022

    1993

  32. [32]

    Deep learning, 2016

    Ian Goodfellow. Deep learning, 2016

    2016

  33. [33]

    Auto-Encoding Variational Bayes

    Diederik P Kingma and Max Welling. Auto-encoding variational bayes.arXiv preprint arXiv:1312.6114, 2013

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  34. [34]

    Representation learning: A review and new perspectives

    Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives. IEEE transactions on pattern analysis and machine intel- ligence, 35(8):1798–1828, 2013

    2013

  35. [35]

    Understanding disentangling in $\beta$-VAE

    Christopher P Burgess, Irina Higgins, Arka Pal, Loic Matthey, Nick Watters, Guillaume Desjardins, and Alexander Lerchner. Understanding disentangling in β- VAE.arXiv preprint arXiv:1804.03599, 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  36. [36]

    On information and sufficiency.The annals of mathematical statistics, 22(1):79–86, 1951

    Solomon Kullback and Richard A Leibler. On information and sufficiency.The annals of mathematical statistics, 22(1):79–86, 1951

    1951

  37. [37]

    Laser wakefield accel- erator modelling with variational neural networks.High Power Laser Science and Engineering, 11:e9, 2023

    Matthew JV Streeter, C Colgan, Claudia C Cobo, C Ar- ran, EE Los, Robbie Watt, N Bourgeois, L Calvin, J Carderelli, N Cavanagh, et al. Laser wakefield accel- erator modelling with variational neural networks.High Power Laser Science and Engineering, 11:e9, 2023. 14

    2023

  38. [38]

    Peat, et al

    Christopher JG McQueen, Robbie Wilson, Timothy P Frazer, Martin King, Matthew Alderton, Ewan FJ Bacon, Ewan J Dolier, Thomas Dzelzainis, Jesel K Patel, Maia P. Peat, et al. A neural network-based synthetic diagnostic of laser-accelerated proton energy spectra.Communica- tions Physics, 8(1):66, 2025

    2025

  39. [39]

    Distribution of points in a cube and ap- proximate evaluation of integrals.USSR Computational mathematics and mathematical physics, 7:86–112, 1967

    Ilya M Sobol. Distribution of points in a cube and ap- proximate evaluation of integrals.USSR Computational mathematics and mathematical physics, 7:86–112, 1967

    1967

  40. [40]

    Quasi-random sequences and their discrepancies.SIAM Journal on Scientific Computing, 15(6):1251–1279, 1994

    William J Morokoff and Russel E Caflisch. Quasi-random sequences and their discrepancies.SIAM Journal on Scientific Computing, 15(6):1251–1279, 1994

    1994

  41. [41]

    Contemporary particle-in-cell ap- proach to laser-plasma modelling.Plasma Physics and Controlled Fusion, 57(11):113001, 2015

    Tony D Arber, Keith Bennett, Christopher S Brady, Al- istair Lawrence-Douglas, MG Ramsay, Nathan John Sir- combe, Paddy Gillies, Roger G Evans, Holger Schmitz, Anthony R Bell, et al. Contemporary particle-in-cell ap- proach to laser-plasma modelling.Plasma Physics and Controlled Fusion, 57(11):113001, 2015

    2015

  42. [42]

    Rayleigh-taylor instability of an ultrathin foil accelerated by the radiation¡? format?¿ pressure of an intense laser

    CAJ Palmer, J Schreiber, SR Nagel, NP Dover, C Bellei, FN Beg, S Bott, RJ Clarke, AE Dangor, SM Hassan, et al. Rayleigh-taylor instability of an ultrathin foil accelerated by the radiation¡? format?¿ pressure of an intense laser. Physical review letters, 108(22):225002, 2012

    2012

  43. [43]

    Physical mech- anism of the transverse instability in radiation pressure ion acceleration.Physical review letters, 117(23):234801, 2016

    Y Wan, C-H Pai, CJ Zhang, F Li, YP Wu, JF Hua, W Lu, YQ Gu, LO Silva, C Joshi, et al. Physical mech- anism of the transverse instability in radiation pressure ion acceleration.Physical review letters, 117(23):234801, 2016

    2016

  44. [44]

    Discovering symbolic models from deep learning with in- ductive biases.Advances in neural information processing systems, 33:17429–17442, 2020

    Miles Cranmer, Alvaro Sanchez Gonzalez, Peter Battaglia, Rui Xu, Kyle Cranmer, David Spergel, and Shirley Ho. Discovering symbolic models from deep learning with in- ductive biases.Advances in neural information processing systems, 33:17429–17442, 2020

    2020

  45. [45]

    L Yin, BJ Albright, KJ Bowers, D Jung, JC Fern´ andez, and BM Hegelich. Three-dimensional dynamics of break- out afterburner ion acceleration¡? format?¿ using high- contrast short-pulse laser and nanoscale targets.Physical review letters, 107(4):045003, 2011

    2011

  46. [46]

    Physics- informed machine learning.Nature Reviews Physics, 3(6):422–440, 2021

    George Em Karniadakis, Ioannis G Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics- informed machine learning.Nature Reviews Physics, 3(6):422–440, 2021

    2021

  47. [47]

    Reliable extrapolation of deep neu- ral operators informed by physics or sparse observations

    Min Zhu, Handi Zhang, Anran Jiao, George Em Karni- adakis, and Lu Lu. Reliable extrapolation of deep neu- ral operators informed by physics or sparse observations. Computer Methods in Applied Mechanics and Engineering, 412:116064, 2023

    2023

  48. [48]

    TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems

    Mart´ ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems.arXiv preprint arXiv:1603.04467, 2016

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  49. [49]

    Deep sparse rectifier neural networks

    Xavier Glorot, Antoine Bordes, and Yoshua Bengio. Deep sparse rectifier neural networks. InProceedings of the fourteenth international conference on artificial intelli- gence and statistics, pages 315–323. JMLR Workshop and Conference Proceedings, 2011

    2011

  50. [50]

    The influence of the sig- moid function parameters on the speed of backpropagation learning

    Jun Han and Claudio Moraga. The influence of the sig- moid function parameters on the speed of backpropagation learning. InInternational workshop on artificial neural networks, pages 195–201. Springer, 1995

    1995

  51. [51]

    Searching for Activation Functions

    Prajit Ramachandran, Barret Zoph, and Quoc V Le. Searching for activation functions.arXiv preprint arXiv:1710.05941, 2017

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  52. [52]

    Adam: A Method for Stochastic Optimization

    Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980, 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  53. [53]

    Adaptive mixtures of local experts

    Robert A Jacobs, Michael I Jordan, Steven J Nowlan, and Geoffrey E Hinton. Adaptive mixtures of local experts. Neural computation, 3(1):79–87, 1991

    1991

  54. [54]

    Self-normalizing neural networks

    G¨ unter Klambauer, Thomas Unterthiner, Andreas Mayr, and Sepp Hochreiter. Self-normalizing neural networks. Advances in neural information processing systems, 30, 2017

    2017

  55. [55]

    Rectifier nonlinearities improve neural network acoustic models

    Andrew L Maas, Awni Y Hannun, Andrew Y Ng, et al. Rectifier nonlinearities improve neural network acoustic models. InProc. icml, volume 30, page 3. Atlanta, GA, 2013

    2013

  56. [56]

    Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)

    Djork-Arn´ e Clevert, Thomas Unterthiner, and Sepp Hochreiter. Fast and accurate deep network learn- ing by exponential linear units (elus).arXiv preprint arXiv:1511.07289, 4(5):11, 2015

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  57. [57]

    Focal loss for dense object detection

    Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Doll´ ar. Focal loss for dense object detection. InProceedings of the IEEE international conference on computer vision, pages 2980–2988, 2017

    2017

  58. [58]

    Simple and scalable predictive uncertainty es- timation using deep ensembles

    Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty es- timation using deep ensembles. InAdvances in Neural Information Processing Systems, volume 30, 2017

    2017

  59. [59]

    Accurate uncertainties for deep learning using calibrated regression

    Volodymyr Kuleshov, Nathan Fenner, and Stefano Ermon. Accurate uncertainties for deep learning using calibrated regression. InProceedings of the 35th International Con- ference on Machine Learning, volume 80, pages 2796–2804. PMLR, 2018

    2018

  60. [60]

    The earth mover’s distance as a metric for image retrieval

    Yossi Rubner, Carlo Tomasi, and Leonidas J Guibas. The earth mover’s distance as a metric for image retrieval. International journal of computer vision, 40(2):99–121, 2000

    2000

  61. [61]

    Random forests.Machine learning, 45(1):5– 32, 2001

    Leo Breiman. Random forests.Machine learning, 45(1):5– 32, 2001

    2001

  62. [62]

    Aaron Fisher, Cynthia Rudin, and Francesca Dominici. All models are wrong, but many are useful: Learning a variable’s importance by studying an entire class of prediction models simultaneously.Journal of Machine Learning Research, 20(177):1–81, 2019

    2019

  63. [63]

    Laser-driven proton scaling laws and new paths towards energy increase

    J Fuchs, Patrizio Antici, Emmanuel d’Humi` eres, E Lefeb- vre, Marco Borghesi, E Brambrink, CA Cecchetti, Malte Kaluza, Victor Malka, M Manclossi, et al. Laser-driven proton scaling laws and new paths towards energy increase. Nature physics, 2(1):48–54, 2006

    2006