[102] M. R. Krumholz et al. "Equations and Algorithms for Mixed-frame Flux-limited Diffusion Radiation Hydrodynamics". In: *The Astrophysical Journal* 667.1 (Sept. 2007), p. 626.

[103] D. Angelis, F. Sofos, and T. E. Karakasidis. "Artificial Intelligence in Physical Sciences: Symbolic Regression Trends and Perspectives". In: *Archives of Computational Methods in Engineering* 30.6 (July 2023), pp. 3845-3865.

[104] A. D. Jagtap, E. Kharazmi, and G. E. Karniadakis. "Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems". In: *Computer Methods in Applied Mechanics and Engineering* 365 (2020), p. 113028.

[105] S. Cuomo et al. "Scientific Machine Learning Through Physics-Informed Neural Networks: Where we are and What's Next". In: *Journal of Scientific Computing* 92.3 (July 2022), p. 88.

[106] F. Scarselli et al. "The Graph Neural Network Model". In: *IEEE Transactions on Neural Networks* 20.1 (2009), pp. 61-80.

[107] F. Alet et al. "Graph Element Networks: adaptive, structured computation and memory". In: *Proceedings of the 36th International Conference on Machine Learning*. Ed. by K. Chaudhuri and R. Salakhutdinov. Vol. 97. Proceedings of Machine Learning Research. PMLR, June 2019, p. 212222.

[108] J. Brandstetter, D. E. Worrall, and M. Welling. "Message Passing Neural PDE Solvers". In: *International Conference on Learning Representations*. 2022.

[109] M. Horie and N. Mitsume. "Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditions". In: *Advances in Neural Information Processing Systems*. Ed. by S. Koyejo et al. Vol. 35. Curran Associates, Inc., 2022, p. 2321823229.

[110] Z. Li et al. "Fourier Neural Operator for Parametric Partial Differential Equations". In: *International Conference on Learning Representations*. 2021.

[111] Z. Li et al. "Physics-Informed Neural Operator for Learning Partial Differential Equations". In: *ACM / IMS J. Data Sci.* 1.3 (May 2024).

[112] M. Kurz and A. Beck. "A machine learning framework for LES closure terms". In: *Electron. Trans. Numer. Analysis* 56 (2022), pp. 117-137.
[113] S. Taghizadeh, F. D. Witherden, and S. S. Girimaji. "Turbulence closure modeling with data-driven techniques: physical compatibility and consistency considerations". In: *New Journal of Physics* 22.9 (Sept. 2020), p. 093023.

[114] Z. Chen and J. Deng. "Data-driven RANS closures for improving mean field calculation of separated flows". In: *Frontiers in Physics* 12 (2024).

[115] A. Harada, S. Nishikawa, and S. Yamada. "Deep Learning of the Eddington Tensor in Core-collapse Supernova Simulation". In: *The Astrophysical Journal* 925.2 (Jan. 2022), p. 117.

[116] D. Kochkov et al. "Machine learning-accelerated computational fluid dynamics". In: *Proceedings of the National Academy of Sciences* 118.21 (2021), e2101784118.

[117] F. Rosenblatt. "The perceptron: a probabilistic model for information storage and organization in the brain." In: *Psychological review* 65 6 (1958), pp. 386-408.

[118] F. Della Santa and S. Pieraccini. "Discontinuous neural networks and discontinuity learning". In: *Journal of Computational and Applied Mathematics* 419 (2023), p. 114678.

[119] K. Hornik, M. Stinchcombe, and H. White. "Multilayer feedforward networks are universal approximators". In: *Neural Networks* 2.5 (1989), pp. 359-366.

[120] D. P. Kingma and J. Ba. "Adam: A Method for Stochastic Optimization". In: *CoRR* abs/1412.6980 (2014).

[121] P. Baldi and P. Sadowski. "The dropout learning algorithm". In: *Artificial Intelligence* 210 (2014), pp. 78-122.

[122] N. M. H. Vaytet et al. "The influence of frequency-dependent radiative transfer on the structures of radiative shocks". In: *Journal of Quantitative Spectroscopy and Radiative Transfer* 125 (2013), pp. 105-122.

[123] M. Schmidt and H. Lipson. "Distilling Free-Form Natural Laws from Experimental Data". In: *Science* 324.5923 (2009), pp. 81-85.

[124] T. Hida and M. Hitsuda. *Gaussian processes*. Vol. 120. American Mathematical Soc., 1993.

[125] Godard, B. et al. "Models of irradiated molecular shocks". In: *Astronomy & Astrophysics* 622 (2019), A100.

[126] Y. B. Zel'Dovich and Y. P. Raizer. *Physics of shock waves and high-temperature hydrodynamic phenomena*. Courier Corporation, 2002.
[127] R. P. Drake. *High-Energy-Density Physics: Fundamentals, Inertial Fusion, and Experimental Astrophysics*. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006.

[128] C. Michaut et al. "Numerical study of the vishniac instability in supernova remnants". In: *The Astrophysical Journal* 759.2 (Oct. 2012), p. 78.

[129] Fadeyev, Yu. A., Le Coroller, H., and Gillet, D. "The structure of radiative shock waves - IV. Effects of electron thermal conduction". In: *Astronomy & Astrophysics* 392.2 (2002), pp. 735-740.

[130] P. Kervella, A. Mérand, and A. Gallenne. "The circumstellar envelopes of the Cepheids *ℓ* Carinae and RS Puppis. Comparative study in the infrared with Spitzer, VLT/VISIR, and VLTI/MIDI". In: *Astronomy & Astrophysics* 498.2 (May 2009), pp. 425-443.

[131] S. W. Stahler, F. H. Shu, and R. E. Taam. "The evolution of protostars. I-Global formulation and results". In: *Astrophysical Journal, Part 11* 241 (Oct. 1980). NSF-supported research., p. 637654.

[132] A. C. Raga et al. "3D Transfer of the Diffuse Ionizing Radiation in ISM Flows and the Preionization of a Herbig-Haro Working Surface". In: *Rev. Mex. Astron. Astrofis.* 35 (Oct. 1999), p. 123.

[133] H. Inoue. "Steady jet ejections from the innermost region of advection-dominated accretion flow around a black hole". In: *Publications of the Astronomical Society of Japan* (Oct. 2022).

[134] B. Loupias et al. "Astrophysical Jet Experiment". In: *Protostellar Jets in Context*. Ed. by K. Tsinganos, T. Ray, and M. Stute. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009, p. 567569.

[135] Gillet, D. and Fokin, A. B. "Emission lines and shock waves in RR Lyrae stars". In: *Astronomy & Astrophysics* 565 (2014), A73.

[136] Gillet, D. "Atmospheric dynamics in long-period Cepheids - Hα profile variations". In: *Astronomy & Astrophysics* 568 (2014), A72.

[137] J. N. Imamura. "On the stability properties of white dwarf radiative shocks". In: *Astrophysical Journal, Part 1* 296 (Sept. 1985). DOE-sponsored research., p. 128141.

[138] S. Bouquet, R. Teyssier, and J. P. Chieze. "Analytical Study and Structure of a Stationary Radiative Shock". In: *The Astrophysical Journal Supplement Series* 127.2 (Apr. 2000), p. 245.
[139] C. Michaut et al. "Jump conditions in hypersonic shocks". In: *The European Physical Journal D-Atomic, Molecular, Optical and Plasma Physics* 28.3 (2004), p. 381392.

[140] R. P. Drake. "Theory of radiative shocks in optically thick media". In: *Physics of Plasmas* 14.4 (Apr. 2007). 043301.

[141] R. B. Lowrie and J. D. Edwards. "Radiative shock solutions with grey nonequilibrium diffusion". In: *Shock Waves* 18.2 (July 2008), pp. 129–143.

[142] J. C. Bozier et al. "Experimental Observation of a Radiative Wave Generated in Xenon by a Laser-Driven Supercritical Shock". In: *Phys. Rev. Lett.* 57 (11 Sept. 1986), pp. 1304–1307.

[143] P. A. Keiter et al. "Observation of a Hydrodynamically Driven, Radiative-Precursor Shock". In: *Phys. Rev. Lett.* 89 (16 Sept. 2002), p. 165003.

[144] S. Bouquet et al. "Observation of Laser Driven Supercritical Radiative Shock Precursors". In: *Phys. Rev. Lett.* 92 (22 June 2004), p. 225001.

[145] M. Koenig et al. "Radiative shocks: An opportunity to study laboratory astrophysics". In: *Physics of Plasmas* 13.5 (May 2006), p. 056504.

[146] R. P. Drake. "Laboratory experiments to simulate the hydrodynamics of supernova remnants and supernovae". In: *Journal of Geophysical Research: Space Physics* 104.A7 (1999), pp. 14505–14515.

[147] R. A. Chevalier and J. N. Imamura. "Linear analysis of an oscillatory instability of radiative shock waves". In: *The Astrophysical Journal* 261 (Oct. 1982), pp. 543–549.

[148] E. T. Vishniac. "The dynamic and gravitational instabilities of spherical shocks". In: *Astrophysical Journal*, Part 1 274 (Nov. 1983), pp. 152–167.

[149] C. Michaut et al. "Classification of and recent research involving radiative shocks". In: *Astrophysics and Space Science* 322.1 (Aug. 2009), pp. 77–84.

[150] G. Radureau and C. Michaut. "Impact of frequency-dependent radiation on the dynamics and structure of radiative shocks". In: *Phys. Rev. E* 111 (4 Apr. 2025), p. 045213.

[151] R. S. Sutherland. "A new computational fluid dynamics code I: Fyris Alpha". In: *Astrophysics and Space Science* 327 (2010), p. 173206.

[152] L. Ensman. "Test problems for radiation and radiation-hydrodynamics codes". In: *Astrophysical Journal*, Part 1 424.1 (1994), p. 275291.
[153] L. Lu et al. "Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators". In: *Nature Machine Intelligence* 3.3 (Mar. 2021), pp. 218-229.

[154] N. Kovachki et al. "Neural operator: learning maps between function spaces with applications to PDEs". In: *J. Mach. Learn. Res.* 24.1 (Jan. 2023).

[155] J. Hesthaven and S. Ubbiali. "Non-intrusive reduced order modeling of nonlinear problems using neural networks". In: *Journal of Computational Physics* 363 (2018), pp. 55-78.

[156] S. Fresca, L. Dede', and A. Manzoni. "A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs". In: *Journal of Scientific Computing* 87.2 (Apr. 2021), p. 61.

[157] O. San and R. Maulik. "Extreme learning machine for reduced order modeling of turbulent geophysical flows". In: *Phys. Rev. E* 97 (4 Apr. 2018), p. 042322.

[158] A. D. Jagtap, K. Kawaguchi, and G. E. Karniadakis. "Adaptive activation functions accelerate convergence in deep and physics-informed neural networks". In: *Journal of Computational Physics* 404 (2020), p. 109136.

[159] Z. Mao, A. D. Jagtap, and G. E. Karniadakis. "Physics-informed neural networks for high-speed flows". In: *Computer Methods in Applied Mechanics and Engineering* 360 (2020), p. 112789.

[160] A. Papados. *Solving hydrodynamic shock-tube problems using weighted physics-informed neural networks with domain extension*. 2021.

[161] S. Mishra and R. Molinaro. "Physics informed neural networks for simulating radiative transfer". In: *Journal of Quantitative Spectroscopy and Radiative Transfer* 270 (2021), p. 107705.

[162] Z. Zhao, X. Ding, and B. A. Prakash. "PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks". In: *The Twelfth International Conference on Learning Representations*. 2024.

[163] T. Wagenaar. "Physics-informed neural networks for highly compressible flows: assessing and enhancing shock-capturing capabilities". MA thesis. Delft University of Technology, Sept. 2023.

[164] E. Kharazmi, Z. Zhang, and G. E. Karniadakis. "hp-VPINNs: Variational physics-informed neural networks with domain decomposition". In: *Computer Methods in Applied Mechanics and Engineering* 374 (2021), p. 113547.
[165] T. De Ryck, S. Mishra, and R. Molinaro. "wPINNs: Weak Physics Informed Neural Networks for Approximating Entropy Solutions of Hyperbolic Conservation Laws". In: *SIAM Journal on Numerical Analysis* 62.2 (2024), pp. 811–841.

[166] L. Liu et al. "Discontinuity Computing Using Physics-Informed Neural Networks". In: *Journal of Scientific Computing* 98.1 (Dec. 2023), p. 22.

[167] A. Ferrer-Sánchez et al. "Gradient-annihilated PINNs for solving Riemann problems: Application to relativistic hydrodynamics". In: *Computer Methods in Applied Mechanics and Engineering* 424 (2024), p. 116906.

[168] E. J. R. Coutinho et al. "Physics-informed neural networks with adaptive localized artificial viscosity". In: *Journal of Computational Physics* 489 (2023), p. 112265.

[169] N. Zhou and Z. Ma. *Capturing Shock Waves by Relaxation Neural Networks*. 2024. arXiv: 2404.01163 [math.NA].

[170] S. Wang, S. Sankaran, and P. Perdikaris. "Respecting causality for training physics-informed neural networks". In: *Computer Methods in Applied Mechanics and Engineering* 421 (2024), p. 116813.

[171] C. Rao, H. Sun, and Y. Liu. "Physics-informed deep learning for incompressible laminar flows". In: *Theoretical and Applied Mechanics Letters* 10.3 (2020), pp. 207–212.

[172] C. Wu et al. "A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks". In: *Computer Methods in Applied Mechanics and Engineering* 403 (2023), p. 115671.

[173] L. Lu et al. "DeepXDE: A Deep Learning Library for Solving Differential Equations". In: *SIAM Review* 63.1 (2021), pp. 208–228.

[174] Z. Mao and X. Meng. "Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving partial differential equations with sharp solutions". In: *Applied Mathematics and Mechanics* 44.7 (July 2023), pp. 1069–1084.

[175] M. A. Nabian, R. J. Gladstone, and H. Meidani. "Efficient training of physics-informed neural networks via importance sampling". In: *Computer-Aided Civil and Infrastructure Engineering* 36.8 (2021), pp. 962–977.

[176] K. Tang, X. Wan, and C. Yang. "DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations". In: *Journal of Computational Physics* 476 (2023), p. 111868.
[177] J. Grun et al. "Instability of Taylor-Sedov blast waves propagating through a uniform gas". In: *Phys. Rev. Lett.* 66 (21 May 1991), p. 27382741.

[178] Bonnet-Bidaud, J. M. et al. "Quasi-periodic oscillations in accreting magnetic white dwarfs★ - I. Observational constraints in X-ray and optical". In: *Astronomy & Astrophysics* 579 (2015), A24.

[179] K. Azizzadenesheli et al. "Neural operators for accelerating scientific simulations and design". In: *Nature Reviews Physics* 6.5 (May 2024), pp. 320–328.
<figure><img src="image_92.png" /></figure>