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REVIEW 4 major objections 6 minor 1 cited by

SPINN: Advancing Cosmological Simulations of Fuzzy Dark Matter with Physics Informed Neural Networks

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A physics-informed neural network called SPINN solves the nonlinear Schrödinger–Poisson equations of fuzzy dark matter in 1D and 3D, reproducing gravitational collapse and conserving total mass without a mesh.

desk verdict A first proof-of-concept that PINNs can handle Schrödinger-Poisson for FDM, but the accuracy and scalability claims outrun the quantitative evidence. read the letter →

arxiv 2506.02957 v1 pith:4JFWGEEE submitted 2025-06-03 astro-ph.CO

Ashutosh Kumar Mishra , Emma Tolley This is my paper
classification astro-ph.CO
keywords fuzzydarkmatterSchrödinger-Poissonequationsphysics-informedneuralnetworksgravitationalcollapsecosmologicalsimulationswavemassconservationspectralmethods
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to show that a physics-informed neural network can act as a mesh-free solver for the nonlinear Schrödinger–Poisson equations that govern fuzzy dark matter, reproducing gravitational collapse in one and three dimensions. The network, SPINN, is trained directly on the equations plus initial and boundary conditions, with no simulation data in the unsupervised case, and its density evolution matches a second-order spectral scheme while total mass stays conserved without any explicit conservation constraint. The motivation is concrete: conventional fuzzy-dark-matter simulations must resolve the de Broglie wavelength throughout the box, which limits them to volumes of a few megaparsecs, whereas a trained PINN no longer needs the mesh at all. The paper presents SPINN as a first step toward scalable cosmological fuzzy-dark-matter simulation, demonstrating modest time extrapolation and an accuracy gain when a small amount of simulation data is added.

What carries the argument

The load-bearing object is the network's output choice: SPINN predicts the real part $R$ and imaginary part $I$ of the wavefunction $\Psi$ along with the gravitational potential $V$ as an auxiliary output, instead of the density and phase used in the Madelung fluid formulation. Writing the Schrödinger–Poisson equations in terms of $R$ and $I$ yields coupled real equations, $\partial_t R = -\tfrac{1}{2}\nabla^2 I + V I$ and $\partial_t I = \tfrac{1}{2}\nabla^2 R - V R$, together with the Poisson constraint $\nabla^2 V = R^2 + I^2 - 1$; a network that minimizes these residuals respects the complex-conjugate structure of the Schrödinger equation, which is why mass conservation follows without an explicit normalization loss. Predicting $V$ directly sidesteps an FFT-based Poisson solve at each training step, whose $\mathcal{O}(N^2\log N)$ cost and memory footprint would otherwise cap resolution. The 1D network uses two hidden layers of 32 neurons with sine activations; the 3D network uses five layers of 16 neurons with a sine first layer and wavelet activations; optimization combines Adam pretraining with L-BFGS, and the loss is a mean-squared sum of PDE, boundary, and initial residuals at randomly sampled collocation points.

What would settle it

Train SPINN on the same sinusoidal case but push the comparison to $t = 10$ instead of $t = 3$, and also on a multi-mode initial condition such as a Gaussian perturbation; the paper's own extrapolation result shows divergence beginning near $t = 4$, so measurable growth in the density error or drift of the normalization $\int |\Psi|^2\,dV$ away from 1 over longer times would falsify the claim that the network has learned the physics rather than fitted this single test case.

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Extended reading notes

Core claim

The paper's claim is that a physics-informed neural network can solve the full nonlinear Schrödinger–Poisson system governing fuzzy dark matter directly from the equations, with no training data and no mesh, and reproduce gravitational collapse in both one and three dimensions. With a single sinusoidal Fourier mode as the initial perturbation, a standard test case for Vlasov–Poisson solvers, the network SPINN matches the density evolution computed by a second-order unitary spectral scheme, keeps the wavefunction normalization (total mass) at approximately one with no explicit conservation constraint, and tracks the growth of overdensities in the three-dimensional radial density profile. The paper further claims that the network extrapolates modestly beyond its training time, and that predicting the gravitational potential as a network output avoids the memory and compute cost of solving Poisson's equation by Fourier transform at every training step. The authors present this as a first step toward efficient, possibly scalable cosmological simulation of fuzzy dark matter, explicitly deferring the question of whether the framework scales to realistic initial conditions and long times.

Load-bearing premise

The load-bearing premise is that a single sinusoidal initial condition evolved for a short time ($t = 0$ to $3$) faithfully represents the multi-scale, long-duration dynamics of a cosmological fuzzy-dark-matter box; as the paper itself states in Section 5, the training time was not long enough nor the initial condition complex enough to assess scalability, and without that transfer the broader significance of the method collapses to this one test case.

Editorial extensions

If this is right

  • A PINN trained only on equation residuals and initial/boundary conditions reproduces 1D and 3D gravitational collapse of fuzzy dark matter, matching a spectral method, so the basic form of the method needs no labeled simulation data.
  • Because the network outputs the real and imaginary parts of the wavefunction, wavefunction normalization and hence total mass stay conserved without an explicit constraint term in the loss.
  • The trained network keeps reasonable accuracy beyond its training window, to about $t = 4$ when trained to $t \approx 3$, so time extrapolation is possible, not just interpolation.
  • Adding a small amount of simulation data in semi-supervised training improves the accuracy of the predicted wavefunction phase relative to pure unsupervised training.
  • Predicting the gravitational potential as a network output avoids per-step Fourier Poisson solves, removing a memory and compute bottleneck that would otherwise force lower resolution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The modest extrapolation horizon (accuracy degrading near $t = 4$ after training to $t \approx 3$) suggests time-window chaining or domain-decomposition, mentioned but not implemented in the paper, as the natural route to long cosmological evolutions; whether the horizon grows with training time is directly testable.
  • The semi-supervised result points to a concrete pipeline the paper proposes but does not build: take existing large cold-dark-matter simulations as the data constraint for large scales and let the Schrödinger–Poisson residuals paint in the small-scale wave features of fuzzy dark matter.
  • The failure of the density–phase (Madelung) variant to produce wave splitting is a prediction about any fluid-formulation neural solver: without enforcing the quantization condition that makes the fluid equations equivalent to Schrödinger–Poisson, a network trained on those equations should also miss interference features.
  • A stricter test than the paper's own: compare the density power spectrum, not just radial profiles, against the spectral solution, so that structure suppression, the key fuzzy-dark-matter observable, is quantified directly.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The manuscript introduces SPINN, a physics-informed neural network solver for the nonlinear Schrödinger-Poisson equations describing fuzzy dark matter, and applies it to gravitational collapse from a sinusoidal perturbation in one and three spatial dimensions. The network outputs the real and imaginary parts of the wavefunction together with the gravitational potential, and is trained by minimizing PDE, initial, and boundary residuals without labeled simulation data, with a spectral method used only for validation. The authors report density evolution, mass conservation, extrapolation beyond the training time, and comparisons with Madelung-formulation, Fourier-potential, and semi-supervised variants. The central claim is that SPINN accurately predicts key metrics and offers a scalable path toward FDM cosmological simulations.

Significance. If the accuracy claims are substantiated, this would be a useful proof-of-concept for applying PINNs to self-gravitating Schrödinger-Poisson systems, a setting where traditional solvers face multi-scale resolution challenges. The paper has several strengths: the unsupervised formulation avoids circularity, since the spectral benchmark enters only in validation and not in the loss; the code is publicly available; and the appendices contain useful ablations (Madelung versus SP formulation, external Poisson solver, activation functions, and hard conditioning). At present, however, the accuracy and scalability claims are not yet supported by quantitative measurements, so the significance is primarily methodological promise rather than demonstrated cosmological capability.

major comments (4)
  1. [Section 4.2, Figures 2, 4-6] The abstract and Section 4.2 claim 'accurate predictions' of density profiles and structure suppression, but no quantitative error norm (relative L2, maximum pointwise error, or similar) is reported for the 1D or 3D comparisons; the 'error' panels are evaluated only visually. Since Eq. (26) minimizes PDE residuals at collocation points rather than solution error, small loss values do not by themselves establish accuracy. Please add numerical errors for the density (and ideally for R, I, V) at the shown times, including the radial profiles in Figure 4 and the slices/cubes in Figures 5 and 6.
  2. [Section 3.1 and Figure 3] Mass conservation is claimed on the basis of Figure 3, where the normalization 'remains approximately one,' but no numerical deviations or tolerance are given, and Section 3.1 explicitly states that no normalization constraint is included. Because normalization is not hard-enforced, the conservation statement needs a quantitative bound, such as max |∫|Ψ|² - 1| over the evolution, and should separate the SPINN error from the spectral reference drift. Without that, 'approximately one' is too weak to support the conservation claim in the abstract.
  3. [Section 5 and Abstract] The paper's broader significance rests on efficiency and scalability ('efficient, possibly scalable modeling'), but no wall-clock times, GPU/memory usage, or scaling with domain size or training duration are reported, and Section 5 states that the training time was not long enough and the initial condition not complex enough to assess scalability. The sinusoidal, short-time tests alone cannot support the scalability statements; please either add concrete scaling measurements or explicitly restrict the claims to the studied regime.
  4. [Section 4.3, Figure 7] The extrapolation claim is supported only by visual inspection of heatmaps; the text says the solutions agree 'uptil t = 4' and then deviate, but no error measure or tolerance is provided. Please quantify the extrapolation error over time (for example, the L2 error relative to the spectral solution in the extrapolated interval) so that the reader can judge how much of the solution is usable beyond the training domain.
minor comments (6)
  1. [Throughout] There are several typos and formatting issues: 'sucessfully' in the Introduction, 'demosntrates' in Section 5, 'T able 1' in the Table 1 caption, 'Inital' in the Section 4.1 heading, and 'The numerical solutions is' in Section 4.2.
  2. [Equation (27)] The boundary residual for V is written as |Vθ - Vb| without a square, while the R and I terms are squared; this should presumably be |Vθ - Vb|² for consistency with a mean-squared-error loss.
  3. [Appendix C.2] The text says the errors are slightly higher 'as shown in Fig. (3),' but Figure 3 is the mass-conservation plot; this cross-reference appears to point to the wrong figure.
  4. [Equation (14)] The boundary operator in Eq. (14) is written as B[NN(X,θ); ] with an empty argument, which is incomplete and should read B[NN(X,θ); Λ] or similar.
  5. [References and Code Availability] The Code Availability section gives only '(DeepWaves-SPINN)' with no URL or DOI; a resolved link should be provided. Also, the first reference entry '2006, Line Search Methods' has no author listed.
  6. [Figures 5 and 6] The captions describe the time and slice locations as 'randomly chosen,' but randomness is not essential to the validation and the selection procedure is not described; please state how these frames were selected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: SPINN is trained on PDE residuals plus initial and boundary conditions, while the spectral-method benchmark is used only for post-training validation, so the central accuracy claim is not forced by construction.

full rationale

The paper's derivation chain is self-contained. The SPINN loss in Eq. (26) penalizes residuals of the real and imaginary Schrödinger equations (Eqs. 23-24) and the Poisson equation (Eq. 25) at randomly sampled collocation points, together with boundary and initial condition residuals (Eqs. 27-28). No term in the loss uses the spectral-method solution, and no parameter is fitted to the benchmark. The spectral algorithm of Appendix A is invoked only to generate comparison plots (Figs. 2, 4-6) and is therefore validation, not training input. The claim that mass conservation follows automatically from learning the Schrödinger equation is a physical consequence of the PDE system rather than a fitted ansatz; the authors explicitly state they include no explicit normalization constraint (Section 3.1). The one self-citation involving an author (Korber et al. 2022, cited for CNN-based mapping applications in the introduction) is peripheral and not load-bearing for the SPINN result. The admitted limitations in Section 5, namely short training times and the simplicity of the sinusoidal initial condition, are honest scope restrictions and not evidence of circularity. Concerns about the absence of quantitative error norms are correctness or validation-strength issues, not circularity. The central claim that a PDE-residual-trained network reproduces the spectral solution of the Schrödinger-Poisson system is therefore independent of the benchmark used to check it.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the validity of the SP model, the accuracy of the spectral benchmark, and the network's ability to approximate the solution. No parameters are fitted to external data; λ and α are set by unit choice.

assumptions (4)
  • domain assumption The dimensionless Schrödinger-Poisson equations with α=1, λ=1 describe the fuzzy dark matter system in the chosen units.
    Sections 2.1 and 3.1 set constants to simplify; the physics is assumed to be captured by these equations.
  • domain assumption The spectral split-step method (Mocz et al. 2017) provides an accurate reference solution.
    Appendix A uses this method as the benchmark without independent verification of its accuracy for this problem.
  • standard math A feedforward neural network with sine and wavelet activations can represent the solution and be trained to a sufficiently small PDE residual.
    The paper relies on universal approximation and empirical convergence; no formal guarantee is provided.
  • ad hoc to paper The chosen sinusoidal initial condition is representative of cosmological perturbations.
    The authors note in Section 5 that the initial condition is not complex enough to assess scalability, yet they use it for all conclusions.

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Cite this review

Pith. "Pith review of SPINN: Advancing Cosmological Simulations of Fuzzy Dark Matter with Physics Informed Neural Networks." pith.science (2026). https://pith.science/paper/4JFWGEEE

@misc{pith2026250602957,
  author       = {Pith},
  title        = {Pith review of: SPINN: Advancing Cosmological Simulations of Fuzzy Dark Matter with Physics Informed Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4JFWGEEE}},
  note         = {Machine review of arXiv:2506.02957}
}
read the original abstract

Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving differential equations by integrating physical laws into the learning process. This work leverages PINNs to simulate gravitational collapse, a critical phenomenon in astrophysics and cosmology. We introduce the Schr\"odinger-Poisson informed neural network (SPINN) which solve nonlinear Schr\"odinger-Poisson (SP) equations to simulate the gravitational collapse of Fuzzy Dark Matter (FDM) in both 1D and 3D settings. Results demonstrate accurate predictions of key metrics such as mass conservation, density profiles, and structure suppression, validating against known analytical or numerical benchmarks. This work highlights the potential of PINNs for efficient, possibly scalable modeling of FDM and other astrophysical systems, overcoming the challenges faced by traditional numerical solvers due to the non-linearity of the involved equations and the necessity to resolve multi-scale phenomena especially resolving the fine wave features of FDM on cosmological scales.

Figures

Figures reproduced from arXiv: 2506.02957 by the authors.

Figure 1
Figure 1. Schematic of the PINN-based SPINN framework: The neural network outputs {R, I, V }, which serve as approximate solutions (Re(Ψ) = R,Im(Ψ) = I). These, along with the governing PDEs, ICs, and BCs, define the total loss function. During training, the parameters θ are iteratively optimized to obtain θ ∗ , yielding the final solutions of the system. residuals for boundary and initial condition for each of the outputs at… view at source ↗
Figure 2
Figure 2. Time evolution of the 1-D density distribution under the effect of the gravitational potential, predicted with SPINN: In the top row, density distributions are shown at fixed time frames indicated by the color gradient, with spatial coordinate on the x-axis and density on the y-axis. From left to right, the plots show the SPINN prediction, the corresponding numerical solution with Spectral Method (described in Appen… view at source ↗
Figure 3
Figure 3. Evaluation of a key metric, mass conservation (or equivalently normalization of the state): The plot shows the normalization at fixed time frames, with time on x-axis and |Ψ| 2 integrated over the spatial coordinate on the y-axis. Schr¨odinger–Poisson equations (4–5) over the interval t = 0 to t = 3, for the chosen coupling parameter λ = 1. The top row of the figure displays the predicted density evolution by SPINN … view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Average radial density at different time frames from the (i) SPINN Predictions (left), (ii) Numerical Solution with spectral second-order unitary numerical scheme - See appendix A (middle), and (iii) Error between the two (right). Plots show evolution of average radial…
Figure 5
Figure 5. Figure 5: Evaluation procedure for randomly chosen density cube at a fixed time frame. From left to right, the first panel shows a density cube at randomly chosen t (t = 1.0367) from the numerically computed solution of SP equations in 3D. The second panel exhibits the same as p…
Figure 6
Figure 6. Figure 6: Evaluation procedure for randomly chosen x − y density slice at a fixed z (z = 0.3125) and time frame (t = 1.0367). From left to right, the first panel shows an x − y slice at randomly chosen z and t as predicted by our trained SPINN model in 3D. The second panel exhib…
Figure 7
Figure 7. Figure 7: Extrapolating the SPINN predictions to outside the training time units: the top panel shows the predicted solution to approximately t = 4 and the bottom panel de￾picts the numerical solution for the same time units, both are represented by the 2D heatplot with spatial …
Figure 8
Figure 8. Figure 8: Semi-Supervised Learning: Errors in ρ, R, I are shown here with respect to the analytical solution (See appendix A) [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: Unsupervised Learning: Errors in ρ, R, I are shown here with respect to the analytical solution (See appendix A) not sufficiently long, nor the initial condition complex enough, to fully assess the scalability of the framework for longer times or more intricate scenari…
Figure 10
Figure 10. Figure 10: Time Evolution of Density from PINN with Madelung formalism: density distributions are shown at fixed time frames indicated by the color gradient, with spatial coordinate on the x-axis and density on the y-axis. From left to right, the plots show the PINN prediction a…
Figure 11
Figure 11. Figure 11: Time Evolution of Density from SPINN with po￾tential obtained by solving Poisson equation through Fourier transform: density distributions are shown at fixed time frames shown by the color gradient with spatial coordinate in x-axis, and density in the y-axis [PITH_FU…
Figure 12
Figure 12. Figure 12: Error comparison for a few activation functions: The plot depicts the error in density prediction on the y-axis vs the spatial coordinates at fixed time frames represented by color gradient, calculated w.r.t the numerical solution [PITH_FULL_IMAGE:figures/full_fig_p0…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Cosmo-SPINN: Fuzzy Dark Matter Simulations with Physics-Informed Generative Networks

    astro-ph.CO 2026-07 conditional novelty 6.0 of 10

    Physics-informed generative U-Nets evolve and super-resolve fuzzy dark matter fields under Schrödinger–Poisson constraints with far less supervised data than pure data-driven baselines.

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Works this paper leans on

90 extracted references · 5 canonical work pages · cited by 1 Pith paper

  1. [1]

    JH cBP ]D aW Z *+b tWwͽ 99- /栚Jd1qI,R'@ RNTT8`w w Bg 5-H.8M( B| I @B|jTWC# B\ *ܤi*|'. |, nȳ P N X > υ 9J v : L 3 *># rs sXæ) Q r CA 6(. +13S Gm r. `B<F c0 !ND > h8 U

    thebibliography [1] 20pt to REFERENCES 6pt =0pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command Each re...

    arXiv 2019

  2. [2]

    2006, Line Search Methods (New York, NY: Springer New York), 30--65, 10.1007/978-0-387-40065-5_3

    doi:10.1007/978-0-387-40065-5_3 2006

  3. [4]

    2010, , 81, 123530, 10.1103/PhysRevD.81.123530

    Arvanitaki , A., Dimopoulos , S., Dubovsky , S., Kaloper , N., & March-Russell , J. 2010, , 81, 123530, 10.1103/PhysRevD.81.123530

    doi:10.1103/physrevd.81.123530 2010

  4. [5]

    2024, arXiv e-prints, arXiv:2403.00599, 10.48550/arXiv.2403.00599

    Baty , H. 2024, arXiv e-prints, arXiv:2403.00599, 10.48550/arXiv.2403.00599

    arXiv doi:10.48550/arxiv.2403.00599 2024

  5. [6]

    G., Pearlmutter, B

    Baydin, A. G., Pearlmutter, B. A., Radul, A. A., & Siskind, J. M. 2018, Journal of Machine Learning Research, 18, 1. http://jmlr.org/papers/v18/17-468.html

    2018

  6. [7]

    2002, , 367, 1, 10.1016/S0370-1573(02)00135-7

    Bernardeau , F., Colombi , S., Gazta \ n aga , E., & Scoccimarro , R. 2002, , 367, 1, 10.1016/S0370-1573(02)00135-7

    doi:10.1016/s0370-1573(02)00135-7 2002

  7. [8]

    2021, MNRAS, 509, 1323, 10.1093/mnras/stab3088

    Bernardini, M., Feldmann, R., Anglés-Alcázar, D., et al. 2021, MNRAS, 509, 1323, 10.1093/mnras/stab3088

    doi:10.1093/mnras/stab3088 2021

  8. [9]

    2024, arXiv e-prints, arXiv:2407.20669, 10.48550/arXiv.2407.20669

    Brevi , L., Mandarino , A., & Prati , E. 2024, arXiv e-prints, arXiv:2407.20669, 10.48550/arXiv.2407.20669

    arXiv doi:10.48550/arxiv.2407.20669 2024

Show all 90 references
  1. [10]

    S., & Boylan-Kolchin , M

    Bullock , J. S., & Boylan-Kolchin , M. 2017, , 55, 343, 10.1146/annurev-astro-091916-055313

    2017 doi

  2. [11]

    Cai, S., Wang, Z., Wang, S., Perdikaris, P., & Karniadakis, G. E. 2021, Journal of Heat Transfer, 143, 060801, 10.1115/1.4050542

    2021 doi

  3. [12]

    2024, Physical Review Research, 6, 013282, 10.1103/PhysRevResearch.6.013282

    Cappelli , L., Tacchino , F., Murante , G., Borgani , S., & Tavernelli , I. 2024, Physical Review Research, 6, 013282, 10.1103/PhysRevResearch.6.013282

    2024 doi

  4. [13]

    2019, MNRAS, 490, 1055, 10.1093/mnras/stz2605

    Chardin, J., Uhlrich, G., Aubert, D., et al. 2019, MNRAS, 490, 1055, 10.1093/mnras/stz2605

    2019 doi

  5. [14]

    H., et al

    Clowe , D., Brada c , M., Gonzalez , A. H., et al. 2006, , 648, L109, 10.1086/508162

    2006 doi

  6. [15]

    2021, Proceedings of the National Academy of Sciences, 118, e2020324118, 10.1073/pnas.2020324118

    Dai, B., & Seljak, U. 2021, Proceedings of the National Academy of Sciences, 118, e2020324118, 10.1073/pnas.2020324118

    2021 doi

  7. [16]

    2020, Physics of the Dark Universe, 28, 100503, https://doi.org/10.1016/j.dark.2020.100503

    De Martino , I., Broadhurst, T., Henry Tye , S.-H., Chiueh, T., & Schive, H.-Y. 2020, Physics of the Dark Universe, 28, 100503, https://doi.org/10.1016/j.dark.2020.100503

    2020

  8. [17]

    2024, Computer Methods in Applied Mechanics and Engineering, 429, 117116, 10.1016/j.cma.2024.117116

    Dolean , V., Heinlein , A., Mishra , S., & Moseley , B. 2024, Computer Methods in Applied Mechanics and Engineering, 429, 117116, 10.1016/j.cma.2024.117116

    2024

  9. [18]

    C., & B \"u rger , D

    Du , X., Schwabe , B., Niemeyer , J. C., & B \"u rger , D. 2018, , 97, 063507, 10.1103/PhysRevD.97.063507

    2018 doi

  10. [19]

    S., & White , S

    Frenk , C. S., & White , S. D. M. 2012, Annalen der Physik, 524, 507, 10.1002/andp.201200212

    2012 doi

  11. [20]

    2010, in Proceedings of Machine Learning Research, Vol

    Glorot, X., & Bengio, Y. 2010, in Proceedings of Machine Learning Research, Vol. 9, Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, ed. Y. W. Teh & M. Titterington (Chia Laguna Resort, Sardinia, Italy: PMLR), 249--256. https://...

    2010

  12. [21]

    2019, Proceedings of the National Academy of Sciences, 116, 13825, 10.1073/pnas.1821458116

    He, S., Li, Y., Feng, Y., et al. 2019, Proceedings of the National Academy of Sciences, 116, 13825, 10.1073/pnas.1821458116

    2019 doi

  13. [22]

    G., Geiger, B

    Hoffer, J. G., Geiger, B. C., Ofner, P., & Kern, R. 2021, Applied Sciences, 11, 10.3390/app11209411

    2021 doi

  14. [23]

    2000, Phys

    Hu, W., Barkana, R., & Gruzinov, A. 2000, Phys. Rev. Lett., 85, 1158, 10.1103/PhysRevLett.85.1158

    2000 doi

  15. [24]

    2021, , 59, 247, 10.1146/annurev-astro-120920-010024

    Hui , L. 2021, , 59, 247, 10.1146/annurev-astro-120920-010024

    2021 doi

  16. [25]

    P., Tremaine , S., & Witten , E

    Hui , L., Ostriker , J. P., Tremaine , S., & Witten , E. 2017, , 95, 043541, 10.1103/PhysRevD.95.043541

    2017 doi

  17. [26]

    D., Mao , Z., Adams , N., & Karniadakis , G

    Jagtap , A. D., Mao , Z., Adams , N., & Karniadakis , G. E. 2022, Journal of Computational Physics, 466, 111402, 10.1016/j.jcp.2022.111402

    2022

  18. [27]

    Jin, X., Cai, S., Li, H., & Karniadakis, G. E. 2021, Journal of Computational Physics, 426, 109951, https://doi.org/10.1016/j.jcp.2020.109951

    2021

  19. [28]

    2020, in 34th Conference on Neural Information Processing Systems

    Kasmanoff, N., Villaescusa-Navarro, F., Tinker, J., & Ho, S. 2020, in 34th Conference on Neural Information Processing Systems . 2012.00186

    2020 arXiv

  20. [29]

    2022, The Astrophysical Journal, 930, 115, 10.3847/1538-4357/ac5c4a

    Kaushal, N., Villaescusa-Navarro, F., Giusarma, E., et al. 2022, The Astrophysical Journal, 930, 115, 10.3847/1538-4357/ac5c4a

    2022 doi

  21. [30]

    Y., & Park , N

    Kim , J., Lee , K., Lee , D., Jin , S. Y., & Park , N. 2020, arXiv e-prints, arXiv:2012.02681, 10.48550/arXiv.2012.02681

    2020 doi arXiv

  22. [31]

    P., & Ba , J

    Kingma , D. P., & Ba , J. 2014, arXiv e-prints, arXiv:1412.6980, 10.48550/arXiv.1412.6980

    2014 doi arXiv

  23. [32]

    2022, arXiv e-prints, arXiv:2208.13803

    Korber , D., Bianco , M., Tolley , E., & Kneib , J.-P. 2022, arXiv e-prints, arXiv:2208.13803. 2208.13803

    2022 arXiv

  24. [33]

    2023, arXiv e-prints, arXiv:2308.12312, 10.48550/arXiv.2308.12312

    Kumar , J., Zarzoso , D., Grandgirard , V., Ebert , J., & Kesselheim , S. 2023, arXiv e-prints, arXiv:2308.12312, 10.48550/arXiv.2308.12312

    2023 doi arXiv

  25. [34]

    Y., Schive , H.-Y., Huang , H., & Liao , P.-Y

    Kunkel , A., Jowett Chan , H. Y., Schive , H.-Y., Huang , H., & Liao , P.-Y. 2024, arXiv e-prints, arXiv:2411.17288, 10.48550/arXiv.2411.17288

    2024 doi arXiv

  26. [35]

    2015, Nature, 521, 436, 10.1038/nature14539

    Lecun, Y., Bengio, Y., & Hinton, G. 2015, Nature, 521, 436, 10.1038/nature14539

    2015 doi

  27. [36]

    Li , X., Hui , L., & Bryan , G. L. 2019, , 99, 063509, 10.1103/PhysRevD.99.063509

    2019 doi

  28. [37]

    2025, , 111, 023535, 10.1103/PhysRevD.111.023535

    Liu , R., Hu , W., & Xiao , H. 2025, , 111, 023535, 10.1103/PhysRevD.111.023535

    2025 doi

  29. [38]

    1927, Zeitschrift fur Physik, 40, 322, 10.1007/BF01400372

    Madelung , E. 1927, Zeitschrift fur Physik, 40, 322, 10.1007/BF01400372

    1927 doi

  30. [39]

    D., & Karniadakis , G

    Mao , Z., Jagtap , A. D., & Karniadakis , G. E. 2020, Computer Methods in Applied Mechanics and Engineering, 360, 112789, 10.1016/j.cma.2019.112789

    2020

  31. [40]

    Marsh , D. J. E. 2015, arXiv e-prints, arXiv:1510.07633, 10.48550/arXiv.1510.07633

    2015 doi arXiv

  32. [41]

    2024, arXiv e-prints, arXiv:2408.10011, 10.48550/arXiv.2408.10011

    Matthews , J., & Bihlo , A. 2024, arXiv e-prints, arXiv:2408.10011, 10.48550/arXiv.2408.10011

    2024 doi arXiv

  33. [42]

    2021, , 506, 2603, 10.1093/mnras/stab1764

    May , S., & Springel , V. 2021, , 506, 2603, 10.1093/mnras/stab1764

    2021 doi

  34. [43]

    F., & Winther , H

    Mina , M., Mota , D. F., & Winther , H. A. 2020, arXiv e-prints, arXiv:2007.04119, 10.48550/arXiv.2007.04119

    2020 doi arXiv

  35. [44]

    2021, Journal of Quantitative Spectroscopy and Radiative Transfer, 270, 107705, https://doi.org/10.1016/j.jqsrt.2021.107705

    Mishra, S., & Molinaro, R. 2021, Journal of Quantitative Spectroscopy and Radiative Transfer, 270, 107705, https://doi.org/10.1016/j.jqsrt.2021.107705

    2021

  36. [45]

    2024, Symmetry, 16, 201, 10.3390/sym16020201

    Misiaszek , M., & Rossi , N. 2024, Symmetry, 16, 201, 10.3390/sym16020201

    2024 doi

  37. [46]

    2018, , 97, 083519, 10.1103/PhysRevD.97.083519

    Mocz , P., Lancaster , L., Fialkov , A., Becerra , F., & Chavanis , P.-H. 2018, , 97, 083519, 10.1103/PhysRevD.97.083519

    2018 doi

  38. [47]

    2021, , 910, 29, 10.3847/1538-4357/abe6ac

    Mocz , P., & Szasz , A. 2021, , 910, 29, 10.3847/1538-4357/abe6ac

    2021 doi

  39. [48]

    H., et al

    Mocz, P., Vogelsberger, M., Robles, V. H., et al. 2017, Monthly Notices of the Royal Astronomical Society, 471, 4559, 10.1093/mnras/stx1887

    2017 doi

  40. [49]

    2021, arXiv e-prints, arXiv:2109.04304, 10.48550/arXiv.2109.04304

    Moya , C., & Lin , G. 2021, arXiv e-prints, arXiv:2109.04304, 10.48550/arXiv.2109.04304

    2021 doi arXiv

  41. [50]

    2015, Neural Networks and Deep Learning (Determination Press)

    Nielsen, M. 2015, Neural Networks and Deep Learning (Determination Press). https://books.google.ch/books?id=STDBswEACAAJ

    2015

  42. [51]

    1980, Mathematics of Computation, 35, 773

    Nocedal, J. 1980, Mathematics of Computation, 35, 773. http://www.jstor.org/stable/2006193

    1980

  43. [52]

    2018, , 478, 3935, 10.1093/mnras/sty1224

    Nori , M., & Baldi , M. 2018, , 478, 3935, 10.1093/mnras/sty1224

    2018 doi

  44. [53]

    A., Navarro , J

    Oman , K. A., Navarro , J. F., Fattahi , A., et al. 2015, , 452, 3650, 10.1093/mnras/stv1504

    2015 doi

  45. [54]

    2022, Universe, 8, 76, 10.3390/universe8020076

    Paduroiu , S. 2022, Universe, 8, 76, 10.3390/universe8020076

    2022 doi

  46. [55]

    2019, arXiv e-prints, arXiv:1912.01703, 10.48550/arXiv.1912.01703

    Paszke , A., Gross , S., Massa , F., et al. 2019, arXiv e-prints, arXiv:1912.01703, 10.48550/arXiv.1912.01703

    2019 doi arXiv

  47. [56]

    D., & Quinn, H

    Peccei, R. D., & Quinn, H. R. 1977, Phys. Rev. Lett., 38, 1440, 10.1103/PhysRevLett.38.1440

    1977 doi

  48. [57]

    2020, , 641, A6, 10.1051/0004-6361/201833910

    Planck Collaboration , Aghanim , N., Akrami , Y., et al. 2020, , 641, A6, 10.1051/0004-6361/201833910

    2020 doi

  49. [58]

    2018, arXiv e-prints, arXiv:1806.08734, 10.48550/arXiv.1806.08734

    Rahaman , N., Baratin , A., Arpit , D., et al. 2018, arXiv e-prints, arXiv:1806.08734, 10.48550/arXiv.1806.08734

    2018 doi arXiv

  50. [59]

    Raissi , M., Perdikaris , P., & Karniadakis , G. E. 2019, Journal of Computational Physics, 378, 686, 10.1016/j.jcp.2018.10.045

    2019 doi

  51. [60]

    Raissi , M., Yazdani , A., & Karniadakis , G. E. 2018, arXiv e-prints, arXiv:1808.04327, 10.48550/arXiv.1808.04327

    2018 doi arXiv

  52. [61]

    2024, arXiv e-prints, arXiv:2402.01868, 10.48550/arXiv.2402.01868

    Rathore , P., Lei , W., Frangella , Z., Lu , L., & Udell , M. 2024, arXiv e-prints, arXiv:2402.01868, 10.48550/arXiv.2402.01868

    2024 doi arXiv

  53. [62]

    2024, Physics of Fluids, 36, 036129, 10.1063/5.0200384

    Ren, X., Hu, P., Su, H., Zhang, F., & Yu, H. 2024, Physics of Fluids, 36, 036129, 10.1063/5.0200384

    2024 doi

  54. [63]

    T., Singh, S., & Guestrin, C

    Ribeiro, M. T., Singh, S., & Guestrin, C. 2016, in Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '16 (New York, NY, USA: Association for Computing Machinery), 1135–1144, 10.1145/2939672.2939778

    2016

  55. [64]

    G., Filippenko , A

    Riess , A. G., Filippenko , A. V., Challis , P., et al. 1998, , 116, 1009, 10.1086/300499

    1998 doi

  56. [65]

    C., Ford , Jr., W

    Rubin , V. C., Ford , Jr., W. K., & Thonnard , N. 1980, , 238, 471, 10.1086/158003

    1980 doi

  57. [66]

    V., Wetzel , A., & Fattahi , A

    Sales , L. V., Wetzel , A., & Fattahi , A. 2022, Nature Astronomy, 6, 897, 10.1038/s41550-022-01689-w

    2022 doi

  58. [67]

    G., Sc \'o ccola , C

    S \'a nchez , A. G., Sc \'o ccola , C. G., Ross , A. J., et al. 2012, , 425, 415, 10.1111/j.1365-2966.2012.21502.x

    2012

  59. [68]

    2014, Nature Physics, 10, 496, 10.1038/nphys2996

    Schive , H.-Y., Chiueh , T., & Broadhurst , T. 2014, Nature Physics, 10, 496, 10.1038/nphys2996

    2014 doi

  60. [69]

    2016, , 818, 89, 10.3847/0004-637X/818/1/89

    Schive , H.-Y., Chiueh , T., Broadhurst , T., & Huang , K.-W. 2016, , 818, 89, 10.3847/0004-637X/818/1/89

    2016 doi

  61. [70]

    2024, arXiv e-prints, arXiv:2408.06650, 10.48550/arXiv.2408.06650

    Shuai , H., & Li , F. 2024, arXiv e-prints, arXiv:2408.06650, 10.48550/arXiv.2408.06650

    2024 doi arXiv

  62. [71]

    2021, Reviews of Modern Physics, 93, 015004, 10.1103/RevModPhys.93.015004

    Sikivie , P. 2021, Reviews of Modern Physics, 93, 015004, 10.1103/RevModPhys.93.015004

    2021 doi

  63. [72]

    2022, Phys

    Silk, J., & Stodolsky, L. 2022, Phys. Rev. D, 105, 063506, 10.1103/PhysRevD.105.063506

    2022 doi

  64. [73]

    Sitzmann , V., Martel , J. N. P., Bergman , A. W., Lindell , D. B., & Wetzstein , G. 2020, arXiv e-prints, arXiv:2006.09661, 10.48550/arXiv.2006.09661

    2020 doi arXiv

  65. [74]

    2020, Computer Methods in Applied Mechanics and Engineering, 361, 112732, 10.1016/j.cma.2019.112732

    Sun , L., Gao , H., Pan , S., & Wang , J.-X. 2020, Computer Methods in Applied Mechanics and Engineering, 361, 112732, 10.1016/j.cma.2019.112732

    2020

  66. [75]

    P., Mildenhall , B., et al

    Tancik , M., Srinivasan , P. P., Mildenhall , B., et al. 2020, arXiv e-prints, arXiv:2006.10739, 10.48550/arXiv.2006.10739

    2020 doi arXiv

  67. [76]

    Wallstrom, T. C. 1994, Phys. Rev. A, 49, 1613, 10.1103/PhysRevA.49.1613

    1994 doi

  68. [77]

    2021, SIAM Journal on Scientific Computing, 43, A3055, 10.1137/20M1318043

    Wang , S., Teng , Y., & Perdikaris , P. 2021, SIAM Journal on Scientific Computing, 43, A3055, 10.1137/20M1318043

    2021 doi

  69. [78]

    2020, arXiv e-prints, arXiv:2007.14527, 10.48550/arXiv.2007.14527

    Wang , S., Yu , X., & Perdikaris , P. 2020, arXiv e-prints, arXiv:2007.14527, 10.48550/arXiv.2007.14527

    2020 doi arXiv

  70. [79]

    H., Bullock , J

    Weinberg , D. H., Bullock , J. S., Governato , F., Kuzio de Naray , R., & Peter , A. H. G. 2015, Proceedings of the National Academy of Science, 112, 12249, 10.1073/pnas.1308716112

    2015 doi

  71. [80]

    White , S. D. M., & Rees , M. J. 1978, , 183, 341, 10.1093/mnras/183.3.341

    1978 doi

  72. [81]

    1969, SIAM Review, 11, 226, 10.1137/1011036

    Wolfe, P. 1969, SIAM Review, 11, 226, 10.1137/1011036

    1969 doi

  73. [82]

    1971, SIAM Review, 13, 185, 10.1137/1013035

    ---. 1971, SIAM Review, 13, 185, 10.1137/1013035

    1971 doi

  74. [83]

    C., Ooi , C., Gupta , A., & Ong , Y.-S

    Wong , J. C., Ooi , C., Gupta , A., & Ong , Y.-S. 2021, arXiv e-prints, arXiv:2109.09338, 10.48550/arXiv.2109.09338

    2021 doi arXiv

  75. [84]

    2025, arXiv e-prints, arXiv:2502.00803, 10.48550/arXiv.2502.00803

    Wu , H., Ma , Y., Zhou , H., et al. 2025, arXiv e-prints, arXiv:2502.00803, 10.48550/arXiv.2502.00803

    2025 doi arXiv

  76. [85]

    2025, arXiv e-prints, arXiv:2502.00318, 10.48550/arXiv.2502.00318

    Xu , C., Liu , D., Hu , Y., et al. 2025, arXiv e-prints, arXiv:2502.00318, 10.48550/arXiv.2502.00318

    2025 doi arXiv

  77. [86]

    Yang , L., Meng , X., & Karniadakis , G. E. 2021, Journal of Computational Physics, 425, 109913, 10.1016/j.jcp.2020.109913

    2021

  78. [87]

    2024, Journal of Computational Physics, 515, 113284, https://doi.org/10.1016/j.jcp.2024.113284

    Yuan, B., Wang, H., Heitor, A., & Chen, X. 2024, Journal of Computational Physics, 515, 113284, https://doi.org/10.1016/j.jcp.2024.113284

    2024

  79. [88]

    2018, Frontiers in Astronomy and Space Sciences, 5, 48, 10.3389/fspas.2018.00048

    Zhang , J., Liu , H., & Chu , M.-C. 2018, Frontiers in Astronomy and Space Sciences, 5, 48, 10.3389/fspas.2018.00048

    2018

  80. [89]

    H., Mo, H

    Zhao, D. H., Mo, H. J., Jing, Y. P., & Börner, G. 2003, Monthly Notices of the Royal Astronomical Society, 339, 12, 10.1046/j.1365-8711.2003.06135.x

    2003

  81. [90]

    2023, arXiv e-prints, arXiv:2307.11833, 10.48550/arXiv.2307.11833

    Zhao , Z., Ding , X., & Aditya Prakash , B. 2023, arXiv e-prints, arXiv:2307.11833, 10.48550/arXiv.2307.11833

    2023 doi arXiv

  82. [91]

    2019, Journal of Computational Physics, 394, 56, 10.1016/j.jcp.2019.05.024

    Zhu , Y., Zabaras , N., Koutsourelakis , P.-S., & Perdikaris , P. 2019, Journal of Computational Physics, 394, 56, 10.1016/j.jcp.2019.05.024

    2019 doi

Pith tools

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