Pith. sign in

REVIEW 3 major objections 5 minor 56 references

Physics-informed neural networks with new techniques can solve the Hamiltonian constraint for generic binary black hole initial data at high accuracy.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-08 20:31 UTC pith:BZ4NCRIU

load-bearing objection PINNs plus residual-weighting tricks can solve the BBH puncture Hamiltonian constraint and match classical NR on shown cases; the leap to “generic / all BBH” is the soft spot, not the core method. the 3 major comments →

arxiv 2607.06002 v1 pith:BZ4NCRIU submitted 2026-07-07 gr-qc

Solving Hamiltonian Constraint Equation with Physics-Informed Neural Networks

Yu-Chen Zhou , Hao Ma , Zhoujian Cao , Tailin Wu , Hong-Bo Jin , Xuefeng Feng , Shuanglin Huang , Zhi-Chao Zhao
show 1 more author
Yue-Liang Wu
This is my paper
classification gr-qc
keywords physics-informed neural networksHamiltonian constraintbinary black holesnumerical relativityinitial dataEinstein equationselliptic PDEPINN
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that a physics-informed neural network (PINN), equipped with a set of new residual-weighting, architectural, and training-schedule techniques, can solve the highly nonlinear elliptic Hamiltonian constraint equation that supplies initial data for binary black hole systems in numerical relativity. Traditional finite-difference, spectral, and finite-element methods already solve this problem, but neural-network PDE methods had not been shown to handle it for generic binaries. The authors validate their PINN solutions against traditional numerical-relativity results and report high accuracy and robustness across the tested systems. If the approach generalizes as claimed, it opens a path to a PINN-based initial-data solver usable for the full range of binary black hole configurations needed by gravitational-wave source modelling.

Core claim

A physics-informed neural network together with residual weighting, architecture choices, and a training schedule tailored to the problem successfully solves the Hamiltonian constraint equation for generic binary black hole systems, reproducing traditional numerical-relativity initial data to high accuracy.

What carries the argument

A Physics-Informed Neural Network (PINN) that minimizes the residual of the Hamiltonian constraint as a loss, stabilized by new residual-weighting, network-architecture, and training-schedule techniques that make the highly nonlinear elliptic PDE tractable for gradient-based learning.

Load-bearing premise

The residual weighting, architecture, and training schedule that work on the validated binary systems will generalize without problem-specific retuning to arbitrary mass ratios, spins, separations, and puncture locations while still reaching production numerical-relativity accuracy.

What would settle it

Run the identical PINN configuration, with no retuning of residual weights or schedule, on a binary black hole system outside the validation set (for example mass ratio 1:100 or near-extremal spins) and check whether the Hamiltonian residual and agreement with a traditional NR solver remain at the same high accuracy.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A PINN-based solver can generate binary black hole initial data for numerical relativity without classical elliptic PDE solvers.
  • The same residual-weighting and training techniques make conventional PINNs workable on this class of highly nonlinear elliptic constraints.
  • Validation against traditional NR results indicates the method already reaches high accuracy and robustness on the tested generic systems.
  • The approach reveals a path toward a single PINN-based initial-data solution covering all binary black hole systems needed for numerical relativity.

Where Pith is reading between the lines

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

  • The same residual-weighting and schedule tricks may transfer to the momentum constraints or the full XCTS elliptic system used for more general initial data.
  • If the network can be warm-started from nearby mass-ratio or spin configurations, the method could amortize the cost of generating large catalogs of initial data.
  • Differentiability of the PINN solution with respect to puncture parameters could enable gradient-based searches for initial data with prescribed ADM quantities or horizon properties.
  • Systematic failure on extreme mass ratios or near-extremal spins would show whether the current architecture still needs case-by-case retuning before production use.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript develops a Physics-Informed Neural Network (PINN) solver for the Hamiltonian constraint equation of binary-black-hole (BBH) puncture initial data in numerical relativity. The target PDE is the highly nonlinear elliptic equation Δψ + (1/8) ψ^{-7} Ã_{ij}Ã^{ij} = 0. Conventional PINN training is reported to fail on this problem; the authors introduce residual reweighting, architectural choices, and a training schedule that together allow the network residual to be driven down. Solutions are validated against traditional numerical-relativity initial-data results and are claimed to achieve high accuracy and robustness for generic BBH systems, with the stated longer-term goal of a PINN-based initial-data pipeline for all BBH configurations.

Significance. A mesh-free, differentiable solver for the puncture Hamiltonian constraint that reaches production-level accuracy on generic BBH free data would be a genuine addition to the NR toolkit and would open a new route to constraint-satisfying initial data, especially for configurations that are awkward for spectral or finite-difference elliptic solvers. The external validation against classical NR solutions is the right benchmark. The work is therefore of clear interest if the genericity and accuracy claims hold at the level asserted. The free parameters of the method (architecture, residual weights, schedule) remain part of the scientific content and must be shown not to require case-by-case retuning if the “all BBH systems” claim is to stand.

major comments (3)
  1. The central claim that the method solves the Hamiltonian constraint for generic BBH systems (and ultimately “all BBH systems”) is load-bearing and currently under-supported. The abstract and validation narrative leave open that the high-accuracy runs cover only a modest subset of the free-data space (near-equal mass, moderate spin, fixed separation/puncture placement). The residual-weighting scheme, network architecture and training schedule are free parameters of the method; without a systematic transfer study across mass ratio q, dimensionless spins χ_i and separation D at fixed residual tolerances, the leap from “working examples” to “generic / all BBH” does not follow. A table or figure that reports residual norms, ADM-mass error and L^∞ error for a grid of (q, χ, D) with frozen hyperparameters is required to underwrite the claim.
  2. Near the punctures the factor ψ^{-7} amplifies any local residual. A globally small collocation residual can still leave large pointwise or ADM-mass errors that seed constraint violations under evolution. The manuscript must report L^∞ residuals (or equivalent puncture-neighbourhood diagnostics) and the relative error in ADM mass (or other asymptotic charges) for the validated configurations, not only a domain-averaged residual. Without these diagnostics the claim of “high accuracy” competitive with classical elliptic solvers remains incomplete.
  3. The axiom that a residual-minimizing network with the proposed weighting converges to a classical weak/strong solution of the elliptic constraint is asserted rather than demonstrated. At minimum the paper should show residual decay under refinement of collocation density (or network capacity) and consistency of the recovered ψ with an independent classical solver on the same free data, including a brief discussion of the function space in which convergence is claimed. Absent such evidence the method remains an empirical residual fitter whose relation to the continuum solution is unclear.
minor comments (5)
  1. The abstract’s phrasing “generic BBH systems” and “all BBH systems” should be aligned with the actual parameter ranges demonstrated once those ranges are clarified; overstated scope language should be tempered.
  2. Notation for the conformal factor ψ, the traceless extrinsic-curvature piece Ã_{ij}, and the precise free-data choices (Bowen–York or otherwise) should be fixed early and used consistently; a short table of free-data parameters for each validated run would help reproducibility.
  3. The new residual-weighting and training-schedule techniques should be stated as explicit algorithms (pseudocode or numbered steps) so that independent groups can reimplement them without reverse-engineering from prose.
  4. References to the classical puncture-initial-data literature (Brandt–Brügmann, Ansorg et al., TwoPunctures, etc.) and to prior PINN-for-elliptic-PDE work should be checked for completeness so that the novelty claim is properly situated.
  5. If code or trained weights are to be released, a short reproducibility statement (software stack, random seeds, collocation sampling) would strengthen the paper; if not, that limitation should be noted.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The three major comments correctly identify where the manuscript over-reaches or under-documents its claims: the scope of “generic/all BBH,” the need for puncture-local and ADM diagnostics beyond domain-averaged residuals, and the missing continuum-consistency evidence. We accept these points. In the revised manuscript we will (i) restrict and quantify the free-data domain actually validated, with frozen hyperparameters and an expanded configuration table; (ii) report L^∞ residuals near the punctures and relative ADM-mass errors for every validated case; and (iii) add residual-decay under collocation/capacity refinement together with direct comparison to a classical elliptic solver on the same free data, plus a brief statement of the function-space setting. We do not claim that a single revision can exhaustively map the entire BBH free-data space; the language of the abstract and conclusions will be aligned with what is demonstrated.

read point-by-point responses

  1. Referee: The central claim that the method solves the Hamiltonian constraint for generic BBH systems (and ultimately “all BBH systems”) is load-bearing and currently under-supported. Without a systematic transfer study across mass ratio q, dimensionless spins χ_i and separation D at fixed residual tolerances, the leap from “working examples” to “generic / all BBH” does not follow. A table or figure that reports residual norms, ADM-mass error and L^∞ error for a grid of (q, χ, D) with frozen hyperparameters is required.

    Authors: We agree that the present wording overstates what is shown. The abstract’s “generic BBH systems” and the longer-term “all BBH systems” phrasing are not underwritten by a systematic free-data scan; the validated runs are a limited but nontrivial subset (near-equal to moderate mass ratio, moderate spins, fixed puncture placement conventions). In revision we will: (1) replace “all BBH systems” with an explicit statement of the free-data domain tested and of the method as a candidate pipeline rather than a completed one; (2) freeze architecture, residual-weighting schedule and training protocol across all reported cases and state this clearly; (3) add a table/figure of residual norms, L^∞ diagnostics and ADM-mass errors over a grid in (q, χ, D) spanning the configurations we can rigorously retrain under those frozen hyperparameters. We cannot honestly claim a complete map of the entire free-data space in one revision cycle; the scientific claim will be reduced to what that table supports. revision: yes

  2. Referee: Near the punctures the factor ψ^{-7} amplifies any local residual. A globally small collocation residual can still leave large pointwise or ADM-mass errors. The manuscript must report L^∞ residuals (or equivalent puncture-neighbourhood diagnostics) and the relative error in ADM mass (or other asymptotic charges) for the validated configurations, not only a domain-averaged residual.

    Authors: The referee is correct: domain-averaged residual alone is insufficient for a nonlinear elliptic constraint with a ψ^{-7} source near punctures, and is not enough to claim competitiveness with classical NR initial-data solvers. The revised manuscript will report, for every validated configuration: (i) L^∞ residual (and residual restricted to puncture neighbourhoods); (ii) relative error in ADM mass (and, where available, other asymptotic charges) against the same free data solved by a traditional elliptic solver. These diagnostics will be placed alongside the existing residual and profile comparisons so that “high accuracy” is judged by the quantities that matter for subsequent evolution. revision: yes

  3. Referee: The axiom that a residual-minimizing network with the proposed weighting converges to a classical weak/strong solution of the elliptic constraint is asserted rather than demonstrated. At minimum the paper should show residual decay under refinement of collocation density (or network capacity) and consistency of the recovered ψ with an independent classical solver on the same free data, including a brief discussion of the function space in which convergence is claimed.

    Authors: We accept this criticism. The current text demonstrates empirical residual reduction and profile agreement but does not establish continuum consistency. In revision we will add: (1) residual-decay plots under successive refinement of collocation density and/or network capacity at fixed free data; (2) pointwise and integral comparison of the recovered conformal factor ψ against an independent classical puncture solver on identical free data; (3) a short discussion of the intended function-space setting (conformal factor of puncture type, with the known singular behaviour factored or regularized as in the classical formulation) and of what residual decay under collocation refinement does and does not imply for strong/weak solutions. We do not claim a full numerical-analysis proof of PINN convergence for this nonlinear elliptic problem; we will present the empirical continuum checks that make the method more than an uncalibrated residual fitter. revision: yes

Circularity Check

0 steps flagged

No significant circularity: external PDE solved by PINN and checked against independent classical NR benchmarks

full rationale

The paper’s central claim is that a Physics-Informed Neural Network, augmented with residual weighting, architecture choices and a training schedule, can solve the Hamiltonian constraint equation (the standard nonlinear elliptic PDE for puncture BBH initial data) for binary black hole free data. That equation and the free-data setup are external mathematical objects taken from the numerical-relativity literature; they are not defined by the network. Accuracy is assessed by comparison with solutions produced by traditional elliptic solvers / NR codes, which constitute independent external benchmarks rather than quantities constructed from the PINN residual itself. There is no self-definitional loop (the network does not redefine the constraint it is asked to satisfy), no fitted parameter that is then re-presented as a prediction of a closely related observable, no load-bearing uniqueness theorem imported solely from the authors’ prior work, no ansatz smuggled in via self-citation that itself rests on an unstated ansatz, and no mere renaming of a known empirical pattern. Any self-citations that may appear for technical details of PINN practice are not load-bearing for the claim that the residual of the Hamiltonian constraint is small and that the solution agrees with classical NR. The derivation chain is therefore self-contained against external benchmarks; circularity score is 0 and the steps list is empty. (Over-claim of “generic / all BBH systems” without exhaustive parameter-space transfer tests is a scope/correctness concern, not a circularity concern under the stated criteria.)

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The work rests on standard Einstein-constraint theory (York/Lichnerowicz-York conformal method for BBH initial data), standard PINN residual minimization, and domain assumptions about puncture or conformal free data. Free parameters are the usual PINN hyperparameters (network size, loss weights, sampling). No new physical entities are invented; the contribution is numerical technique, not new physics.

free parameters (2)
  • PINN architecture and loss weights
    Network depth/width, activation, residual/boundary loss coefficients, and sampling density near punctures are chosen by hand or tuned to make training succeed; they are not derived from first principles.
  • Training schedule and residual reweighting scheme
    The paper introduces techniques (reweighting near holes, training strategies) whose numerical coefficients and schedules are free choices that affect whether the nonlinear elliptic solve converges.
axioms (3)
  • domain assumption Einstein Hamiltonian constraint in conformal/puncture form is the correct elliptic equation for BBH initial data
    Standard NR assumption (Lichnerowicz-York / puncture method); invoked as the PDE the PINN must satisfy.
  • ad hoc to paper A neural network residual minimizer with suitable weighting converges to a classical weak/strong solution of the elliptic constraint
    Not proved; assumed so that low training residual plus boundary match implies a valid NR initial-data solution.
  • standard math Standard calculus and elliptic PDE theory for existence/uniqueness of conformal factor given free data
    Background mathematical infrastructure for the constraint equation.

pith-pipeline@v0.9.1-grok · 7611 in / 2444 out tokens · 127629 ms · 2026-07-08T20:31:08.471785+00:00 · methodology

0 comments
read the original abstract

Numerical relativity (NR), solving Einstein equation numerically, plays an important role in source modelling for gravitational wave astronomy. Traditional methods for NR including finite difference method, spectral method and finite element method have been well developed. But newly developed neural network methods for partial differential equations (PDE) have not been well studied yet for NR. We present a Physics-Informed Neural Network (PINN) method to solve the Hamiltonian constraint equation for binary black hole (BBH) initial data in NR. This equation is a highly non-linear elliptic PDE, posing significant challenges for conventional PINN approaches. To overcome these difficulties, we introduce a set of new techniques. We show that our PINN together with these techniques can successfully solve the Hamiltonian constraint equation for generic BBH systems. Validation against the traditional results demonstrates the high accuracy and robustness of our method, revealing the immense potential of constructing a PINN-based initial data solution to all BBH systems for NR.

Figures

Figures reproduced from arXiv: 2607.06002 by Hao Ma, Hong-Bo Jin, Shuanglin Huang, Tailin Wu, Xuefeng Feng, Yu-Chen Zhou, Yue-Liang Wu, Zhi-Chao Zhao, Zhoujian Cao.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic overview of the designed Physics-Informe [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Behavior of our designed scheme about the solution of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Similar to Fig. 2 but for the scheme without the analyt [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Same to Fig. 3 but for the scheme with an alternative an [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Same to Fig. 3 but for the scheme with an alternative ge [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Same to Fig. 3 but for the scheme with [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Same to Fig. 3 but for the scheme without the loss balan [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Similar to Fig. 2 but for an equal-mass spinning binar [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Similar to Fig. 2 but for an unequal-mass non-spinnin [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Similar to Fig. 2 but for an unequal-mass, spinning b [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

56 extracted references · 56 canonical work pages · 24 internal anchors

  1. [1]

    (4) is a highly non-linear elliptic partial dif- ferential equation, it is very difficult to learn the solu- tion u directly by PINN

    Guided hard-enforcement ansatz Since Eq. (4) is a highly non-linear elliptic partial dif- ferential equation, it is very difficult to learn the solu- tion u directly by PINN. To reduce the difficulty of net- work training, we introduce the analytical formula for u from [40] as a guidance and assume that the solution uθ learned by the neural network takes the ...

    work page

  2. [2]

    The network parameters θ are optimized by minimizing the loss function L = LPDE + LBC

    Training method After determining the value of κ, we proceed to train hθ(x) with the neural network. The network parameters θ are optimized by minimizing the loss function L = LPDE + LBC. (11) For the PDE residual, we adopt the following form LPDE =w2L2 +w∞ soft-L∞ . (12) Corresponding to the residual of the equation Ri at each point, L2 is defined as the ...

    work page

  3. [3]

    When combined directly, the largest magnitude term may dominate the optimiza- tion and hinder the reduction of other constraints

    Loss balancing strategy The total loss function contains three components in- cluding Lk = {L2, soft-L∞ , LBC}, which may differ by several orders of magnitude. When combined directly, the largest magnitude term may dominate the optimiza- tion and hinder the reduction of other constraints. To mitigate this scale imbalance, we dynamically rescale each indiv...

    work page

  4. [4]

    Strategy summary We summarize the architecture of our PINN method for Hamiltonian constraint equations in Fig. 1. The net- 5 FIG. 3: Similar to Fig. 2 but for the scheme without the analyt ical guidance. The TP reference solution is exactly the same as that shown in Fig. 2, so here panel (c) only shows two subplots . work consists of 3 hidden layers. Each...

    work page

  5. [5]

    In addition, we find that the guidance form is also important

    Impact of the correction form We have seen the important effect of the analytical guidance to the numerical solution above. In addition, we find that the guidance form is also important. To show this effect, we replace the ansatz (5) with an alternative form uθ, add(x) = κu g(x) +cW (x)hθ (x). (20) As indicated in the third row of Table I, the alterna- tive ...

    work page

  6. [6]

    0363 ± 0. 0087. Once again 5 random runs are used. While it significantly outperforms the scheme without a guiding solution (19), its error is significantly higher than that of our designed scheme. Similar to Fig. 3, we show one example of the 5 runs in Fig. 4. The L2 loss decreases rapidly at the beginning and continues to decay throughout training, with s...

    work page

  7. [7]

    Impact of the Window Function In our designed scheme, we use a window function based on the magnitude of the guiding solution. To evalu- ate the necessity of this approach, we compare it against a general smooth window W ′(x), which is constructed through the superposition of individual smooth window functions Sn for each puncture Sn(rn) = 1 2 [ 1 − tanh ...

    work page

  8. [8]

    As reported in the fifth row of Table I, κ = 1 leads to an average L2RE of 0

    Impact of the factor κ In order to investigate the impact of the factor κ in our designed scheme, we simply set κ = 1 and train hθ(x) directly. As reported in the fifth row of Table I, κ = 1 leads to an average L2RE of 0 . 5357 ± 0. 0470, which is significantly higher than that of our designed scheme. Again, one example evolution of the 5 runs for L2 loss f...

    work page

  9. [9]

    25 at x = ± 8) and an unequal-mass spinning system ( m+ = 3 , m − = 1, P y + = 0

    2, P y − = − 0. 25 at x = ± 8) and an unequal-mass spinning system ( m+ = 3 , m − = 1, P y + = 0 . 3, P y − = − 0. 2, Sz + = 0 . 1, Sz − = 0 . 2 at x = ± 5). We take 5 independent runs for each case. As shown in Table I, the average value and the standard deviation are presented before and after the ± sign. Configuration L2RE Equal-mass spinning 0 . 0838 ±...

    work page

  10. [10]

    25, P x − = 0

    1,P y + = 0. 25, P x − = 0. 2,P y − = − 0. 25 at x = ± 8. The learning rate is set to 5 × 10− 4 for 5000 training steps. Other hyperparameters are set to c = 1 . 0, w2 = 1 . 0, w∞ = 0. 5, β = 10 and wrob = 1. 0. As reported in the second row of Table II, the result yields highly robust and accurate solutions across 5 in- dependent runs, achieving an extre...

    work page

  11. [11]

    Raissi, P

    M. Raissi, P. Perdikaris, and G. Karniadakis, Journal of Computational Physics 378, 686 (2019), ISSN 0021- 9991, URL https://www.sciencedirect.com/science/ article/pii/S0021999118307125

    work page 2019

  12. [12]

    S. Cai, Z. Mao, Z. Wang, M. Yin, and G. E. Karniadakis, arXiv preprint arXiv:2105.09506 (2021)

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  13. [13]

    C. Rao, H. Sun, and Y. Liu, Theoretical and Ap- plied Mechanics Letters 10, 207 (2020), ISSN 2095- 0349, URL https://www.sciencedirect.com/science/ article/pii/S2095034920300350

    work page 2020

  14. [14]

    EPINN-NSE: Enhanced Physics-Informed Neural Networks for Solving Navier-Stokes Equations

    A. Farkane, M. Ghogho, M. Oudani, and M. Boutayeb, ArXiv abs/2304.03689 (2023), URL https://api. semanticscholar.org/CorpusID:258041233

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  15. [15]

    Donnelly, A

    J. Donnelly, A. Daneshkhah, and S. Abolfathi, Science of the Total Environment 912, 168814 (2024)

    work page 2024

  16. [16]

    S. Hu, M. Liu, S. Zhang, S. Dong, and R. Zheng, Engineering Applications of Artificial Intelli- gence 128, 107453 (2024), ISSN 0952-1976, URL https://www.sciencedirect.com/science/article/ pii/S0952197623016378

    work page 2024

  17. [17]

    C. Zhou, T. Li, C. Lan, R. Du, G. Xin, W. Li, G. Wang, X. Liu, and H. Yang, in The Thirty-ninth Annual Conference on Neural Information P rocessing Systems (2025), URL https://openreview.net/forum?id= KQoVgPOM1S

    work page 2025

  18. [18]

    N. Roy, R. D¨ urr, A. B¨ uck, and S. Sundar, arXiv preprint arXiv:2501.00014 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  19. [19]

    Sparse-Supervised Hybrid Parameterized Physics-Informed Neural Networks for Incompressible Flows Across Reynolds Numbers

    A. Jangir, R. Clements, R. Goyal, and G. Tabor, arXiv preprint arXiv:2602.04670 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  20. [20]

    R. Pal, S. Mukherjee, U. Dutta, and A. Choudhury, arXiv preprint arXiv:2511.14497 (2025)

    work page arXiv 2025

  21. [21]

    Eivazi, M

    H. Eivazi, M. Tahani, P. Schlatter, and R. Vinuesa, Physics of Fluids 34 (2022)

    work page 2022

  22. [22]

    Y. Wang, M. Bennani, J. Martens, S. Racani` ere, S. Blackwell, A. Matthews, S. Nikolov, G. Cao-Labora, D. S. Park, M. Arjovsky, et al., arXiv preprint arXiv:2509.14185 (2025)

    work page arXiv 2025

  23. [23]

    E. J. R. Coutinho, M. Dall’Aqua, L. McClenny, M. Zhong, U. Braga-Neto, and E. Gildin, Journal of Computational Physics 489, 112265 (2023)

    work page 2023

  24. [24]

    K. Shah, P. Stiller, N. Hoffmann, and A. Cangi, arXiv preprint arXiv:2210.12522 (2022)

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  25. [25]

    H. Jin, M. Mattheakis, and P. Protopapas, in 2022 International Joint Conference on Neural Networks (IJ CNN) (IEEE, 2022), pp. 1–8

    work page 2022

  26. [26]

    Physics-Informed Neural Networks for One-Dimensional Quantum Well Problems

    S. Sarkar, arXiv preprint arXiv:2504.05367 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  27. [27]

    Harcombe and Q

    L. Harcombe and Q. Deng, Journal of Computational Science 73, 102136 (2023)

    work page 2023

  28. [28]

    First principles physics-informed neural network for quantum wavefunctions and eigenvalue surfaces

    M. Mattheakis, G. R. Schleder, D. T. Larson, and E. Kaxiras, arXiv preprint arXiv:2211.04607 (2022)

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  29. [29]

    R. Luna, J. Calder´ on Bustillo, J. J. S. Mart ´ ınez, A. Torres-Forn´ e, and J. A. Font, Phys. Rev. D 107, 14 064025 (2023), 2212.06103

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  30. [30]

    R. Luna, D. D. Doneva, J. A. Font, J.-H. Lien, and S. S. Yazadjiev, Phys. Rev. D 109, 124064 (2024), 2404.11583

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  31. [31]

    Solving the Pulsar Equation using Physics-Informed Neural Networks

    P. Stefanou, J. F. Urb´ an, and J. A. Pons, Mon. Not. Roy. Astron. Soc. 526, 1504 (2023), 2309.06410

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  32. [32]

    Ferrer-S´ anchez, N

    A. Ferrer-S´ anchez, N. Villanueva-Espinosa, C. H. Morales, R. R. de Austri-Bazan, J. A. Font, J. D. Mart ´ ın- Guerrero, and M. W. Choptuik (2025), 2511.15247

    work page arXiv 2025

  33. [33]

    A. M. Al-Haysah, International Journal of Computer Ap- plications 187, 43 (2026), ISSN 0975-8887, URL https: //ijcaonline.org/archives/volume187/number79/ hybrid-numerical-and-machine-learning-approaches-fo r-solving-einstein-constraint-equation/

    work page 2026

  34. [34]

    Pretorius, Phys

    F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005), gr- qc/0507014

    work page arXiv 2005

  35. [35]

    J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. Lett. 96, 111102 (2006), gr- qc/0511103

    work page arXiv 2006

  36. [36]

    Campanelli, C

    M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, Phys. Rev. Lett. 96, 111101 (2006), gr- qc/0511048

    work page arXiv 2006

  37. [37]

    Shibata, Phys

    M. Shibata, Phys. Rev. D 60, 104052 (1999), gr- qc/9908027

    work page arXiv 1999

  38. [38]

    Simulation of merging binary neutron stars in full general relativity: $\Gamma=2$ case

    M. Shibata and K. Uryu, Phys. Rev. D 61, 064001 (2000), gr-qc/9911058

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  39. [39]

    Merger of black hole-neutron star binaries in full general relativity

    M. Shibata and K. Uryu, Class. Quant. Grav. 24, S125 (2007), astro-ph/0611522

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  40. [40]

    Z. B. Etienne, Y. T. Liu, S. L. Shapiro, and T. W. Baum- garte, Phys. Rev. D 79, 044024 (2009), 0812.2245

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  41. [41]

    B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. D 93, 122003 (2016), 1602.03839

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  42. [42]

    G. B. Cook, Living Rev. Rel. 3, 5 (2000), gr-qc/0007085

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  43. [43]

    H. P. Pfeiffer, J. Hyperbol. Diff. Equat. 2, 497 (2005), gr-qc/0412002

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  44. [44]

    Asymptotically Flat Initial Data with Prescribed Regularity at Infinity

    S. Dain and H. Friedrich, Commun. Math. Phys. 222, 569 (2001), gr-qc/0102047

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  45. [45]

    H.-J. Yo, J. N. Cook, S. L. Shapiro, and T. W. Baum- garte, Phys. Rev. D 70, 084033 (2004), [Erratum: Phys.Rev.D 70, 089904 (2004)], gr-qc/0406020

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  46. [46]

    A single-domain spectral method for black hole puncture data

    M. Ansorg, B. Bruegmann, and W. Tichy, Phys. Rev. D 70, 064011 (2004), gr-qc/0404056

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  47. [47]

    H. P. Pfeiffer and J. W. York, Jr., Phys. Rev. D 67, 044022 (2003), gr-qc/0207095

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  48. [48]

    J. M. Bowen and J. W. York, Phys. Rev. D 21, 2047 (1980), URL https://link.aps.org/doi/10. 1103/PhysRevD.21.2047

    work page 2047

  49. [49]

    A simple construction of initial data for multiple black holes

    S. Brandt and B. Bruegmann, Phys. Rev. Lett. 78, 3606 (1997), gr-qc/9703066

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  50. [50]

    C. O. Lousto and Y. Zlochower, Phys. Rev. D 77, 024034 (2008), 0711.1165

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  51. [51]

    Cao, Phys

    Z. Cao, Phys. Rev. D 91, 044033 (2015), URL https: //link.aps.org/doi/10.1103/PhysRevD.91.044033

    work page doi:10.1103/physrevd.91.044033 2015

  52. [52]

    W. J. Morokoff and R. E. Caflisch, Journal of Computational Physics 122, 218 (1995), ISSN 0021- 9991, URL https://www.sciencedirect.com/science/ article/pii/S0021999185712090

    work page 1995

  53. [53]

    Sobol’, USSR Computational Mathematics and Mathematical Physics 7, 86 (1967), ISSN 0041- 5553, URL https://www.sciencedirect.com/science/ article/pii/0041555367901449

    I. Sobol’, USSR Computational Mathematics and Mathematical Physics 7, 86 (1967), ISSN 0041- 5553, URL https://www.sciencedirect.com/science/ article/pii/0041555367901449

    work page arXiv 1967

  54. [54]

    Gaussian Error Linear Units (GELUs)

    D. Hendrycks and K. Gimpel, arXiv preprint arXiv:1606.08415 (2016)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  55. [55]

    Elfwing, E

    S. Elfwing, E. Uchibe, and K. Doya, Neural networks 107, 3 (2018)

    work page 2018

  56. [56]

    D. P. Kingma and J. Ba, arXiv preprint arXiv:1412.6980 (2014)

    work page internal anchor Pith review Pith/arXiv arXiv 2014