gr-qcNov 1, 2026

Black Hole Love Numbers with Physics-Informed Neural Networks

Authors: Maria J. Rodriguez, Luis Fernando Temoche, Pavlos Protopapas

Organizations: Department of Physics, Utah State University, 4415 Old Main Hill Road, UT 84322, USA · Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid, 13-15 Calle Nicolás Cabrera, 28049 Madrid, Spain · John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

Abstract

We develop and validate a physics-informed neural network (PINN) framework for computing the tidal Love numbers of black holes directly from their perturbation equations. We consider static perturbations of the Kerr black hole, whose exact hypergeometric solutions provide a controlled benchmark for both the numerical solver and the extraction of the physical response. The framework combines exact enforcement of the horizon boundary conditions with an H2H^2-Sobolev loss that incorporates derivatives of the differential-equation residual. The Love number is then obtained by extracting the asymptotic coefficients of the learned solution, without using the analytic connection formula. Across the tested scalar, vector, and tensor perturbations, the method recovers the finite dissipative response with relative errors below 10−510^{-5}. Controlled comparisons show that the higher-derivative loss terms improve solution accuracy by orders of magnitude, while independent tests of the extraction procedure distinguish solver errors from errors in recovering the response coefficients. The conservative response, which vanishes analytically, is constrained by the numerical error floor. These results demonstrate that PINNs can accurately compute black-hole tidal response and establish a validated framework for future extensions to frequency-dependent and non-separable perturbation problems.

Figures & tables

Figure 1: Trained pinn output (red dashed) versus the Frobenius series (black solid), together with the first and second zz -derivatives. Left column: u=Re⁡y~u=\operatorname{Re}\tilde{y} and right column: v=Im⁡y~v=\operatorname{Im}\tilde{y} . Configuration s=2s=2 , ℓint=2\ell_{\rm int}=2 , m=2m=2 , ε=10−5\varepsilon=10^{-5} and 25002500 epochs.
Figure 2: Pointwise error of the trained solution and of its first two derivatives against the exact series, on a logarithmic scale, for the configuration of Figure 1 . Left: real part and right: imaginary part.
MethodFrobeniusPINN’s
 Re⁡kℓ\operatorname{Re}k_{\ell}Im⁡kℓ\operatorname{Im}k_{\ell}Re⁡kℓ\operatorname{Re}k_{\ell}Im⁡kℓ\operatorname{Im}k_{\ell}
19.67×10−59.67\times 10^{-5}−3.04×10−2-3.04\times 10^{-2}6.86×10−56.86\times 10^{-5}−3.07×10−2-3.07\times 10^{-2}
29.74×10−59.74\times 10^{-5}−3.04×10−2-3.04\times 10^{-2}1.35×10−41.35\times 10^{-4}−3.07×10−2-3.07\times 10^{-2}
3−1.21×103-1.21\times 10^{3}2.50×1032.50\times 10^{3}8.75×1048.75\times 10^{4}3.50×1043.50\times 10^{4}
49.68×10−59.68\times 10^{-5}−3.04×10−2-3.04\times 10^{-2}−1.30×10−3-1.30\times 10^{-3}−3.16×10−2-3.16\times 10^{-2}
51.00×10−41.00\times 10^{-4}−3.04×10−2-3.04\times 10^{-2}2.82×10−32.82\times 10^{-3}−2.74×10−2-2.74\times 10^{-2}
Table 1: Love number value kℓk_{\ell} for ℓ=m=s=2\ell=m=s=2 , with ϵ=1×10−3\epsilon=1\times 10^{-3} and using only the ADAM optimizer. The analytical reference is kℓ=9.63×10−5−3.04×10−2ik_{\ell}=9.63\times 10^{-5}-3.04\times 10^{-2}i . The pinn values therefore differ from Table 8 , which reports the median over three seeds at 50005000 epochs.
MethodFrobeniusPINN’s
 ∣δA∣|\delta A|∣δB∣|\delta B|∣δA∣|\delta A|∣δB∣|\delta B|
11.63×10−41.63\times 10^{-4}5.97×10−75.97\times 10^{-7}5.66×10−35.66\times 10^{-3}5.07×10−35.07\times 10^{-3}
22.90×10−42.90\times 10^{-4}5.31×10−75.31\times 10^{-7}4.57×10−34.57\times 10^{-3}5.04×10−35.04\times 10^{-3}
39.15×1049.15\times 10^{4}4.58×10−74.58\times 10^{-7}3.09×1063.09\times 10^{6}5.19×10−35.19\times 10^{-3}
41.07×10−31.07\times 10^{-3}2.98×10−72.98\times 10^{-7}5.71×10−25.71\times 10^{-2}5.32×10−35.32\times 10^{-3}
51.32×10−31.32\times 10^{-3}3.67×10−73.67\times 10^{-7}1.37×10−11.37\times 10^{-1}5.18×10−35.18\times 10^{-3}
Table 2: Error values for AdecA_{dec} and BgrowB_{grow} for ℓ=m=s=2\ell=m=s=2 , with ϵ=1×10−3\epsilon=1\times 10^{-3} and using only the ADAM optimizer. The analytical reference is kℓ=9.63×10−5−3.04×10−2ik_{\ell}=9.63\times 10^{-5}-3.04\times 10^{-2}i .
ε\varepsilon∣Adec∣|A_{\rm dec}|∣BgrowcN∣|B_{\rm grow}c_{N}|∣cN∣|c_{N}|∣Aeff∣/∣Adec∣|A_{\rm eff}|/|A_{\rm dec}|∣cN+kℓ∣|c_{N}+k_{\ell}|
10−210^{-2}0.2163100.2163100.2194160.2194160.0298930.0298931.58×10−21.58\times 10^{-2}4.67×10−44.67\times 10^{-4}
10−410^{-4}0.2181980.2181980.2182290.2182290.0305080.0305081.60×10−41.60\times 10^{-4}4.88×10−64.88\times 10^{-6}
10−510^{-5}0.2182160.2182160.2182190.2182190.0305140.0305141.60×10−51.60\times 10^{-5}4.88×10−74.88\times 10^{-7}
10−810^{-8}0.2182180.2182180.2182180.2182180.0305140.0305141.60×10−81.60\times 10^{-8}4.88×10−104.88\times 10^{-10}
Table 3: The quantity returned by the local-derivative extraction methods at s=ℓint=m=2s=\ell_{\rm int}=m=2 ( N=5N=5 ). cNc_{N} is the coefficient of (1−z)N(1-z)^{N} in f2f_{2} , Eq. ( 45 ). The magnitude ∣cN∣|c_{N}| remains finite as ε→0\varepsilon\to 0 , indicating that the contamination arises from cancellation rather than divergence. The last column shows cN→−kℓc_{N}\to-k_{\ell} linearly in ε\varepsilon , while ∣Aeff∣/∣Adec∣=1.60 ε|A_{\rm eff}|/|A_{\rm dec}|=1.60\,\varepsilon for ε\varepsilon from 10−210^{-2} to 10−810^{-8} .
 integer exponent NNexact exponent NεN_{\varepsilon}
ε\varepsilon∣δA∣|\delta A|∣δB∣|\delta B|∣δA∣|\delta A|∣δB∣|\delta B|
10−110^{-1}1.623×10−21.623\times 10^{-2}4.165×10−54.165\times 10^{-5}2.1×10−162.1\times 10^{-16}2.0×10−162.0\times 10^{-16}
10−210^{-2}1.667×10−31.667\times 10^{-3}5.589×10−65.589\times 10^{-6}5.5×10−165.5\times 10^{-16}2.7×10−162.7\times 10^{-16}
10−310^{-3}1.671×10−41.671\times 10^{-4}5.785×10−75.785\times 10^{-7}1.3×10−161.3\times 10^{-16}3.5×10−163.5\times 10^{-16}
10−410^{-4}1.672×10−51.672\times 10^{-5}5.805×10−85.805\times 10^{-8}8.4×10−168.4\times 10^{-16}6.3×10−166.3\times 10^{-16}
10−510^{-5}1.672×10−61.672\times 10^{-6}5.807×10−95.807\times 10^{-9}3.9×10−163.9\times 10^{-16}6.2×10−176.2\times 10^{-17}
Table 4: Relative error of the integral-residual extraction applied to the Frobenius series at s=ℓint=m=2s=\ell_{\rm int}=m=2 ( N=5N=5 ), computed in mpmath at 5050 digits. Left: the superseded integer exponent (1−z)N(1-z)^{N} . Right: the exact exponent (1−z)Nε(1-z)^{N_{\varepsilon}} . The apparent “conditioning floor” of the connection problem reported previously is an artifact of the exponent mismatch ( 48 ).
variantloss ∗ϵy\epsilon_{y}seed range in ϵy\epsilon_{y}ϵA\epsilon_{A}
H2H^{2} (0.1, 0.01)(0.1,\,0.01)1.17×10−41.17\times 10^{-4}9.97×10−59.97\times 10^{-5}[0.71, 1.85]×10−4[0.71,\,1.85]\times 10^{-4}1.08×10−41.08\times 10^{-4}
H1H^{1} (0.1, 0)(0.1,\,0)4.59×10−34.59\times 10^{-3}2.33×10−12.33\times 10^{-1}[2.22, 3.16]×10−1[2.22,\,3.16]\times 10^{-1}2.36×10−12.36\times 10^{-1}
L2L^{2} (0, 0)(0,\,0)1.11×10−21.11\times 10^{-2}4.89×10−14.89\times 10^{-1}[4.88, 5.81]×10−1[4.88,\,5.81]\times 10^{-1}4.91×10−14.91\times 10^{-1}
Table 5: Effect of the Sobolev order at s=ℓint=m=2s=\ell_{\rm int}=m=2 , ε=10−3\varepsilon=10^{-3} , 50005000 epochs, extraction by Method 1, three seeds per row. Median over seeds, with the seed range for ϵy\epsilon_{y} in brackets. The loss column is not comparable between rows.
λ1\lambda_{1}λ2\lambda_{2}ϵy\epsilon_{y}seed range in ϵy\epsilon_{y}ϵA\epsilon_{A}
0.10.10.010.011.22×10−41.22\times 10^{-4}[0.76, 2.18]×10−4[0.76,\,2.18]\times 10^{-4}1.12×10−41.12\times 10^{-4}
0.010.010.010.011.11×10−41.11\times 10^{-4}[1.07, 1.67]×10−4[1.07,\,1.67]\times 10^{-4}1.10×10−41.10\times 10^{-4}
110.010.011.57×10−41.57\times 10^{-4}[1.25, 2.16]×10−4[1.25,\,2.16]\times 10^{-4}1.54×10−41.54\times 10^{-4}
0.10.10.0010.0013.18×10−43.18\times 10^{-4}[2.34, 4.23]×10−4[2.34,\,4.23]\times 10^{-4}3.40×10−43.40\times 10^{-4}
0.10.10.10.13.71×10−53.71\times 10^{-5}[3.25, 4.67]×10−5[3.25,\,4.67]\times 10^{-5}2.73×10−52.73\times 10^{-5}
Table 6: Sensitivity to the Sobolev coefficients at s=ℓint=m=2s=\ell_{\rm int}=m=2 , ε=10−3\varepsilon=10^{-3} , 50005000 epochs, extraction by Method 1, three seeds per row; median over seeds, with the seed range for ϵy\epsilon_{y} in brackets. The first row is the default. It was run separately from Table 5 , and the ∼20%\sim 20\% difference from the corresponding row there reflects run-to-run variation.
Figure 3: Solution error ϵy\epsilon_{y} (circles; bars show the range over three seeds) and coefficient error ϵA\epsilon_{A} (squares) when varying λ1\lambda_{1} at λ2=0.01\lambda_{2}=0.01 (left) and λ2\lambda_{2} at λ1=0.1\lambda_{1}=0.1 (right). Same runs as Table 6 .
Sobolevweightlossϵy\epsilon_{y}ϵA\epsilon_{A}ϵy\epsilon_{y} spread
H2H^{2}1/[z(1−z)]1/[z(1-z)]5.28×10−45.28\times 10^{-4}2.15×10−42.15\times 10^{-4}2.06×10−42.06\times 10^{-4}[1.3, 3.0]×10−4[1.3,\,3.0]\times 10^{-4}
H2H^{2}112.88×10−52.88\times 10^{-5}4.47×10−44.47\times 10^{-4}4.63×10−44.63\times 10^{-4}[3.8, 5.2]×10−4[3.8,\,5.2]\times 10^{-4}
L2L^{2}1/[z(1−z)]1/[z(1-z)]2.40×10−22.40\times 10^{-2}5.63×10−15.63\times 10^{-1}5.67×10−15.67\times 10^{-1}[5.1, 6.1]×10−1[5.1,\,6.1]\times 10^{-1}
L2L^{2}111.21×10−31.21\times 10^{-3}6.05×10−16.05\times 10^{-1}6.11×10−16.11\times 10^{-1}[5.6, 6.5]×10−1[5.6,\,6.5]\times 10^{-1}
Table 7: Sobolev order crossed with endpoint weight, at s=ℓint=m=2s=\ell_{\rm int}=m=2 , ε=10−3\varepsilon=10^{-3} , 30003000 epochs, two seeds, extraction by Method 1. Median over seeds. The loss column is the value of a different functional in every row and is not comparable between them.
 kℓk_{\ell} ( pinn )kℓk_{\ell} (analytical) 
ε\varepsilonRe⁡kℓ\operatorname{Re}k_{\ell}Im⁡kℓ\operatorname{Im}k_{\ell}Re⁡kℓ\operatorname{Re}k_{\ell}Im⁡kℓ\operatorname{Im}k_{\ell}∣ΔRe⁡kℓ∣|\Delta\operatorname{Re}k_{\ell}|
10−110^{-1}+7.057×10−3+7.057\times 10^{-3}−2.1512×10−2-2.1512\times 10^{-2}+7.0495×10−3+7.0495\times 10^{-3}−2.15166×10−2-2.15166\times 10^{-2}7.5×10−67.5\times 10^{-6}
10−210^{-2}+9.377×10−4+9.377\times 10^{-4}−2.9453×10−2-2.9453\times 10^{-2}+9.3337×10−4+9.3337\times 10^{-4}−2.94547×10−2-2.94547\times 10^{-2}4.3×10−64.3\times 10^{-6}
10−310^{-3}+1.017×10−4+1.017\times 10^{-4}−3.0403×10−2-3.0403\times 10^{-2}+9.6321×10−5+9.6321\times 10^{-5}−3.04064×10−2-3.04064\times 10^{-2}5.4×10−65.4\times 10^{-6}
10−410^{-4}+1.485×10−5+1.485\times 10^{-5}−3.0501×10−2-3.0501\times 10^{-2}+9.6628×10−6+9.6628\times 10^{-6}−3.05034×10−2-3.05034\times 10^{-2}5.2×10−65.2\times 10^{-6}
Table 8: Love number versus ε\varepsilon at s=ℓint=m=2s=\ell_{\rm int}=m=2 , extraction by Method 1, median over three seeds at 50005000 epochs. The analytical columns are exact and convention-independent. The last column is the absolute error in the conservative part; it is flat while Re kℓ\mathrm{Re}\,k_{\ell} itself falls by two orders of magnitude.
     Re⁡kℓ\operatorname{Re}k_{\ell}Im⁡kℓ\operatorname{Im}k_{\ell}rel. error in Im⁡kℓ\operatorname{Im}k_{\ell}
ssℓint\ell_{\rm int}mmdegϵy\epsilon_{y}pinnanalyticalanalyticalpinnN≡0\mathcal{N}\equiv 0
01111.5×10−61.5\times 10^{-6}7.71×10−57.71\times 10^{-5}7.72×10−57.72\times 10^{-5}−2.159×10−2-2.159\times 10^{-2}4.9×10−64.9\times 10^{-6}1.3×10−41.3\times 10^{-4}
02121.5×10−71.5\times 10^{-7}5.39×10−65.39\times 10^{-6}5.39×10−65.39\times 10^{-6}−1.508×10−3-1.508\times 10^{-3}5.5×10−85.5\times 10^{-8}1.01.0
03137.6×10−77.6\times 10^{-7}3.55×10−73.55\times 10^{-7}3.54×10−73.54\times 10^{-7}−9.898×10−5-9.898\times 10^{-5}6.7×10−66.7\times 10^{-6}6.46.4
04142.1×10−72.1\times 10^{-7}2.27×10−82.27\times 10^{-8}2.27×10−82.27\times 10^{-8}−6.359×10−6-6.359\times 10^{-6}5.1×10−65.1\times 10^{-6}3030
11101.6×10−61.6\times 10^{-6}−1.54×10−4-1.54\times 10^{-4}−1.54×10−4-1.54\times 10^{-4}4.316×10−24.316\times 10^{-2}6.2×10−76.2\times 10^{-7}1.3×10−61.3\times 10^{-6}
Table 9: Love numbers at ε=10−3\varepsilon=10^{-3} , aKerr=0.2a_{\rm Kerr}=0.2 , from Method 1. Rows with s≥1s\geq 1 are computed at spin weight −s-s , which gives the same kℓk_{\ell} . “deg” is the polynomial degree ℓint−∣s∣\ell_{\rm int}-|s| of y~\tilde{y} at ε=0\varepsilon=0 for the equation that is solved. The last column is the error obtained with the network output set to zero, y~=1+y~′(0) z\tilde{y}=1+\tilde{y}^{\prime}(0)\,z , i.e. from the hard boundary condition alone; rows where it is already small (deg ≤1\leq 1 ) are consistency checks. Single seed, 750750 Adam epochs ( 10001000 for (2,3,2)(2,3,2) ) followed by L-BFGS.
Figure 4: Validation loss versus wall-clock time for (s,ℓint,m)=(2,3,2)(s,\ell_{\rm int},m)=(2,3,2) , with the equation solved at s=+2s=+2 and at s=−2s=-2 , which give the same kℓk_{\ell} . The dotted lines mark the switch from Adam to L-BFGS ( torch.optim.LBFGS ), at about 2424 min for s=+2s=+2 (blue) and about 44 min for s=−2s=-2 (red). The validation loss is the weighted H2H^{2} residual on a fixed grid of 40004000 points in z∈[0.01,0.99]z\in[0.01,0.99] that is not used for training; times are for one CPU. Final solution errors: ϵy≈0.09\epsilon_{y}\approx 0.09 ( s=+2s=+2 ) and 4×10−74\times 10^{-7} ( s=−2s=-2 ).
Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

 r=r+r=r_{+} ( z=0z=0 )r→∞r\to\infty ( z=1z=1 )r=r−r=r_{-} ( z=∞z=\infty )
R^s\hat{R}_{s}±(s2−imγ)\pm\bigl(\tfrac{s}{2}-im\gamma\bigr)ℓ,  −(ℓ+1)\ell,\;-(\ell+1)—
yy0,  1−c=s−2imγ0,\;1-c=s-2im\gamma0,  c−a−b=−(2ℓ+1)0,\;c-a-b=-(2\ell+1)a, ba,\,b
Appendix
Table 14
More in this topic

Astrophysics

See all 39 papers

Explore similar work

CardsList

  1. Black Hole Black Boxes: Numerical Black Hole Metrics via AInstein Neural Networks

    Jul 6, 2026Tancredi Schettini Gherardini, Edward Hirst, Alexander George StapletonPDE SolvingPhysics-Informed ML

    Instituto de Matemática, Estatística e Computação Científica (IMECC) da Universidade Estadual de Campinas (UNICAMP), 13083-859, Brasil · Mathematisches Institut (MI), University of Bonn, Bonn, Germany · Max Planck Institute for Mathematics, Bonn, Germany +1

  2. Silent Failures in Physics-Informed Neural Networks: Parameter Poisoning and the Limits of Loss-Based Validation

    Jun 23, 2026David McShannon, Nicholas DietrichModel MisspecificationPhysics-Informed ML

  3. An Optimisation Framework for the Well-Conditioned Training of Physics-Informed Neural Networks

    Jul 2, 2026Joseph Webb, Sadok Jerad, Coralia CartisGauss-Newton OptimizationPDE Solving

    Mathematical Institute, University of Oxford, Oxford, United Kingdom