PINNing the pion: conformal deep learning for $F_π(s)$ and the $(g-2)_μ$ hadronic contribution

Mayank Goel; Subhadip Mitra; Monalisa Patra

hep-phSep 30, 2026

PINNing the pion: conformal deep learning for $F_π(s)$ and the $(g-2)_μ$ hadronic contribution

Authors: Mayank Goel, Subhadip Mitra, Monalisa Patra

Organizations: Center for Computational Natural Sciences and Bioinformatics, International Institute of Information Technology, Hyderabad 500 032, India

Abstract

Extracting the pion electromagnetic form factor $F_π(s)$ through phenomenological curve-fitting models introduces model dependence, unphysical artefacts, and kinematic inconsistencies. We introduce a Physics-Informed Neural Network (PINN) embedded in a conformal $z$ -plane that constructs $F_π(s)$ directly from first principles across spacelike and timelike domains: charge normalisation and Schwarz reflection are enforced by construction, while Cauchy-Riemann analyticity, dispersion relations, Watson's theorem, and perturbative QCD asymptotics enter through the loss functional. Thus, the fundamental S-matrix principles dictate the form factor's behaviour while data act as constraints. Mapping the cut complex plane onto the unit disk bounds the Hessian norm and prevents Neural Tangent Kernel spectral starvation, two known failure modes of deep-learning optimisation. Besides $e^+e^-$ scattering data, we also incorporate $τ$ -decay data through a switch that isolates the pure isovector form factor natively, bypassing model-dependent isospin-breaking pre-corrections. The network organically yields an interior zero-free form factor, while the framework tests experimental tensions around the $ρ(770)$ peak against analyticity and dispersion constraints. We obtain model-independent estimates of the pion charge radius, $\langle r_π^2 \rangle = 0.435 \pm 0.008_{\text{stat}} \pm 0.007_{\text{cali}}$ fm $^2$ , the second-sheet pole parameters, $m_ρ^{\text{pole}} = 761.72\pm 1.04$ MeV and $Γ_ρ^{\text{pole}} = 135.99 \pm 1.20$ MeV, and the two-pion contribution to the muon anomalous magnetic moment, $a_μ^{ππ} = (506.48 \pm 2.02_{\text{stat}} \pm 1.70_{\text{cali}}) \times 10^{-10}$ .

Figures & tables

Figure 1: The conformal map

z(s)

compactifying the physical

\mathcal{S}

plane onto the unit disk. The physical branch cut

[4m_{\pi}^{2},\infty)

maps to the unit circle boundary. The

\rho(770)

resonance pole lies on the second Riemann sheet, outside the unit disk at

z_{\rho(770)}\approx 0.8-0.7i

, ensuring

F_{\pi}

is holomorphic throughout the interior

|z|<1

Figure 2: Network schematic: PINN architecture for the pion form factor $F_{\pi}(s)$ in the conformal $\mathcal{Z}$ plane. The physical variable $s$ is first mapped to the unit disk via the conformal bijection $z(s)$ . The network takes three inputs – $\text{Re}\,z$ , $\text{Im}\,z$ , and a channel switch distinguishing the $e^{+}e^{-}$ and $\tau$ -decay datasets – and passes them through four hidden layers of $128$ neurons each with adaptive sine activations. The output $\mathcal{N}(z)$ is combined with the structural ansatz $F_{\pi}(z)=1+s(z)\mathcal{N}(z)$ , which hardwires the charge normalisation $F_{\pi}(0)=1$ . Training minimises a composite loss $\mathcal{L}=\lambda_{\rm data}\mathcal{L}_{\rm data}+\lambda_{\rm phys}\mathcal{L}_{\rm phys}$ , where the physics loss enforces unitarity (Watson’s theorem), analyticity (Cauchy-Riemann conditions via stochastic collocation), the Schwarz reflection principle, the twice-subtracted dispersion relation, and perturbative QCD asymptotics simultaneously.

Scattering channel	Experiment	Data points	Energy range (in GeV 2 )
Spacelike data			$Q^{2}=-s$
$e\pi\to e\pi$	NA7 [ 6 ]	$45$	$0.015-0.253$
$e\pi\to e\pi$	Fermilab [ 36 ]	20	$0.03-0.07$
$e^{-}p\to e^{-}\pi^{+}n$	CEA [ 26 ]	5	$0.176-1.188$
	Cornell ’71 data [ 17 ]	6	$0.62-2.015$
	JLab [ 21 ]	8	$0.60-2.45$

Table 1: Experimental datasets used for training the neural network.

Panel of Figure 3 — Figure 3: (a) The pion form factor analytically continued on the complex $\mathcal{S}$ plane; (b) its argument validating the Schwarz relation. The peak is at the $\rho(770)$ resonance. The discontinuity across the branch cut is clearly visible in the argument.

Panel of Figure 4 — Figure 4: Heat map of $|F_{\pi}(s)|$ confirming that the predicted form factor contains no zeros in the interior of the $\mathcal{S}$ plane.

Figure 6: The predicted pion electromagnetic form factor squared $|F_{\pi}(s)|^{2}$ (black curve) with its associated $1\sigma$ uncertainty band (imperceptible at this scale as $\sigma\sim 10^{-2}$ ), plotted alongside experimental data across both spacelike ( $s<0$ ) and timelike ( $s>4m_{\pi}^{2}$ ) kinematic regions. The plot highlights the continuous transition through the low-energy region, the primary $\rho(770)$ resonance peak, and the narrow $\rho-\omega$ interference structure around $s\approx 0.6\text{ GeV}^{2}$ .

Figure 7: PINN prediction for the timelike region with the $e^{+}e^{-}$ datasets showing the $\rho$ resonance along with the dip-bump pattern from the $\rho$ – $\omega$ interference.

Figure 8: PINN prediction for the spacelike region, overlaid on the experimental data.

Figure 9: Asymptotic scaling of the pion form factor multiplied by the squared momentum transfer, $Q^{2}|F_{\pi}(Q^{2})|$ , in the spacelike domain up to $Q^{2}=-s=20\text{ GeV}^{2}$ . The solid black curve shows the PINN framework prediction. For comparison, Curve A (blue dashed) represents a QCD theoretical model bridging large and short distance scales; Curve B (orange dash-dotted) corresponds to a standard monopole parametrisation; and Curves C (green dash-dotted) and D (red dotted) illustrate high-energy short-distance quark-gluon approaches valid for $Q^{2}\gtrsim 7\text{ GeV}^{2}$ .

Figure 10: The extracted phase of the form factor (black-solid) compared with the Roy equation data (orange-dashed) from Ref. [ 34 ] . The blue dotted curve shows the PINN prediction with Watson loss disabled ( $\lambda_{\text{Watson}}=0$ ), illustrating the systematic phase drift that results once the loss is removed.

Observable	Previous estimation (in fm n )	Our result (in fm n )
$\langle r_{\pi}^{2}\rangle$	Lattice [ 40 ] : $0.42\pm 0.02$	$0.435\pm 0.008\pm 0.007$
$\langle r_{\pi}^{2}\rangle$	PDG [ 49 ] : $0.434\pm 0.005$	$0.435\pm 0.008\pm 0.007$
$\langle r_{\pi}^{4}\rangle$	$2$ -loop ChPT [ 19 ] : $0.7\pm 0.11$	$0.748\pm 0.057\pm 0.048$
$\langle r_{\pi}^{4}\rangle$	Ref. [ 9 ] : $0.68-0.72$	$0.748\pm 0.057\pm 0.048$
$\langle r_{\pi}^{6}\rangle$	Ref. [ 9 ] : $2.95-3.11$	$4.01\pm 0.95\pm 0.90$
$\langle r_{\pi}^{6}\rangle$	Ref. [ 58 ] : $2.89\pm 0.12$	$4.01\pm 0.95\pm 0.90$

Table 2: Comparison of the extracted pion charge radius squared

\langle r_{\pi}^{2}\rangle

, fourth-order radius

\langle r_{\pi}^{4}\rangle

, and sixth-order radius

\langle r_{\pi}^{6}\rangle

from our PINN framework against existing theoretical, lattice QCD, ChPT, and empirical estimates from the literature. All values are given in units of

\text{fm}^{n}

(where

n=2,4,6

). Errors for our work are presented in the form

\hat{\mathcal{O}}\pm\sigma_{\text{stat}}\pm\sigma_{\text{cali}}

as defined in Sec. V.5

(GeV)	Previous estimations	Our estimation
Energy range	Previous estimations	Our estimation
$[2m_{\pi},0.63]$	$133.258\pm 0.723$ [ 7 ]	$130.66\pm 0.67\pm 1.20$
$[2m_{\pi},0.63]$	$130.47\pm 0.64$ [ 34 ]	$130.66\pm 0.67\pm 1.20$
$[2m_{\pi},1.0]$	$495.0\pm 2.58$ [ 34 ]	$496.37\pm 2.08\pm 1.70$
$[2m_{\pi},\infty]$	$513.2\pm 3.8$ [ 37 ]	$506.48\pm 2.02\pm 1.70$
$[2m_{\pi},\infty]$	$507.9\pm 3.29$ [ 12 ]	$506.48\pm 2.02\pm 1.70$

Table 3: Comparison of the two-pion hadronic vacuum polarisation contribution to the muon anomalous magnetic moment,

a_{\mu}^{\pi\pi}\times 10^{10}

, integrated over various centre-of-mass energy intervals (

\sqrt{s}

in GeV). Our PINN estimations are benchmarked against existing dispersive and data-driven determinations from the literature. Errors for our work are presented in the form

\hat{\mathcal{O}}\pm\sigma_{\text{stat}}\pm\sigma_{\text{cali}}

as defined in Sec. V.5 .

Data ablation (baseline sets)
Configuration	$\langle r_{\pi}^{2}\rangle$ ( $\text{fm}^{2}$ )	$\langle r_{\pi}^{4}\rangle$ ( $\text{fm}^{4}$ )	$\langle r_{\pi}^{6}\rangle$ ( $\text{fm}^{6}$ )	$a_{\mu}^{\pi\pi}\times 10^{10}$			$\|\Delta\delta_{1}^{1}(s)\|$ in degrees
Configuration	$\langle r_{\pi}^{2}\rangle$ ( $\text{fm}^{2}$ )	$\langle r_{\pi}^{4}\rangle$ ( $\text{fm}^{4}$ )	$\langle r_{\pi}^{6}\rangle$ ( $\text{fm}^{6}$ )	$[2m_{\pi},0.63]$	$[2m_{\pi},1.0]$	$[2m_{\pi},\infty]$	$s=s_{0}$	$s=s_{1}$
Set A (no CMD-3)	$\mathbf{0.435}$	$\mathbf{0.748}$	$\mathbf{4.01}$	$\mathbf{130.66}$	$\mathbf{496.37}$	$\mathbf{506.48}$	$\mathbf{0.25}$	$\mathbf{1.72}$
Set B (CMD-3 only)	$0.447$	$0.738$	$3.335$	$137.71$	$520.08$	$532.27$	$0.08$	$5.08$
Set C raw ( $\tau$ -decay)	$0.421$	$0.665$	$2.709$	$131.36$	$502.34$	$512.79$	$1.15$	$6.65$
Set C ${}_{\rm R_{IB}}$ ( $\tau$ -decay)	$0.437$	$0.817$	$5.458$	$127.40$	$492.51$	$502.83$	$0.19$	$6.28$

Table 4: Performance comparison of pion form factor moments

\langle r_{\pi}^{2n}\rangle

, integrated HVP contributions

a_{\mu}^{\pi\pi}\times 10^{10}

, and P-wave phase-shift deviations

|\Delta\delta_{1}^{1}(s)|

from the Roy equation at

s_{0}=(0.8\text{ GeV})^{2}

and

s_{1}=(1.15\text{ GeV})^{2}

across dataset variations (Sets A-D) and physics-loss ablation studies on Set A.

Panel of Figure 11 — Figure 11: Reconstruction of $|F_{\pi}(s)|^{2}$ using $\tau$ -decay data in the timelike region under two distinct treatment schemes: (a) Set $C_{\rm raw}$ , where raw $\tau$ -decay data are directly fitted while isolating the pure $I=1$ isovector form factor $F_{\pi}^{I=1}(s)$ by setting the $\rho-\omega$ mixing parameter to zero ( $\alpha_{\rho-\omega}=0$ ); and (b) Set $C_{R_{\rm IB}}$ , where raw $\tau$ data are explicitly pre-corrected for isospin-breaking effects using the point-by-point $R_{IB}(s)$ factor before training. Both panels display the resulting PINN predictions overlaid against experimental data points from BELLE and CLEO.

Panel of Figure 12 — Figure 12: Discrepancy between the full PINN prediction and its $[4,4]$ Padé approximation across the complex $s$ -plane. (a) Absolute difference in magnitude, $\big||F_{\pi}^{\rm PINN}(s)|-|F_{\pi}^{\text{Pad\'{e}}}(s)|\big|$ : the 3D surface (left) zooms into the region marked by the yellow box in the 2D heat map (right), which shows the relative error over the full sampled domain. (b) Corresponding phase difference, $\big|\mathrm{Arg}(F_{\pi}^{\rm PINN})-\mathrm{Arg}(F_{\pi}^{\text{ Pad\'{e}}})\big|$ in degrees, with the same left/right correspondence.

Figure 13: Comparison of $|F_{\pi}(s)|$ from the full PINN (solid black) and the [4/4] Padé approximant (red dashed) along the physical axis, across the spacelike and timelike regions. The two curves are visually indistinguishable through the $\rho(770)$ peak.

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Taming the Loss Landscape of PINNs with Noisy Feynman-Kac Supervision: Operator Preconditioning and Non-Asymptotic Error Bounds

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Parametric Physics-Informed Neural Network Partial Differential Equations Spectral Preconditioning

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Physics-Informed Neural Networks (PINNs) often train slowly or fail to converge on challenging partial differential equations (PDEs), a behavior recently linked to severely ill-conditioned loss landscapes inherited from the underlying differential operator. We study PINNs augmented with a pointwise data-fidelity term, added at a few points in the domain to the standard residual and boundary losses. We show that this supervision term acts as an operator-level preconditioner: for suitable weights, our comparison bounds guarantee a substantially smaller condition number than under the standard PINN loss, independently of how the pointwise labels are obtained. For a broad class of PDEs admitting a Feynman-Kac (FK) representation, we generate such labels by Monte Carlo averages of the FK functional, resulting in what we call ``FK-PINNs", and using the excess risk decomposition approach, we derive non-asymptotic

L^2(Ω)

-error bounds for FK-PINNs with

\tanh

activation trained by finitely many steps of gradient descent. Along the way, we establish pseudo-dimension bounds for first- and second-order derivatives of

\tanh

neural networks, which are of independent interest and, to the best of our knowledge, new. Numerical experiments on Poisson, Schrödinger, mean exit time, and committor problems corroborate the theory, and show that FK-PINNs can successfully solve PDEs for which standard PINNs exhibit severe failure modes.

Abstract

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