S-matrix informed neural networks for amplitude analysis

Wyatt A. Smith; Arkaitz Rodas; Marius D. Thomas; César Fernández-Ramírez; Giorgio Foti; Lin Qiu; Adam P. Szczepaniak; Alessandro Pilloni

hep-phAug 24, 2026

S-matrix informed neural networks for amplitude analysis

Authors: Wyatt A. Smith, Arkaitz Rodas, Marius D. Thomas, César Fernández-Ramírez, Giorgio Foti, Lin Qiu, Adam P. Szczepaniak, Alessandro Pilloni

Organizations: Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA · Department of Physics, University of California, Berkeley, California 94720, USA · Department of Physics, The College of William & Mary, Williamsburg, Virginia 23187, USA · Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universit`a degli Studi di Messina, Messina I-98166, Italy · Theory Center, Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA · Department of Physics, Old Dominion University, Norfolk, Virginia 23529, USA · Departamento de F´ısica Interdisciplinar, Universidad Nacional de Educaci´on a Distancia (UNED), Madrid E-28040, Spain · INFN Sezione di Catania, Catania I-95123, Italy · Center for Exploration of Energy and Matter, Indiana University, Bloomington, Indiana 47403, USA · Department of Physics, Indiana University, Bloomington, Indiana 47405, USA

Abstract

Reconstructing scattering amplitudes from finite, noisy, and mutually inconsistent measurements is an ill-posed inverse problem common to many reactions relevant to particle physics. We introduce S-matrix informed neural networks (SINNs), and demonstrate their ability to learn scattering amplitudes directly from data while respecting first principles. We further develop a novel data selection procedure, which uses the response of constrained neural network ensembles to identify a set of experiments compatible with first principles, and with each other. We apply this framework to $ππ$ scattering, producing reusable amplitudes and correlated uncertainties without relying on a fixed functional form. We validate our results against residual model dependencies and training biases through closure tests and ablations. We find negligible impact of model architecture on our results. Our workflow unifies physics-constrained representation learning, data selection, and uncertainty quantification. Our strategy is transferable to other scattering processes, and other constrained physics problems limited by inconsistent data.

Figures & tables

Figure 1: Experimental input for the

S_{0}

wave before data selection. The figure includes all

S_{0}

phase-shift and inelasticity data studied in this work. The spread between experiments illustrates the data tensions that motivate the selection analysis described in Section V .

Figure 2: SINN architecture employed in our analysis. Each bubble in the Trunk and Heads parts corresponds to a neuron. Postprocessing bubbles correspond to the exact analytic maps of Section IV.1 : threshold factors and the inelasticity map. Color scheme corresponds to the one used for the observables in subsequent figures.

Partial wave	$c$	$\sqrt{s_{c}}\,(\mathrm{GeV})$	$\ell_{c}$
$S_{0}$	$K\bar{K}$	$0.9914$	$0$
$P_{1}$	$\pi\omega$	$0.9222$	$1$
$S_{2}$	effective	$1.1000$	$1$
$D_{0}$	$K\bar{K}$	$0.9914$	$2$
$F_{1}$	$\pi\omega$	$0.9222$	$2$

Table 1: Inelastic-channel assignments used in the postprocessing map of Eq. 14 . The

S_{2}

threshold uses an effective mass

m_{\text{eff}}=0.55~\mathrm{GeV}

. The

D_{2}

and

G_{2}

waves are treated as elastic.

Phase	Epochs	$\lambda_{\mathrm{data}}$	$\lambda_{\mathrm{Roy}}$	$\lambda_{\mathrm{con}}$	$w_{\chi}$
I	0–400	1	0	0	0.25
II	400–800	1	0.01	0.01	0.5
III	800–1000	1	0.5	0.1	0.75
IV	1000–1200	1	1	1	1
V	1200–1500	1	1	10	1
VI	1500–2000	1	2	—	1

Table 2: Curriculum weight schedule. In the final phase, the auxiliary trainable subtraction constants are replaced by network-derived values and

\mathcal{L}_{\mathrm{con}}

is dropped.

\lambda_{\mathrm{SL}}

is not listed because it is held fixed at

1/N_{\mathrm{data}}

in every phase.

Figure 3: Data selection workflow via spectral response clustering. See text for details. $\mathbb{E}$ and $\mathbb{S}$ stand for the sets of all experiments and SINNs, respectively.

Figure 4: Group-level response after selection. The maximum surviving between-group response is $+0.3$ .

Figure 5: Ward-clustering dendrogram for the experiment response matrix. The cut at the largest merge-height gap produces ten groups.

Observable	Included experiments	Excluded experiments
$\|a_{0}^{0}-a_{0}^{2}\|$	DIRAC
$S_{0}$ phase	Grayer (A, B), Kamiński, $K$ decay, ${\rm NA}_{48}$	Grayer (C, D, E), Hyams ( $-$ $-$ $-$ , $-$ $+$ $-$ ), Protopopescu (VI, XIII)
$S_{0}$ inelasticity	Cohen, Etkin, Grayer B, Kamiński, Martin-Ozmutlu, Polychronakos, Wetzel	Hyams ( $-$ $-$ $-$ , $-$ $+$ $-$ ), Protopopescu (VI, XIII)
$P_{1}$ phase	Estabrooks, Hyams 73	Hyams ( $-$ $-$ $-$ , $-$ $+$ $-$ ), Protopopescu (VI, XIII)
$P_{1}$ inelasticity	Hyams 73	Hyams ( $-$ $-$ $-$ , $-$ $+$ $-$ ), Protopopescu (VI, XIII)
$S_{2}$ phase	Cohen, Durusoy (OPE, OPEDP), Hoogland (A,B), Losty (A,B)	—

Table 3: Experimental data used in the final fit. The “Excluded” column lists experiments removed by the selection procedure.

Figure 6: Force-directed tension graph of the experiment groups, rendered as a 2D representation of the experiment embeddings. Nodes are individual experiments within their groups (shaded central circles). The layout pushes (in)compatible groups (apart) closer. Red edges connect groups in tension and blue edges connect compatible groups. Retained groups are drawn with solid outlines and filled markers, while rejected groups are drawn with dashed outlines.

Figure 7:

S_{0}

P_{1}

, and

S_{2}

phase shifts and inelasticities. Solid lines are ensemble medians. Dark and light bands are central 68% and 95% replica intervals.

Figure 8: SINN

t_{\ell}^{I}

amplitudes (red) compared to Roy-reconstructed

\tilde{t}_{\ell}^{I}

amplitudes (blue, dashed) for

S_{0}

P_{1}

, and

S_{2}

, with the residual below each panel. Agreement at the percent level throughout the Roy validity range confirms that crossing symmetry is well satisfied.

Figure 9: (Left panel) Joint distribution of the isospin scattering lengths

a_{0}^{0}

and

a_{0}^{2}

across the

10{,}000

replicas of the production SINN ensemble. (Right panel) the same distribution for the

1{,}000

-network architecture ensemble of Section VII.1 , spanning

100

distinct architectures. In both panels the solid ellipse has half-extents equal to the central

68\%

interval of that ensemble. In the left panel the dashed ellipse adds the spread of the

100

per-architecture medians in quadrature, as an approximation of systematic uncertainty in the production interval. Reference determinations from Refs. [ 33 , 47 ] are shown for comparison.

	This work	Ref. [ 33 ]	Ref. [ 47 ]
$a_{0}^{0}$	$0.222^{+0.014}_{-0.014}$	$0.220\pm 0.005$	$0.220\pm 0.008$
$a_{0}^{2}$	$-0.041^{+0.005}_{-0.006}$	$-0.0444\pm 0.0010$	$-0.042\pm 0.004$

Table 4: Scattering lengths in units of

m_{\pi}^{-1}

. The uncertainty on this work is the central

68\%

replica interval.

Observable	Production	Model average
$a_{0}^{0}$ ( $m_{\pi}^{-1}$ )	$0.222^{+0.014}_{-0.014}$	$0.222^{+0.013}_{-0.017}$
$a_{0}^{2}$ ( $m_{\pi}^{-1}$ )	$-0.041^{+0.005}_{-0.006}$	$-0.041^{+0.005}_{-0.009}$

Table 5: Architecture dependence of the scattering lengths. Values are medians with the central

68\%

interval taken over every member of each ensemble, the full

10{,}000

-replica production ensemble and the

1{,}000

-network architecture ensemble.

Observable	Target	SINN Closure	$z$
$a_{0}^{0}$	$0.2207$	$0.2201^{+0.0102}_{-0.0104}$	$-0.06$
$a_{0}^{2}$	$-0.0432$	$-0.0421^{+0.0028}_{-0.0032}$	$+0.37$
$\|a_{0}^{0}-a_{0}^{2}\|$	$0.2639$	$0.2623^{+0.0079}_{-0.0075}$	$-0.22$

Table 6: Closure-test results for the

1{,}000

-replica CFD ensemble. Scattering lengths are in units of

m_{\pi}^{-1}

Figure 10: Closure test, real-axis reconstruction of the

S_{0}

P_{1}

, and

S_{2}

waves. Solid lines are ensemble medians with

68\%

and

95\%

bands; dashed lines are the CFD amplitude used to generate the pseudo-data. Upper row, phase shifts; lower row, inelasticities.

	Production	Ablation
$\mathcal{L}_{\text{data}}$	$2.93^{+0.21}_{-0.19}$	$2.93^{+0.23}_{-0.21}$
$\mathcal{L}_{\text{Roy}}$	$0.067^{+0.013}_{-0.011}$	$0.74^{+0.30}_{-0.41}$
$a_{0}^{0}$ ( $m_{\pi}^{-1}$ )	$0.222^{+0.014}_{-0.014}$	$0.166^{+0.035}_{-0.027}$
$a_{0}^{2}$ ( $m_{\pi}^{-1}$ )	$-0.041^{+0.005}_{-0.006}$	$-0.097^{+0.042}_{-0.019}$

Table 7: Production versus ablated SINN ensembles,

10{,}000

and

1{,}000

replicas. Intervals are central

68\%

. The ablated ensemble carries no total-loss cut.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 12: Experiment-level symmetrized response matrix $C_{ij}$ for the second selection round, ordered by Ward group. Positive entries indicate tension and negative entries indicate compatibility; the group-averaged criterion and rejection threshold are described in Section V.1 .

Figure 13: Eigenvalue spectrum of the response matrix

C_{ij}

. The first 11 eigenvalues capture 90% of the total absolute spectral weight and are retained for the clustering.

Figure 14: Leading eigenvector loadings for each experiment. Experiments with similar loading patterns are grouped together by the Ward clustering.

Group	Members
G1	Grayer D
G2	Grayer E
G3	Cohen, Martin-Ozmutlu, Wetzel
G4	Durusoy OPEDP, Estabrooks, Etkin, Hoogland A, Hyams 73, Losty (A,B)
G5	Grayer (A,B), Hoogland B, Kamiński
G6	Polychronakos

Appendix

Table 8: Ward groups for the second selection round before the rejection step.

Figure 15:

D_{0}

phase shift and inelasticity. Bands and data as in Fig. 7 .

Figure 16:

F_{1}

phase shift and inelasticity. Bands and data as in Fig. 7 . The

F_{1}

partial is underspecified due to poor data coverage above threshold, and also underconstrained as it is not controlled by the Roy equations. The large fluctuation in the

95\%

uncertainty band is an artifact of the truncation of the dispersive integral.

Figure 17:

D_{2}

phase shift (elastic).

Figure 18:

G_{2}

phase shift (elastic).

Figure 19: Loss components across the

10{,}000

-replica production ensemble, every member satisfying

\mathcal{L}_{\text{total}}\leq 4.0

Figure 20: Training loss evolution for a representative replica: total (black), data (blue), Roy (red), consistency (green), and DIRAC (orange) terms of Eq. 16 .

\mathcal{L}_{\mathrm{SL}}

carries its fixed

1/N_{\mathrm{data}}

normalization; the other components are shown unweighted and only the total carries the curriculum weights, which are annotated per phase.

Figure 21: Distribution of trained network weights across the retained ensemble. The peak near zero indicates substantial effective sparsity.

Component	$\ell$ , $I$	Phase layers	Inel. layers
Trunk	—	[8, 12]	—
$S_{0}$ head	0, 0	[8, 4]	[4]
$P_{1}$ head	1, 1	[4, 2]	[2]
$S_{2}$ head	0, 2	[4, 2]	[2]
$D_{0}$ head	2, 0	[2, 2]	[2]
$D_{2}$ head	2, 2	[4, 2]	—

Appendix

Table 9: Network layer dimensions. Numbers are hidden neurons before the final single-neuron output. The head subnetworks use

\mathrm{SiLU}

activations, while the shared trunk uses the fixed activation pattern discussed below.

Optimization setting	Value
Learning rate	$0.005$
Weight decay	$0.001$
$(\beta_{1},\beta_{2},\epsilon)$	$(0.9,0.999,10^{-8})$
Gradient clipping	norm $\leq 1.0$
LR scheduler	cosine annealing
Scheduler step / factor	100 epochs / 0.5

Appendix

Table 10: Optimization settings used in production that are not specified explicitly elsewhere in the manuscript.

Figure 22: As Fig. 10 , for the

D_{0}

D_{2}

, and

F_{1}

waves.

Figure 23: Closure test, joint distribution of

a_{0}^{0}

and

a_{0}^{2}

over the CFD ensemble, with the CFD target marked.

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Learning the S-matrix from data: Rediscovering gravity from gauge theory via symbolic regression

Nathan Moynihan

Queen Mary, University of London, Mile End Road, London E1 4NS, UK

High-Energy Physics Symbolic Regression

Open paper AskPDF arXiv

We demonstrate that modern machine-learning methods can autonomously reconstruct several flagship analytic structures in scattering amplitudes directly from numerical on-shell data. In particular, we show that the Kawai--Lewellen--Tye (KLT) relations can be rediscovered using symbolic regression applied to colour-ordered Yang--Mills amplitudes with Mandelstam invariants as input features. Using standard feature-selection techniques, specifically column-pivoted QR factorisation, we simultaneously recover the Kleiss--Kuijf and Bern--Carrasco--Johansson (BCJ) relations, identifying a minimal basis of partial amplitudes without any group-theoretic input. We obtain the tree-level KLT relations with high numerical accuracy up to five external legs, using only minimal theoretical priors, and we comment on the obstacles to generalising the method to higher multiplicity. Our results establish symbolic regression as a practical tool for exploring the analytic structure of the scattering-amplitude landscape, and suggests a general data-driven strategy for uncovering hidden relations in general theories. For comparison, we benchmark this general approach with a recently introduced neural-network based method.

Abstract

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