Fast and Precise Learned Charged-Particle Trajectory Regression at the Large Hadron Collider

Jonathan Renusch; Benjamin Huth; Daniel Murnane; Eleni Xochelli; Doğa Elitez; Paul Gessinger-Befurt; Andreas Stefl; Jeremy Couthures

cs.LGSep 26, 2026

Fast and Precise Learned Charged-Particle Trajectory Regression at the Large Hadron Collider

Authors: Jonathan Renusch, Benjamin Huth, Daniel Murnane, Eleni Xochelli, Doğa Elitez, Paul Gessinger-Befurt, Andreas Stefl, Jeremy Couthures, +4 more

Organizations: CERN, Geneva, Switzerland · Technical University of Munich, Germany · Niels Bohr Institute, University of Copenhagen, Denmark · Lawrence Berkeley National Laboratory, USA · Universitat Autònoma de Barcelona, Spain · Johannes Gutenberg-Universität Mainz, Germany · LAPP, Université Savoie Mont Blanc, CNRS/IN2P3, Annecy, France · SLAC National Accelerator Laboratory, USA

Abstract

We propose a training recipe that treats charged-particle trajectory parameter regression on high-energy physics detector data as a sequence-modeling task. Kalman filters and linearized least-squares fits have been the classical standard approach for this task: they are optimal estimators for sparsely sampled linear-Gaussian data and are commonly used for trajectory parameter regression (fitting). The classical fitting techniques implemented for this domain reach a final precision of one part in $10^5$ through detailed modeling of detector geometry, material, detection effects and precise numerical integration of the equations of motion through the detector's inhomogeneous magnetic field. With this study, we demonstrate that using a bidirectional gated linear recurrent encoder, one is able to reproduce the full precision of classical track fitting techniques. Using a custom kernel, we also achieve significantly higher throughput during GPU inference, compared to classical fitting software running on similarly priced multi-core CPU servers representing typically employed hardware. Such a speedup would lead to considerable cost savings for the pattern recognition at the Large Hadron Collider. To our knowledge, this is the first end-to-end learned track fit to reach the full precision and, at the same time, offer the opportunity to reduce the computing costs.

Figures & tables

Figure 1: Charged-particle track fitting as sequence regression. A charged particle’s hits across the silicon layers of the tracking detector [ 7 ] form an ordered sequence of feature vectors

\vec{\mathbf{x}}_{N}

. An analytic helix through the first three hits gives a seed estimate

\vec{\mathbf{p}}_{\text{seed}}

; each hit carries measured coordinates plus three seed residuals:

\Delta u

\Delta v

, compressed with

\operatorname{asinh}

, and

s_{\mathrm{helix}}

. A bidirectional minGRU encoder reads the sequence and predicts a correction

\Delta\vec{\mathbf{p}}

to the seed, giving the perigee parameters

\vec{\mathbf{p}}=(d_{0},z_{0},\varphi,\theta,q/p)

at closest approach to the collision axis. Schematic, not to scale.

Figure 2: Architecture of the seed-guided bidirectional minGRU model. An analytic three-hit helix seed (top, no learned parameters) supplies residual features to the encoder input and, via the skip connection, anchors the output. The encoder (

0.64

M parameters) predicts per-parameter quantile corrections

\Delta p

(7 quantiles each, median as the point estimate); the final estimate is

\hat{p}=p_{\mathrm{seed}}+\Delta p

, additive for

d_{0},z_{0},\varphi,\theta

and a scale-free residual for

q/p

	clipped core (iter- $3\sigma$ )					un-clipped (with tails)
sample	$d_{0}$	$z_{0}$	$\varphi$	$\theta$	$q/p$	$d_{0}$	$z_{0}$	$\varphi$	$\theta$	$q/p$
$\mu$ , 2 GeV	0.992(2)	0.991(3)	0.989(2)	0.993(2)	1.006(4)	0.989(2)	0.981(4)	0.993(3)	0.984(6)	0.959(4)
$\mu$ , 10 GeV	0.993(1)	0.995(2)	0.992(2)	0.991(2)	1.003(3)	0.968(5)	0.950(5)	0.969(5)	0.986(2)	0.987(3)
$\mu$ , 50 GeV	0.998(1)	0.999(1)	0.977(2)	0.994(3)	1.013(2)	0.993(1)	0.957(4)	0.977(1)	0.9968(8)	1.018(4)
$\mu$ , 1–70 GeV	0.9955(3)	0.9981(3)	0.9815(6)	0.9958(5)	1.0154(7)	0.980(3)	0.960(5)	0.975(7)	0.98(1)	0.96(2)

Table 1: Resolution ratio minGRU / truth-seeded KF per perigee parameter and test sample (

|\eta|\leq 2

): iteratively

3\sigma

-clipped

\mathrm{RMS}

(left) and un-clipped

\mathrm{RMS}

including all tails (right). Values

<1

mean the minGRU achieves better resolutions than the reference. The reference is the truth-seeded KF shipped with the dataset in every row. Brackets are

1\sigma

bootstrap uncertainties on the last digit, from

400

paired replicas per sample.

Figure 3: Core resolution versus pseudorapidity on the

10\text{\,}\mathrm{GeV}

muon sample (

|\eta|\leq 2

): iterative-

3\sigma

-clipped

\mathrm{RMS}

of each perigee parameter for the minGRU (blue) and the truth KF (red), with the minGRU/KF ratio beneath each panel. Each legend gives the unbinned

\mathrm{RMS}

; the title states the total number of trajectories fitted by both estimators. Bands are the analytic

\mathrm{RMS}

standard error.

Figure 4: Core resolution versus transverse momentum on the uniform

1

–

70

GeV muon sample (

|\eta|\leq 2

, equal-width

p_{\mathrm{T}}

bins): iterative-

3\sigma

-clipped

\mathrm{RMS}

of each perigee parameter for the minGRU (blue) and the truth KF (red), with the minGRU/KF ratio beneath each panel. Each legend gives the unbinned

\mathrm{RMS}

; the title states the total number of trajectories fitted by both estimators. Bands are the analytic

\mathrm{RMS}

standard error.

device	configuration	tracks / s	approximate cost [$]	ratio
32-core CPU	ACTS Kalman filter fit (reference)	$\sim$ 170 k	2 000	86
RTX 5000 Ada	this work, deployment path	1.51 M	4 000	378
H100 NVL	this work, deployment path	5.41 M	30 000	180

Table 2: The throughput and throughput per device dollar. GPU rows: this work’s model on the deployment path, at the saturating batch size (Fig. 5 ). Prices are the device-only list prices (GPU board / CPU chip [ 36 ] ; hosts excluded on every row). The gain of the kernel treatment for each encoder is given in Table A.1 .

Figure 5: Inference throughput versus batch size (blue, top axis) on a datacenter H100 NVL, for the seed-guided bidirectional minGRU trajectory parameter estimator at fp16 inference. The on-GPU seed calculation is allowed for in every throughput estimate. The red curve is the ACTS KF fit on an AMD Threadripper 3970X versus the number of threads it was given (bottom axis), reaching

\sim

170 k tracks per second at 64 threads; the dotted line is ideal linear scaling from the single-thread measurement. The dashed blue curve is the same model and benchmark on a workstation RTX 5000 Ada.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

encoder	default kernel H100	optimized H100	gain H100	optimized RTX 5000 Ada
minGRU	$0.88$ M	$5.41$ M	$6.1\times$	$1.51$ M
Transformer	$0.48$ M	$3.18$ M	$6.6\times$	$0.855$ M
Mamba-2	$0.58$ M	$2.06$ M	$3.6\times$	$0.649$ M

Appendix

Table A.1: Single-H100 (NVL) inference throughput, in tracks/s, at the saturating batch size (

131\,000

tracks/batch). The kernel’s default configuration is: strict IEEE fp32, standard recurrent and SSM kernels, full-attention for the Transformer (flash attention does not give any gains for short sequences). Deployed is the fastest physics-gated path: packed batches, a fused Triton kernel for the token mixing, a fused Fourier encoding, and fp16 projections for all three encoders. All three encoders have the same parameter count within 3 %. RTX 5000 Ada reports the same deployed path on a single RTX 5000 Ada GPU.

Figure A.1: Trunk-gradient cosine similarity of the paper model. Mean cosine between the per-parameter loss gradients on the shared trunk ( $0.63$ M parameters, output head excluded), averaged over $450$ minibatches of $2048$ tracks ( $0.92$ M tracks); each cell shows the mean $\pm$ its standard error. The four geometry parameters $(d_{0},z_{0},\varphi,\theta)$ are mutually aligned, most strongly on the transverse perigee pair $(d_{0},\varphi)$ , while $q/p$ is nearly orthogonal to all of them.

Figure A.2: Encoder comparison. Resolution ratio to the truth-seeded KF for each perigee parameter (panels) and each muon test sample (rows) at

|\eta|\leq 2

, for four encoders trained with the same recipe, data, schedule, random seed and parameter budget (

\sim

0.63

M) and read out through the same seed, features and quantile heads. These are first-stage models, trained for

25

epochs at one shared learning rate and without the second-stage training of the deployed model. Filled markers are the iterative-

3\sigma

-clipped core, open markers the un-clipped

\mathrm{RMS}

including all tails. The dotted line marks parity with the reference. Three of the four reach the same precision. The non-selective diagonal state-space model falls behind, most clearly in

q/p

. Single seed per encoder; shared learning rate.

Figure A.3: single muons,

2

GeV , iterative-

3\sigma

-clipped

\mathrm{RMS}

versus

\eta

(minGRU/KF ratio beneath each panel; clip fractions in the legends, total in the title).

Figure A.4: single muons,

2

GeV , residual distributions; each legend gives the iterative-

3\sigma

\mathrm{RMS}

and the clipped fraction.

Figure A.5: single muons,

10

GeV , residual distributions; each legend gives the iterative-

3\sigma

\mathrm{RMS}

and the clipped fraction. (The

10\text{\,}\mathrm{GeV}

resolution-versus-

\eta

curves are the main-text Fig. 3 .)

Figure A.6: single muons,

50

GeV , iterative-

3\sigma

-clipped

\mathrm{RMS}

versus

\eta

(minGRU/KF ratio beneath each panel; clip fractions in the legends, total in the title).

Figure A.7: single muons,

50

GeV , residual distributions; each legend gives the iterative-

3\sigma

\mathrm{RMS}

and the clipped fraction.

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Charged-particle tracking -- reconstructing trajectories from sparse detector measurements -- is a fundamental high-energy-physics inference problem and a canonical example of learning under extreme combinatorial ambiguity. At the High-Luminosity Large Hadron Collider (HL-LHC), tracking must remain accurate and efficient despite unprecedented collision densities. Graph neural networks perform strongly, but incur substantial costs from graph construction and processing, while transformer-based approaches rely on auxiliary stages that prevent end-to-end optimization. To address this, we present HEPTv2, an end-to-end point-transformer architecture that reconstructs tracks from detector hits in one trainable pipeline. HEPTv2 combines a locality-aware point encoder with a track decoder that predicts complete trajectories without graph-building, clustering, or filtering. The encoder uses locality-sensitive hashing in detector coordinate space to preserve tracking-relevant geometry while enabling efficient local attention. The decoder resolves ambiguities through sectorized decoding and direct hit-to-track prediction under joint encoder-decoder supervision, allowing the full pipeline to be optimized end-to-end. On TrackML, HEPTv2 achieves 98.6% double-majority tracking efficiency at a 0.8% fake rate, while requiring only

\sim

15~~ms inference time and 0.4~~GB peak memory per event on a NVIDIA A100 GPU. Latency and memory scale approximately linearly for events with up to

5\times10^5

hits. HEPTv2 establishes a new state of the art in the accuracy-latency trade-off, improving efficiency by 4.5% over the strongest prior transformer and by 1.1--2.2% over optimized graph-based pipelines, while reducing latency by factors of 7 and 38--52, respectively. These results show end-to-end transformers can deliver the accuracy and efficiency required for real-time particle reconstruction at the HL-LHC.

Abstract

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