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Exploring phase space with Neural Importance Sampling

Enrico Bothmann, Timo Janßen, Max Knobbe, Tobias Schmale, Steffen Schumann

arXiv:2001.05478v3hep-phhep-ex

TL;DR

High-multiplicity scattering makes phase-space sampling a potential bottleneck, especially as dimensionality and integrand complexity grow. The paper applies neural-network-trained coupling layers to importance sampling, with full phase-space coverage. Its sampler improves performance on simpler studied cases, but does not outperform VEGAS for the four-jet case.

  • Problem

    Phase-space sampling can limit matrix-element generators: poor mappings create large event-weight fluctuations, and performance tends to deteriorate with dimensionality and integrand complexity.

    Source: §1 · p. 2

  • Method

    The paper uses neural networks to train coupling-layer maps for importance sampling, aiming to preserve full phase-space coverage.

    Source: §1 · p. 2, §5 · p. 14

  • Results

    The unweighting-efficiency gain factor fell from 2.3 for 3-jet production to 1.1 for 4-jet production; the method improved sampling for simpler examples up to d = 5 but did not outperform VEGAS for 4-jet production.

    Source: §5 · p. 14

  • Takeaways & Limitations

    The top-quark and 3-jet results indicate that neural-network approaches can potentially outperform conventional optimisers such as VEGAS on more complex problems.

    Source: §5 · p. 15

    For 4-jet production with d = 8, limited computing resources constrained the network architecture and training batches, and the method did not outperform VEGAS.

    Source: §5 · pp. 14–15

Abstract

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We present a novel approach for the integration of scattering cross sections and the generation of partonic event samples in high-energy physics. We propose an importance sampling technique capable of overcoming typical deficiencies of existing approaches by incorporating neural networks. The method guarantees full phase space coverage and the exact reproduction of the desired target distribution, in our case given by the squared transition matrix element. We study the performance of the algorithm for a few representative examples, including top-quark pair production and gluon scattering into three- and four-gluon final states.

1. Introduction

High-multiplicity hard-scattering processes are important in collider analyses, but their matrix elements are computationally demanding and phase-space sampling can limit performance. The paper introduces a neural-network-based importance sampler and benchmarks it on representative high-energy physics processes.

1. Introduction

  • High-multiplicity processes, including V + jets and top-quark pairs with an associated boson, require evaluating matrix elements with thousands of diagrams.The relevant scattering matrix elements can feature up to eight final-state particles.Source: §1 · p. 1
  • Poor phase-space mappings cause large event-weight fluctuations and require more target-function evaluations to generate unit-weight events.Sampling performance also tends to deteriorate with particle multiplicity and integrand complexity, including resonances, singularities, interference, and kinematic cuts.Source: §1 · p. 2
  • Although phase-space sampling has seen fewer conceptual developments than amplitude construction, recent work has explored Monte Carlo and machine-learning approaches.The paper situates its approach among existing sampling work and recent interest in machine-learning techniques.Source: §1 · p. 2
  • The proposed sampler uses neural-network-trained bijective coupling layers for importance sampling and is benchmarked on top-quark and multi-gluon processes.The applications include top-quark pair production and gluon scattering into three- and four-gluon final states.Source: §1 · p. 2

2. Phase space sampling: existing approaches

The section reviews Monte Carlo integration and phase-space sampling, then sets out requirements for an improved sampler. It explains importance sampling and VEGAS before discussing how neural-network mappings can fail to cover the target phase space.

2. Phase space sampling: existing approaches

  • The sampler must reproduce the target distribution, cover the full physical phase space, adapt to new integrands, and produce uncorrelated events.The section frames integration and event generation as tasks over a positive target on the unit hypercube, with phase-space dimension d = 3n − 4.Source: §2 · pp. 2–3

2.1. Importance Sampling

  • Importance sampling reduces estimator variance by drawing from a density shaped like the target; mixtures of channels can approximate multimodal targets.For strongly structured targets, variance reduction matters for integral estimates and unweighting efficiency.Source: §2.1 · pp. 3–4

2.2. VEGAS algorithm

  • VEGAS adapts a product of piecewise-constant coordinate densities by iteratively updating bin widths to reduce sample variance.Equal-probability bins become narrow around peaks and wider in flatter regions; VEGAS can struggle when multimodal peaks are not aligned with coordinate axes.Source: §2.2 · p. 4

2.3. Existing proposals for neural-network based sampling and possible pitfalls

  • Earlier neural samplers train mappings to approximate the target, but bounded inputs or hidden-layer activations can prevent those mappings from being surjective.Artificial boundaries may persist despite training, leaving tails undersampled, peaks enhanced, and integration results wrong; the cited [27] structure avoids this issue using surjective functions and unbounded Gaussian inputs.Source: §2.3 · pp. 5–6
  • In a two-dimensional Gaussian test, non-surjective networks showed asymptotic coverage, while a surjective network, uniform samples, and VEGAS coverage increased with sample size.The non-surjective samples also undershot the tails and put too many events in the peak.Source: §2.3 · p. 6
  • The section measures coverage by the convex hull of generated events, making coverage a distinct check alongside agreement with the target distribution.The comparison includes uniform and VEGAS samples as well as neural-network variants.Source: §2.3 · p. 6

3. Neural-Network assisted Importance Sampling

The approach uses trainable, invertible maps to remap random numbers for importance sampling, preserving the channel mappings needed by event generators. Coupling layers make the mappings tractable, and piecewise quadratic transforms provide the sampling map used in the algorithm.

3. Neural-Network assisted Importance Sampling

  • The sampler adopts Neural Importance Sampling because its adaptive model must be invertible for use in multi-channel sampling.The trainable mapping redistributes the random numbers entering sample generation, similar to a way VEGAS is used in practice.Source: §3 · p. 6

3.1. Coupling Layers

  • Each coupling layer transforms one input subset while leaving the other unchanged, yielding a triangular Jacobian whose determinant is cheaper to compute than for arbitrary matrices.The paper states that arbitrary-matrix determinant computation can be obstructive; triangular structure makes the determinant calculation scale linearly with dimension.Source: §3.1 · p. 8, §3 · p. 8

3.2. Piecewise Quadratic Coupling Layers

  • Piecewise quadratic coupling transforms are built by integrating normalized, piecewise-linear probability densities represented with adaptive bins.The component transforms act as cumulative distribution functions over the unit hypercube.Source: §3.2 · pp. 8–9

3.3. Importance Sampling with Coupling Layers

  • The sampler remaps uniform random numbers through layered coupling maps before a channel mapping generates target-domain points, with event weights incorporating the relevant Jacobians.The neural network is not used to generate target-domain points directly; channel mappings can implement four-momentum conservation and mass-shell conditions and map known peak structures.Source: §3.3 · p. 9
  • Empirically, mean squared error distance performs better than Kullback–Leibler divergence for variance reduction and unweighting efficiency increase in the studied applications.Minimising the Pearson χ2-divergence is also linked to minimising Monte Carlo estimator variance [33].Source: §3.3 · p. 10
  • For multi-channel sampling, mappings can be trained simultaneously so different channels are aware of one another and do not adapt to the same features..Source: §3.3 · p. 10

4. Results

The study benchmarks NN-assisted importance sampling against VEGAS for top-quark decays and pair production, and for three- and four-gluon production. NN sampling improves several top-quark and three-jet results, while four-jet performance is similar to VEGAS.

4.1. Top quarks

  • For the same number of generated points, NN sampling has a standard deviation 8 times smaller than VEGAS and unweighting efficiency 0.84, versus 0.50 for VEGAS and 0.59 for Uniform sampling.Source: §4.1 · p. 11
  • For top-pair production, NN sampling has a standard deviation 5 times smaller than VEGAS and unweighting efficiency of 78% versus 40%.The NN unweighting efficiency is reduced by 7% relative to the single top-decay setup.Source: Table 1 · p. 11, §4.1 · p. 12
  • For top-pair observables, VEGAS oversamples the mee tails while NN weights are nearly constant; both methods have nearly equal MC errors across the observable ranges.Both methods show weight distortions at low θe−¯b, but in different directions.Source: §4.1 · p. 12, Figure 3 · p. 13

4.2. Gluon-induced multi-jet production

  • In gluon production, NN optimization doubles VEGAS’s unweighting efficiency for three jets but gives no significant improvement for four jets.VEGAS and NN optimization produce similar improvements in the standard-deviation estimates for both multiplicities.Source: §4.2 · p. 13, Table 2 · p. 14
  • For three-jet production, NN weights stay near one up to about 240 GeV, while four-jet NN weights depart from unity above 80 GeV and decrease.In four-jet production, VEGAS weights are closer to unity than NN weights above 100 GeV.Source: §4.2 · p. 14
  • In four-jet high-pT bins, NN sampling has smaller relative MC errors despite a worse match to the integrand than VEGAS.The NN sample oversamples this region, generating more events per bin and thus lower variance.Source: §4.2 · p. 14

5. Conclusions

Neural Importance Sampling guarantees full phase-space coverage and improves sampling performance in simpler examples, but the advantage narrows for four-jet production and scaling remains challenging.

5. Conclusions

  • The proof-of-principle study applies piecewise-quadratic coupling layers while guaranteeing faithful target-distribution sampling and full phase-space coverage.Benchmarks include top decay, leptonic top-pair production with subsequent decays, and gluonic 3-jet and 4-jet production.Source: §5 · p. 14
  • The unweighting-efficiency gain falls from 2.3 for 3-jet production to 1.1 for 4-jet production, where the method does not outperform VEGAS.The comparison covers simpler examples up to d = 5 and the more complex 4-jet case at d = 8.Source: §5 · p. 14
  • Limited NN architecture complexity and training-batch size reflected computing-resource constraints; extending to 5-jet production poses a further scaling challenge.The authors suggest the 4-jet result may improve with more powerful hardware or an optimised implementation.Source: §5 · p. 14
  • Future work aims to extend final-state multiplicity while keeping training costs acceptable, including study of multi-channel integration and alternative coupling layers.The authors also identify adversarial training and better handling of phase-space cuts as directions to investigate.Source: §5 · p. 15

A. Auxiliary jet p⊥distributions in multi-jet production

The appendix presents auxiliary jet-transverse-momentum distributions for three- and four-gluon production and compares VEGAS-optimised with NN-optimised sampling.

A. Auxiliary jet p⊥distributions in multi-jet production

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