
Machine Learning for the LHC
LHC physics, like many other field in and around particle physics is living through an exciting, datadriven era. As theorists we can come up with ideas for physics beyond the Standard Model, motivated by dark matter or the matterantimatter asymmetry or whatever else keeps us up at night, and immediately try it on LHC data. Even if we do not have access to the actual data we can devise search strategies using a simulation chain based on first principles and then hand our ideas and tools over to our experimental friends. The scientific challenge is that we have to understand the basic structure of LHC data, its (precision) simulation, and the best ways of using huge amounts of data. This immediately brings us to machine learning as a great tool for particle physicists. In our group we develop and apply modern data analysis tools in four ways, where this number should be increasing. Here is a list of papers we published on machine learning applications:
We are especially happy to publish our hepml lecture notes, based on a masterlevel course and with a set of tutorial problems (2022). The notes start with a very basic introduction to modern neural networks for particle physicists and then introduces classification, unsupervised training, generative networks, and conditional generative networks for inverse problems. Some, but not all tools have been in developed in Heidelberg, and we expect the lecture notes to further grow over the coming years. 
Jet and Event Classification
Multivariate analysis methods have a long tradition in particle physics. If we use subjet physics as an example, we want to ask the question what kind of quark or gluon initiated an observed jet. Traditionally, this question is asked in relation to bottom quarks, which can be identified by a displaced decay vertex and one or more leptons inside the hadronic jets. From a theory and experimental perspective the cleanest signature are boosted top quarks. The original HEPTopTagger approach asks for two mass drops, one from the top decay and one from the Wdecay. We know that this tagger can be hugely improved when we include a wealth of kinematic observables, some directly and some algorithmically derived from the energies and the momenta of the subjets. The obvious question is why we need to construct these highlevel observables and if we cannot just feed the 4momenta of the subjets into a classification algorithm. As a matter of fact, we can do exactly that. The next question is if we really need to train our classification on data where we know the truth label, or if we can just look for jets which look as little like QCD as possible. That is the second aspect we are working on these days. Many aspects we have been studying are related to uncertainties, symmetries, latent representations, data augmentations, and can be viewed as explainable AI for particle physicists. This series of papers has been made possible by Sven Bollweg, Anja Butter, Sascha Diefenbacher, Barry Dillon, Friedrich Feiden, Hermann Frost, Manuel Haussmann, Theo Heimel, Gregor Kasieczka, Nicholas Kiefer, Michel Luchmann, Tanmoy Modak, Michael Russell, Torben Schell, Christof Sauer, Peter Sorrenson, Jennie Thompson, Lorenz Vogel, and more students and postdocs.
Modelindependent searches for anomalous events (2023) are one of the big unsolved problems at the LHC, as they define the idea of triggering in the language of data science. We use contrastive learning on augmented background events to define a representation space which allows us to identify anomalous events based on basic properties. This method goes beyond density estimation to define anomalies and attempts to define a modelagnostic implicit bias.  
Resilience is a big problems for LHC classifiers (2022). One reason is that they are usually trained on simulations, which are excellent, but might still lead to a generalization error. This leads to a serious problem if the difference between training and test data is strongly correlated with relevant classification features. We show how networks can be trained on a continuous interpolation between training samples, including an error analysis using Bayesian networks. Continuous interpolation between training and test data might allow us to take a fresh look at calibration strategies.  
Normalizing autoencoder for LHC triggers, tested on dark jets (2022). We develop the first autoencoder which identified jets with higher and with lower complexity than QCD jets and without a significant dependence on the data preprocessing. Its architecture is the same as the usual autoencoder, but with an energybased training. This training uses a combination of latent and reconstruction space Markov chains.  
Comparison of different densitybased anomaly serches for jets (2022). The challenge of anomaly searches at the LHC, for instance applied to jets, is that outofdistribution searches cannot be used to analyze phase space configurations of a quantum process. Instead, we need to identify configurations with a small probability density extracted by unsupervised training. We compare a classic kmeansbased method with a Dirichlet autoencoder and an INN on two new, darkmatterinspired jet datasets.  
The first application of contrastive learning to LHC physics (2021). The idea is to train a network to incorporate symmetries like rotations, translations, and permutations in its representations of QCD jets. In addition, the network also learns infrared and collinear safety as fundamental properties of the underlying quantum field theory. The way these symmetries and augmentations are encoded in the latent representation can be studied using linear classifier tests.  
Using unsupervised learning to reduce the model dependence in searches for anomalous and hardtodetect Higgs decays (2021). For hidden valley models the number of parameters in the signal hypothesis is large, and their impact on the analysis is sizeable. We show how an unsupervised search method based on appropriate observables is an attractive alternative to supervised search methods based on unknown model parameters.  
Constructing a symmetric autoencoder with better latent spaces (2021). We know from our earlier study that it is possible to extract top jets from a QCD jet sample, essentially based on the compressibility of the encoded information. To apply unsupervized classification the other way around, we need to define a nice latent space and access this latent space rather than the reconstruction loss. And then it works.  
Measurement of jet momenta with uncertainties and calibration of regression tools with Bayesian networks (2020). The measurement by the regression network tracks statistical and systematic uncertainties from the training data. We propose to calibrate the network in a straightforward way through the smearing introduced by the measurement of labels.  
Jet and event classification with capsule networks (2019). Such capsule networks are a natural extension of (scalar) convolutional networks and can be used to analyse sparse sets of detector objects, each represented by a sparse calorimeter image.  
Top tagging with uncertainties using Bayesian classification networks (2019). Such a tagger provides a classification output and a jetwise error estimate on the classification outcome. While statistical uncertainties from a limited training sample are easily traced, systematic uncertainties lead to a correlation of the central value and the uncertainties, all the way to adversarial examples.  
Toptagging community paper (2019) comparing a wide range of top taggers, from imagebased to 4vectorbased and theorymotivated tagging approaches. We show their respective performances for a standard data set and find that there are many ways of contructing highly performing taggers ready to hit LHC data.  
LoLabased quarkgluon tagger (2018) showing that the same 4vectorbased architecture can be used to distinguish hard processes with preferably quark or gluon jets. The issue is how to design a highperformance tagger in the presence of detector effects.  
Jet autoencoder (2018) based on the DeepTop and LoLa taggers. We show howtop jets are less compressible than QCD jets, how they can be tagged, and how we can decorrelate the jet mass using an adversary. We also test the autoencoder on new physics in jets.  
4vectorbased LoLa tagger (2017) which allows us to combine information from the calorimeter and the tracker, accounting for the different resolutions. It can be thought of a graph network over Minkowski space, where we use the fact that we know the Minkowski metric. For a cross check this is one of the few papers which quotes the Minkowski metric with an error bar.  
Imagebased DeepTop tagger (2017) showing that we can use convolutional networks on calorimeter images to identify boosted tops. We showed how this deeplearning tagger compares to classic multivariate methods and how it is possible to interpret the intermediate layers and the output of the tagging network. 
Event Generation
Firstprinciple simulations LHC events based on quantum field theory (mostly QCD) is one of the unique features of LHC physics. Many groups are investing a huge effort into computing the underlying predictions in perturbative quantum field theory. These prediction can be included in multipurpose event generators like Pythia, Sherpa, Madgraph, or Herwig. These simulations are based on Monte Carlo simulations and are extremely efficient. The question is if we can use machine learning tools to improve them further or to get access to information that is usually lost in the simulation framework. One example is the use of the hard matrix element in a hypothesis test, usually called the matrix element method. An important part of this research direction is the uncertainty quantification for generative networks. Our group includes Marco Bellagente, Anja Butter, Sascha Diefenbacher, Manuel Haussmann, Theo Heimel, Gregor Kasieczka, Ulli Kothe, Claudius Krause, Michel Luchmann, Armand Rousselot, and Ramon Winterhalder, and it is still expanding.
MadNIS is an MLbased multichannel importance sampling tool for LHC simulations (2022). It replaces learned channel weights with locally defined regression networks and importance sampling with an INN or normalizing flow. Key to an efficient training of the two networks is the combination of online training and buffered training, making optimal use of expensive integrands. In the appendix we discuss a whole range of possible loss functions for this bidirectional training.  
Precision network setup based on Bayesian networks combined with a boosted training for LHC amplitudes (2022). One challenge for precision networks is that neural networks can be too powerful in fitting through key regions of phase space. In this situation it would be beneficial to interpolate through some training data points rather than fit a function over the full phase space. We use a Bayesian network to decide which training data points are not reproduced well enough and reweight these data points in the likelihood loss.  
Community report and review article on machine learning applications in LHC event generation (2022). The report started as a contribution to the US Snowmass process and turned into a detailed description of the state of the art in machine learning for event generators. It includes MLapplications to standard generators, souptonuts MLgenerators, and different approaches to unfolding and inference.  
Proposal to use a generative network to supplement the datascouting trigger (2022). The idea is to train a generative network on LHC data which does not pass the standard triggers and compress the incoming dataset instead of compressing individual LHC events. We compare the discovery potential of such a generative network with the corresponding datascouting performance or triggerlwevel analysis. It turns out that compressing data samples is actually more efficient than reading out a fraction of events using comparable bandwidth.  
First proof that amplification by generative networks works for a detector simulations (Calomplification) in a semirealistic setup (2022). For photon showers in an electromagnetic calorimeter we show that generative models outperform their training dataset statistically by using an implicit bias adapted to the phase space density expected for such a physical process.  
As a side project, the first application of machine learning to loop integrals, (2021). We use a normalizing flow network to describe a contour integration for different 2loop integrals with several mass scales. The network is trained to construct the optimal integration path for the Feynman parameters.  
The first application of a Bayesian precision generator, to LHC event generation (2021). It combines an INNnormalizing flow generator with a discriminator for reweighting and for a combined, GANlike training. The generatordiscriminator coupling follows an alternative approach to the usual adversarial training. Trainingrelated uncertainties are described through a Bayesian network, systematic uncertainties through conditional training.  
Brandnew Bayesian generative network, specifically a normalizing flow network or INN (2021). It learns not only a density map over phase space, but also an uncertainty map, and it provides this information in terms of events with an uncertainty weight. That setup is probably the first time, someone has attached uncertainties to a generative network, a crucial step for LHC applications. Moreover, we can study how the network learns the density and uncertainty maps in parallel and confirm that the flow network really works like some kind of functional fit.  
Study of superresolution on jet images (2020). The mismatch of calorimeter and tracking resolution on jet constituents is one of the leading themes of jet classification. We show how this measurement structure motivates applying a generative superresolution network. As usual, we look at QCD vs topdecay jets and study the model dependence for these two kinds of jets.  
GAN study on event unweighting (2020). The transformation of weighted events into unweighted events is the numerical bottleneck of many LHC simulations. We show how the classic hitandmiss algorithm can be improved significantly by applying a generative network which gets trained on weighted events and produces unweighted events.  
Generative networks review article (2020). We given an overview of the many ways generative networks are developed for LHC event generation. This includes many different architectures and applications.  
GAN study on the statistical gain (2020). An open question, crucial to LHC applications, is how much we can gain beyond the statistical power of the training sample by training a generatve network for more events. We define an amplification factor for simple multidimensional toy distributions and first find that, just like a fit, the structure of the GAN adds information to the discrete set of training events. Second, the GANned events do not have the same individual statistical power as a sampled event. The GAN applification factor becomes larger for sparcely distributed training data in highdimensional phase spaces.  
Conditional INN application to detector and QCD unfolding (2020). Invertible networks based on normalizing flows and built out of coupling layers can also serve as generative networks for LHC events. Their greatest advantage is that in a conditional setup they are built to generate spreadout probability distributions in the target space. For LHC unfolding this means we can construct a probability distribution over partonlevel phase space for a single detectorlevel event. We expand the detector unfolding to also unfold jet radiation to a predefined hard process.  
Fully conditional GAN application to detector unfolding (2019). If we train a conditional GAN on matched event samples we can use it to invert a MonteCarlobased simulation of, for instance, detector effects. This unfolding is not limited to one or lowdimensional distribution, but covers the entire phase space. The matching of local structures in the two data sets reduces the model dependence of the unfolding procedure.  
GAN application to event subtraction (2019). A general problem in dealing with event samples is that there is no efficient way to subtract event sample from each other, or combine events with positive and negative weights. We show how GANs can do that avoid statistical limitations from the usual binning procedure. Applications could be subtraction terms in Monte Carlo simulations or background subtraction in analyses.  
GAN application to event generation (2019). We show that a GAN, supplemented with a dedicated MMD loss, can generate top pair production events at the LHC. We model all kinematic distributions all the way to the three top decay jets. We find that regions with large systematic uncertainties on the GAN are directly linked to sparse training data. 
Inference
Finally, LHC measurements can benefit conceptionally from machine learning methods. To start with, information geometry is not exactly machine learning, but it is a concept which benefits from machine learning when we apply it to LHC physics. The question we are trying to answer is what kind of information is available to an LHC analysis, what observables capture it best, and what the limiting factors in an analysis might be. The machine learning aspect only comes in once we ask these questions beyond the parton level, and two former Heidelberg students (Johann Brehmer and Felix Kling) have worked with Kyle Cranmer's NYU group on developing the corresponding MadMiner program. Other collaborators on this topic include Sally Dawson and Sam Homiller for Higgs applications, or Ulli Kothe and Stefan Radev for invertible networks.
First attempt to enhance the matrix element method with generative networks (2022). To extract a fundamental parameter from individual events or small samples, the socalled matrix element method has been developed at the Tevatron. The challenge of this method is phase space sampling for general transfer functions, which can be solved by combining two normalizing flows, one describing the transfer function and one for the phase space mapping.  
First application of symbolic regression to LHC physics (2021). We compute the optimal observable or score for extracting certain higherdimensional operators from LHC events. Formulas are extracted from standard simulated data sets, where we use MadMiner to compute the score and a slightly modified version of PySR to find appropriate formulas. We test our approach for a simple ZH production toy model in detail and then show how it finds the correct result for CPviolation in weak boson fusion.  
Proposing a measurement of QCD splitting kernels using a conditional INN (2020). We start by defining a new interpretation framework for QCD jet measurements, leading to an actual measurement of QCDparameters. The complexity of lowlevel data and the highly nonlinear relation between splitting kernels and jet data make this measurement an excellent case for a machine learning application.  
Validation of simplfied template cross sections for VH production (2019). We analyse how these observables compare to an analysis of the full phase space, including detector effects and mixxing transverse momentum. This is where machine learning enters.  
Information geometry of Higgs CP in the SMEFT framework (2017). We compare different Higgs production and decay signatures, all based on the amplitude with four additional fermions, in their potential to test the CP properties of the Higgs coupling to intermediate gauge bosons. We link this approach to the established optimal observables.  
Information geometry of Higgs signatures in the SMEFT framework (2016). We compute the information available over the entire partonic phase space of Higgs signatures and compare for instance the impact of the QBF tagging jet kinematics with the Higgs decay kinematics. 
Global SFitter analyses
Obviously, machine learning should be able to improve global analyses of LHC data, where we interpret a large number of measurements in a highdimensional parameter space. Our SFitter tool has serves us well in the last 10 years, so we are working on improving it with machine learning applications...
A big problem for global analyses are theory uncertainties, affecting predictions based on perturbative QCD (2022). While the combined renormalization and factorization scale dependence describes this uncertainty reasonably well, this estimate fails for electroweak processes. We propose a new approach, based on representative classes of processes, and find that it describes the step from LO to NLO QCD predictions well, with very few remaining outliers which can be linked to wellunderstood effects. 