CMSPASMLG24001  
Reweighting of simulated events using machine learning techniques in CMS  
CMS Collaboration  
19 July 2024  
Abstract: Data analyses in particle physics rely on the accurate simulation of particle collisions and a detailed detector simulation to extract physical knowledge from the recorded data. Event generators together with a GEANTbased simulation of the detectors are used to produce large samples of simulated events for analysis by the LHC experiments. These simulations come at a high computational cost, where the detector simulation and reconstruction algorithms have the largest CPU demands. This note describes how machine learning (ML) techniques are used to reweight simulated samples to different model parameters or entirely different models. The ML method avoids the need for simulating the detector response multiple times by incorporating the relevant information in a single sample through event weights. Results are presented for reweightings to model variations and higherorder calculations in simulated top quark pair production at the LHC. This MLbased reweighting is an important element of the future computing model of the CMS Collaboration and will enable precision measurements with the CMS experiment at the HighLuminosity LHC.  
Links: CDS record (PDF) ; Physics Briefing ; CADI line (restricted) ; 
Figures  
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Figure 1:
The normalised differential cross section of $ \mathrm{t} \overline{\mathrm{t}} $ production in pp collisions at 13 TeV as a function of the $ p_{\mathrm{T}} $ (left) and $ \eta $ (right) of the $ {\mathrm{t}\overline{\mathrm{t}}} $ system obtained with the POWHEG program. The standard setting of $ h_\text{{damp}} = 1.379 m_{\mathrm{t}} $ (black) is compared to down (green) and up (magenta) variations in $ h_\text{{damp}} $. The ratios of the predictions with the $ h_\text{{damp}} $ variations to the nominal one are shown in the lower panels. The error bars represent the statistical uncertainties in the MC samples. 
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Figure 1a:
The normalised differential cross section of $ \mathrm{t} \overline{\mathrm{t}} $ production in pp collisions at 13 TeV as a function of the $ p_{\mathrm{T}} $ (left) and $ \eta $ (right) of the $ {\mathrm{t}\overline{\mathrm{t}}} $ system obtained with the POWHEG program. The standard setting of $ h_\text{{damp}} = 1.379 m_{\mathrm{t}} $ (black) is compared to down (green) and up (magenta) variations in $ h_\text{{damp}} $. The ratios of the predictions with the $ h_\text{{damp}} $ variations to the nominal one are shown in the lower panels. The error bars represent the statistical uncertainties in the MC samples. 
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Figure 1b:
The normalised differential cross section of $ \mathrm{t} \overline{\mathrm{t}} $ production in pp collisions at 13 TeV as a function of the $ p_{\mathrm{T}} $ (left) and $ \eta $ (right) of the $ {\mathrm{t}\overline{\mathrm{t}}} $ system obtained with the POWHEG program. The standard setting of $ h_\text{{damp}} = 1.379 m_{\mathrm{t}} $ (black) is compared to down (green) and up (magenta) variations in $ h_\text{{damp}} $. The ratios of the predictions with the $ h_\text{{damp}} $ variations to the nominal one are shown in the lower panels. The error bars represent the statistical uncertainties in the MC samples. 
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Figure 2:
The NN histories of the training for the $ h_\text{{damp}} $ parameter reweighting. Shown are the loss functions for the training data (blue) and the validation data (orange) for the down (left) and up (right) variations of $ h_\text{{damp}} $. 
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Figure 2a:
The NN histories of the training for the $ h_\text{{damp}} $ parameter reweighting. Shown are the loss functions for the training data (blue) and the validation data (orange) for the down (left) and up (right) variations of $ h_\text{{damp}} $. 
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Figure 2b:
The NN histories of the training for the $ h_\text{{damp}} $ parameter reweighting. Shown are the loss functions for the training data (blue) and the validation data (orange) for the down (left) and up (right) variations of $ h_\text{{damp}} $. 
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Figure 3:
The normalised differential cross section as a function of the $ p_{\mathrm{T}} $ (upper) and $ \eta $ (lower) of the $ {\mathrm{t}\overline{\mathrm{t}}} $ system. The black line shows the predictions from the down (left) and up (right) variations in $ h_\text{{damp}} $, and the green line shows the prediction from the nominal sample. The red line shows the nominal sample reweighted to the down (left) and up (right) $ h_\text{{damp}} $ variations using the DCTR method. The ratios to the samples with the target values of $ h_\text{{damp}} $ are shown below the distributions. Vertical bars represent statistical uncertainties. 
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Figure 3a:
The normalised differential cross section as a function of the $ p_{\mathrm{T}} $ (upper) and $ \eta $ (lower) of the $ {\mathrm{t}\overline{\mathrm{t}}} $ system. The black line shows the predictions from the down (left) and up (right) variations in $ h_\text{{damp}} $, and the green line shows the prediction from the nominal sample. The red line shows the nominal sample reweighted to the down (left) and up (right) $ h_\text{{damp}} $ variations using the DCTR method. The ratios to the samples with the target values of $ h_\text{{damp}} $ are shown below the distributions. Vertical bars represent statistical uncertainties. 
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Figure 3b:
The normalised differential cross section as a function of the $ p_{\mathrm{T}} $ (upper) and $ \eta $ (lower) of the $ {\mathrm{t}\overline{\mathrm{t}}} $ system. The black line shows the predictions from the down (left) and up (right) variations in $ h_\text{{damp}} $, and the green line shows the prediction from the nominal sample. The red line shows the nominal sample reweighted to the down (left) and up (right) $ h_\text{{damp}} $ variations using the DCTR method. The ratios to the samples with the target values of $ h_\text{{damp}} $ are shown below the distributions. Vertical bars represent statistical uncertainties. 
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Figure 3c:
The normalised differential cross section as a function of the $ p_{\mathrm{T}} $ (upper) and $ \eta $ (lower) of the $ {\mathrm{t}\overline{\mathrm{t}}} $ system. The black line shows the predictions from the down (left) and up (right) variations in $ h_\text{{damp}} $, and the green line shows the prediction from the nominal sample. The red line shows the nominal sample reweighted to the down (left) and up (right) $ h_\text{{damp}} $ variations using the DCTR method. The ratios to the samples with the target values of $ h_\text{{damp}} $ are shown below the distributions. Vertical bars represent statistical uncertainties. 
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Figure 3d:
The normalised differential cross section as a function of the $ p_{\mathrm{T}} $ (upper) and $ \eta $ (lower) of the $ {\mathrm{t}\overline{\mathrm{t}}} $ system. The black line shows the predictions from the down (left) and up (right) variations in $ h_\text{{damp}} $, and the green line shows the prediction from the nominal sample. The red line shows the nominal sample reweighted to the down (left) and up (right) $ h_\text{{damp}} $ variations using the DCTR method. The ratios to the samples with the target values of $ h_\text{{damp}} $ are shown below the distributions. Vertical bars represent statistical uncertainties. 
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Figure 4:
The normalised differential cross section as a function of $ N_{jet} $ (left) and $ \hat{p_{\mathrm{T}}} $ (right). The black line shows the predictions from the up variation in $ h_\text{{damp}} $ and the green line shows the prediction from the nominal sample. The red line shows the nominal sample reweighted to the $ h_\text{{damp}} $ variation using the DCTR method. The ratios to the target distributions are shown in the pads below. Vertical bars represent statistical uncertainties. 
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Figure 4a:
The normalised differential cross section as a function of $ N_{jet} $ (left) and $ \hat{p_{\mathrm{T}}} $ (right). The black line shows the predictions from the up variation in $ h_\text{{damp}} $ and the green line shows the prediction from the nominal sample. The red line shows the nominal sample reweighted to the $ h_\text{{damp}} $ variation using the DCTR method. The ratios to the target distributions are shown in the pads below. Vertical bars represent statistical uncertainties. 
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Figure 4b:
The normalised differential cross section as a function of $ N_{jet} $ (left) and $ \hat{p_{\mathrm{T}}} $ (right). The black line shows the predictions from the up variation in $ h_\text{{damp}} $ and the green line shows the prediction from the nominal sample. The red line shows the nominal sample reweighted to the $ h_\text{{damp}} $ variation using the DCTR method. The ratios to the target distributions are shown in the pads below. Vertical bars represent statistical uncertainties. 
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Figure 5:
Ratios between the $ h_\text{{damp}} $ target distributions in $ p_{\mathrm{T}}({\mathrm{t}\overline{\mathrm{t}}} ) $ (left) and $ \eta({\mathrm{t}\overline{\mathrm{t}}} ) $ (right), and 50 different reweightings (grey lines). The ratio to the target before the reweighting is shown as a green dashed line and the mean of the different reweightings as a red dashed line. The red band represents the statistical uncertainty of the method obtained from the standard deviation of the 50 reweightings. 
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Figure 5a:
Ratios between the $ h_\text{{damp}} $ target distributions in $ p_{\mathrm{T}}({\mathrm{t}\overline{\mathrm{t}}} ) $ (left) and $ \eta({\mathrm{t}\overline{\mathrm{t}}} ) $ (right), and 50 different reweightings (grey lines). The ratio to the target before the reweighting is shown as a green dashed line and the mean of the different reweightings as a red dashed line. The red band represents the statistical uncertainty of the method obtained from the standard deviation of the 50 reweightings. 
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Figure 5b:
Ratios between the $ h_\text{{damp}} $ target distributions in $ p_{\mathrm{T}}({\mathrm{t}\overline{\mathrm{t}}} ) $ (left) and $ \eta({\mathrm{t}\overline{\mathrm{t}}} ) $ (right), and 50 different reweightings (grey lines). The ratio to the target before the reweighting is shown as a green dashed line and the mean of the different reweightings as a red dashed line. The red band represents the statistical uncertainty of the method obtained from the standard deviation of the 50 reweightings. 
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Figure 6:
Distributions in $ x_{\mathrm{b}} $ (upper) and $ p_{\mathrm{T},{\mathrm{B}}} $ (lower) from $ \mathrm{t} \overline{\mathrm{t}} $ simulations with PYTHIA 8 with value $ r_{\mathrm{b}}= $ 0.855 (dashed green line) and a second value of $ r_{\mathrm{b}} $ (solid black line). The nominal sample reweighted to $ r_{\mathrm{b}}= $ 1.056 (left) and $ r_{\mathrm{b}}= $ 1.252 (right) are is shown as red dotted lines. Below each distribution, the ratios to the target distribution are shown. The vertical bars represent the statistical uncertainties. 
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Figure 6a:
Distributions in $ x_{\mathrm{b}} $ (upper) and $ p_{\mathrm{T},{\mathrm{B}}} $ (lower) from $ \mathrm{t} \overline{\mathrm{t}} $ simulations with PYTHIA 8 with value $ r_{\mathrm{b}}= $ 0.855 (dashed green line) and a second value of $ r_{\mathrm{b}} $ (solid black line). The nominal sample reweighted to $ r_{\mathrm{b}}= $ 1.056 (left) and $ r_{\mathrm{b}}= $ 1.252 (right) are is shown as red dotted lines. Below each distribution, the ratios to the target distribution are shown. The vertical bars represent the statistical uncertainties. 
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Figure 6b:
Distributions in $ x_{\mathrm{b}} $ (upper) and $ p_{\mathrm{T},{\mathrm{B}}} $ (lower) from $ \mathrm{t} \overline{\mathrm{t}} $ simulations with PYTHIA 8 with value $ r_{\mathrm{b}}= $ 0.855 (dashed green line) and a second value of $ r_{\mathrm{b}} $ (solid black line). The nominal sample reweighted to $ r_{\mathrm{b}}= $ 1.056 (left) and $ r_{\mathrm{b}}= $ 1.252 (right) are is shown as red dotted lines. Below each distribution, the ratios to the target distribution are shown. The vertical bars represent the statistical uncertainties. 
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Figure 6c:
Distributions in $ x_{\mathrm{b}} $ (upper) and $ p_{\mathrm{T},{\mathrm{B}}} $ (lower) from $ \mathrm{t} \overline{\mathrm{t}} $ simulations with PYTHIA 8 with value $ r_{\mathrm{b}}= $ 0.855 (dashed green line) and a second value of $ r_{\mathrm{b}} $ (solid black line). The nominal sample reweighted to $ r_{\mathrm{b}}= $ 1.056 (left) and $ r_{\mathrm{b}}= $ 1.252 (right) are is shown as red dotted lines. Below each distribution, the ratios to the target distribution are shown. The vertical bars represent the statistical uncertainties. 
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Figure 6d:
Distributions in $ x_{\mathrm{b}} $ (upper) and $ p_{\mathrm{T},{\mathrm{B}}} $ (lower) from $ \mathrm{t} \overline{\mathrm{t}} $ simulations with PYTHIA 8 with value $ r_{\mathrm{b}}= $ 0.855 (dashed green line) and a second value of $ r_{\mathrm{b}} $ (solid black line). The nominal sample reweighted to $ r_{\mathrm{b}}= $ 1.056 (left) and $ r_{\mathrm{b}}= $ 1.252 (right) are is shown as red dotted lines. Below each distribution, the ratios to the target distribution are shown. The vertical bars represent the statistical uncertainties. 
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Figure 7:
Values of $ \chi^{2}/\text{NDF} $ obtained for distributions in $ x_{\mathrm{b}} $ (circles) and $ p_{\mathrm{T}}^{{\mathrm{B}}} $ (squares), where target distributions for events with different $ r_{\mathrm{b}} $ values are compared to a distribution with the nominal value of $ r_{\mathrm{b}}= $ 0.855 before the reweighting (green) and after the reweighting to the target value of $ r_{\mathrm{b}} $ (red). The lines connecting the markers are shown for illustration purposes only. 
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Figure 8:
Distributions in top quark $ p_{\mathrm{T}} $ (left) and $ \eta $ (right) obtained from simulations at NNLO accuracy (black solid lines), NLO accuracy (green dashed lines), and NLO reweighted to NNLO with the DCTR method (red dotted lines). The ratio to the NNLO predictions is shown in the lower panels, where the vertical bars correspond to the statistical uncertainties. 
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Figure 8a:
Distributions in top quark $ p_{\mathrm{T}} $ (left) and $ \eta $ (right) obtained from simulations at NNLO accuracy (black solid lines), NLO accuracy (green dashed lines), and NLO reweighted to NNLO with the DCTR method (red dotted lines). The ratio to the NNLO predictions is shown in the lower panels, where the vertical bars correspond to the statistical uncertainties. 
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Figure 8b:
Distributions in top quark $ p_{\mathrm{T}} $ (left) and $ \eta $ (right) obtained from simulations at NNLO accuracy (black solid lines), NLO accuracy (green dashed lines), and NLO reweighted to NNLO with the DCTR method (red dotted lines). The ratio to the NNLO predictions is shown in the lower panels, where the vertical bars correspond to the statistical uncertainties. 
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Figure 9:
Distributions in $ p_{\mathrm{T}} $ (upper left), $ \eta $ (upper right), $ \Delta\phi $ (lower left), and mass (lower right) of the $ \mathrm{t} \overline{\mathrm{t}} $ system obtained from simulations at NNLO accuracy (black solid lines), NLO accuracy (green dashed lines), and NLO reweighted to NNLO with the DCTR method (red dotted lines). The ratio to the NNLO predictions is shown in the lower panels, where the vertical bars correspond to the statistical uncertainties. 
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Figure 9a:
Distributions in $ p_{\mathrm{T}} $ (upper left), $ \eta $ (upper right), $ \Delta\phi $ (lower left), and mass (lower right) of the $ \mathrm{t} \overline{\mathrm{t}} $ system obtained from simulations at NNLO accuracy (black solid lines), NLO accuracy (green dashed lines), and NLO reweighted to NNLO with the DCTR method (red dotted lines). The ratio to the NNLO predictions is shown in the lower panels, where the vertical bars correspond to the statistical uncertainties. 
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Figure 9b:
Distributions in $ p_{\mathrm{T}} $ (upper left), $ \eta $ (upper right), $ \Delta\phi $ (lower left), and mass (lower right) of the $ \mathrm{t} \overline{\mathrm{t}} $ system obtained from simulations at NNLO accuracy (black solid lines), NLO accuracy (green dashed lines), and NLO reweighted to NNLO with the DCTR method (red dotted lines). The ratio to the NNLO predictions is shown in the lower panels, where the vertical bars correspond to the statistical uncertainties. 
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Figure 9c:
Distributions in $ p_{\mathrm{T}} $ (upper left), $ \eta $ (upper right), $ \Delta\phi $ (lower left), and mass (lower right) of the $ \mathrm{t} \overline{\mathrm{t}} $ system obtained from simulations at NNLO accuracy (black solid lines), NLO accuracy (green dashed lines), and NLO reweighted to NNLO with the DCTR method (red dotted lines). The ratio to the NNLO predictions is shown in the lower panels, where the vertical bars correspond to the statistical uncertainties. 
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Figure 9d:
Distributions in $ p_{\mathrm{T}} $ (upper left), $ \eta $ (upper right), $ \Delta\phi $ (lower left), and mass (lower right) of the $ \mathrm{t} \overline{\mathrm{t}} $ system obtained from simulations at NNLO accuracy (black solid lines), NLO accuracy (green dashed lines), and NLO reweighted to NNLO with the DCTR method (red dotted lines). The ratio to the NNLO predictions is shown in the lower panels, where the vertical bars correspond to the statistical uncertainties. 
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Figure 10:
Distributions in $ p_{\mathrm{T}} $ of the $ \mathrm{t} \overline{\mathrm{t}} $ system (left) and $ p_{\mathrm{T}} $ of the t (right) obtained from simulations at NNLO accuracy (black solid lines), NLO accuracy (green dashed lines), NLO reweighted to NNLO with the DCTR method (red dotted lines), and NLO reweighted using a twodimensional reweighting in $ p_{\mathrm{T}} $ of the $ \mathrm{t} \overline{\mathrm{t}} $ system and of the t (blue dashdotted line). The ratio to the NNLO predictions is shown in the lower panels, where the vertical bars correspond to the statistical uncertainties. 
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Figure 10a:
Distributions in $ p_{\mathrm{T}} $ of the $ \mathrm{t} \overline{\mathrm{t}} $ system (left) and $ p_{\mathrm{T}} $ of the t (right) obtained from simulations at NNLO accuracy (black solid lines), NLO accuracy (green dashed lines), NLO reweighted to NNLO with the DCTR method (red dotted lines), and NLO reweighted using a twodimensional reweighting in $ p_{\mathrm{T}} $ of the $ \mathrm{t} \overline{\mathrm{t}} $ system and of the t (blue dashdotted line). The ratio to the NNLO predictions is shown in the lower panels, where the vertical bars correspond to the statistical uncertainties. 
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Figure 10b:
Distributions in $ p_{\mathrm{T}} $ of the $ \mathrm{t} \overline{\mathrm{t}} $ system (left) and $ p_{\mathrm{T}} $ of the t (right) obtained from simulations at NNLO accuracy (black solid lines), NLO accuracy (green dashed lines), NLO reweighted to NNLO with the DCTR method (red dotted lines), and NLO reweighted using a twodimensional reweighting in $ p_{\mathrm{T}} $ of the $ \mathrm{t} \overline{\mathrm{t}} $ system and of the t (blue dashdotted line). The ratio to the NNLO predictions is shown in the lower panels, where the vertical bars correspond to the statistical uncertainties. 
Summary 
Particle physics relies on the simulation of events using Monte Carlo (MC) event generators for datatotheory comparisons. Data analyses require the production of several samples simulating the same physical process to estimate systematic uncertainties or the impact of higherorder calculations. To have statistically significant value, these samples have to be very large with millions of events generated and simulated at a high computational cost. Nevertheless, the statistical precision from the finite size of these samples can become a limiting factor in precision analyses. The production of sufficiently large MC samples, such that the statistical precision of these samples is better than the statistical precision of the data, will become increasingly prohibitive at the HighLuminosity LHC (HLLHC) with the expected computing resources. Current estimates assume that 160 billion fully simulated and reconstructed MC events have to be produced per year after the start of the HLLHC. This number may become larger by up to 30% because of negative events in nexttoleading order (NLO) and nexttoNLO (NNLO) simulations, which reduce the statistical precision of the MC samples. In this note, the deep neural network using classification for tuning and reweighting (DCTR) method has been introduced to reweight MC samples used in CMS analyses. The weights calculated with the DCTR model enable the modification of one central sample to mimic other samples obtained with different parameters or different simulation programs. This methodology avoids the need for simulating the detector response for multiple samples by incorporating the relevant variations in a single sample. While dedicated samples have to be generated for the training and validation of the model, these do not need to have the full detector simulation and reconstruction, saving up to 75% of the typical CPU resources needed for the production of MC samples in CMS. In addition, after the training of the DCTR model, the training samples can be deleted saving storage space for multiple billions of events. This method will become an integral part of the computing model of the CMS Collaboration for the HLLHC. The DCTR method has been shown to work reliably for two important sources of modelling uncertainties in the simulation of top quark pair ($ \mathrm{t} \overline{\mathrm{t}} $) production. The systematic uncertainty connected to the matching of radiation from matrix elements and the parton shower previously had to be estimated with dedicated samples. The reweighting of variations in the b quark fragmentation shows that a continuous reweighting in a model parameter is possible, paving the way for the determination of model parameters directly from collision data. Additionally, the method has been extended to reweight an NLO simulation to an NNLO one for $ \mathrm{t} \overline{\mathrm{t}} $ production, which will allow for a fast evaluation of the impact of higherorder corrections on data analyses. The DCTR reweighting can be seamlessly integrated into CMS analyses, providing an elegant solution to address the computational challenges posed by the production of large MC samples, particularly for the HLLHC. 
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