The most serious difficulty in the ME method that has limited its applicability to searches for beyond-the-SM imdin the new physics rech of the Cwhich wi be dominated by increase in integrated The applica tion of the ME method is computationally challenging for two reasons:(1)it involves high- spite the attr rdenof the ME method and promise of futher optimiztion ive featur dern machine lea ed up the the ME method and therefore broaden the applicability of the ME method to the efit of HL-LHC physics is suf ly rich to encod al N of ti he wmple) (DN re strong APwdhatepoiyoaaeeodp imations,poesibly n co unctio ith smart ph can be found i On ea set of DNNs re oodapnoii of th ese DN n be ME method to be both nimble and usb neither of which is true today. The all strategy is to do the sive full me calculatio sible ideally ace for dnn raining and once more for a final pass befor a good appr ture analysis thod with might l ng lk sed through this pro dure As the s through selection and /or sample nRetnticteanetcte,the DNN-base o the time ME method is published.a final pass using full ME calculation would likely be performed both to maximize orty of the ruad to adate teevi the Dbas alone with a com software project for ME calculations in the spirit of This area is very nclaiedformpactlcohboatiouwihg hine learning. sing a DNN trainins from full ME calculations and direct compa ns of the integration accura and DNN-based cal tions should be udertaken.More placed in developing comeling common API,is proposed. 3.6 Matrix Element Machine Learning Method The ma element method is hased in the fact that the physics of article collisions is t-ha r al and side of interest.In exactly the same inputs as in the matrix element method.namely,the matrix elements.parton distribution 10The most serious difficulty in the ME method that has limited its applicability to searches for beyond-the-SM physics and precision measurements is that it is very computationally intensive. If this limitation is overcome, it would enable more widespread use of ME methods for analysis of LHC data. This could be particularly im￾portant for extending the new physics reach of the HL-LHC which will be dominated by increases in integrated luminosity rather than center-of-mass collision energy. The application of the ME method is computationally challenging for two reasons: (1) it involves high￾dimensional integration over a large number of events, signal and background hypotheses, and systematic variations and (2) it involves sharply-peaked integrands2 over a large domain in phase space. Therefore, de￾spite the attractive features of the ME method and promise of further optimization and parallelization, the computational burden of the ME technique will continue to limit is range of applicability for practical data analysis without new and innovative approaches. The primary idea put forward in this section is to utilize modern machine learning techniques to dramatically speed up the numerical evaluations in the ME method and therefore broaden the applicability of the ME method to the benefit of HL-LHC physics. Applying neural networks to numerical integration problems is plausible but not new (see [34–36], for example). The technical challenge is to design a network which is sufficiently rich to encode the complexity of the ME cal￾culation for a given process over the phase space relevant to the signal process. Deep Neural Networks (DNNs) are strong candidates for networks with sufficient complexity to achieve good approximations, possibly in con￾junction with smart phase-space mapping such as described in [37]. Promising demonstration of the power of Boosted Decision Trees [38, 39] and Generative Adversarial Networks [40] for improved Monte Carlo integration can be found in [41]. Once a set of DNNs representing definite integrals is generated to good approximation, evaluation of the ME method calculations via the DNNs will be very fast. These DNNs can be thought of as preserving the essence of ME calculations in a way that allows for fast forward execution. They can enable the ME method to be both nimble and sustainable, neither of which is true today. The overall strategy is to do the expensive full ME calculations as infrequently as possible, ideally once for DNN training and once more for a final pass before publication, with the DNNs utilized as a good approximation in between. A future analysis flow using the ME method with DNNs might look something like the following: One performs a large number of ME calculations using a traditional numerical integration technique like VE￾GAS [42, 43] or FOAM [44] on a large CPU resource, ideally exploiting acceleration on many-core devices. The DNN training data is generated from the phase space sampling in performing the full integration in this initial pass, and DNNs are trained either in situ or a posteriori. The accuracy of the DNN-based ME calculation can be assessed through this procedure. As the analysis develops and progresses through selection and/or sample changes, systematic treatment, etc., the DNN-based ME calculations are used in place of the time-consuming, full ME calculations to make the analysis nimble and to preserve the ME calculations. Before a result using the ME method is published, a final pass using full ME calculation would likely be performed both to maximize the numerical precision or sensitivity of the results and to validate the analysis evolution via the DNN-based approximations. There are several activities which are proposed to further develop the idea of a Sustainable Matrix Element Method. The first is to establish a cross-experiment group interested in developing the ideas presented in this section, along with a common software project for ME calculations in the spirit of [45]. This area is very well-suited for impactful collaboration with computer scientists and those working in machine learning. Using a few test cases (e.g. tt¯ or tth¯ production), evaluation of DNN choices and configurations, developing methods for DNN training from full ME calculations and direct comparisons of the integration accuracy between Monte Carlo and DNN-based calculations should be undertaken. More effort should also be placed in developing compelling applications of the ME method for HL-LHC physics. In the longer term, the possibility of Sustainable-Matrix￾Element-Method-as-a-Service (SMEMaaS), where shared software and infrastructure could be used through a common API, is proposed. 3.6 Matrix Element Machine Learning Method The matrix element method is based in the fact that the physics of particle collisions is encoded in the distri￾bution of the particles’ four-momenta and with their flavors. As noted in the previous section, the fundamental task is to approximate the left-hand side of Eq. (5) for all (exclusive) final states of interest. In the matrix element method, one proceeds by approximating the right-hand side of Eq. (5). But, since the goal is to com￾pute Pξ(x|α), and given that billions of fully simulated events will be available, and that the simulations use exactly the same inputs as in the matrix element method, namely, the matrix elements, parton distribution 2a consequence of imposing energy/momentum conservation in the processes 10