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Bias in causal comparisons has a correspondence with distributional imbalance of covariates between treatment groups. Weighting strategies such as inverse propensity score weighting attempt to mitigate bias by either modeling the treatment assignment mechanism or balancing specified covariate moments. This article introduces a new weighting method, called energy balancing, which instead aims to balance weighted covariate distributions. By directly targeting distributional imbalance, the proposed weighting strategy can be flexibly utilized in a wide variety of causal analyses without the need for careful model or moment specification. Our energy balancing weights (EBW) approach has several advantages over existing weighting techniques. First, it offers a model-free and robust approach for obtaining covariate balance that does not require tuning parameters, obviating the need for modeling decisions of secondary nature to the scientific question at hand. Second, since this approach is based on a genuine measure of distributional balance, it provides a means for assessing the balance induced by a given set of weights for a given dataset. We demonstrate the effectiveness of this EBW approach in a suite of simulation experiments, and in studies on the safety of right heart catheterization and on three additional studies using electronic health record data.

For energy-assisted compression ignition (EACI) engine propulsion at high-altitude operating conditions using sustainable jet fuels with varying cetane numbers, it is essential to develop an efficient engine control system for robust and optimal operation. Control systems are typically trained using experimental data, which can be costly and time consuming to generate due to setup time of experiments, unforeseen delays/issues with manufacturing, mishaps/engine failures and the consequent repairs (which can take weeks), and errors in measurements. Computational fluid dynamics (CFD) simulations can overcome such burdens by complementing experiments with simulated data for control system training. Such simulations, however, can be computationally expensive. Existing data-driven machine learning (ML) models have shown promise for emulating the expensive CFD simulator, but encounter key limitations here due to the expensive nature of the training data and the range of differing combustion behaviors (e.g. misfires and partial/delayed ignition) observed at such broad operating conditions. We thus develop a novel physics-integrated emulator, called the Misfire-Integrated GP (MInt-GP), which integrates important auxiliary information on engine misfires within a Gaussian process surrogate model. With limited CFD training data, we show the MInt-GP model can yield reliable predictions of in-cylinder pressure evolution profiles and subsequent heat release profiles and engine CA50 predictions at a broad range of input conditions. We further demonstrate much better prediction capabilities of the MInt-GP at different combustion behaviors compared to existing data-driven ML models such as kriging and neural networks, while also observing up to 80 times computational speed-up over CFD, thus establishing its effectiveness as a tool to assist CFD for fast data generation in control system training.