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In this paper, we are interested in the acceleration of numerical simulations. We focus on a hypersonic planetary reentry problem whose simulation involves coupling fluid dynamics and chemical reactions. Simulating chemical reactions takes most of the computational time but, on the other hand, cannot be avoided to obtain accurate predictions. We face a trade-off between cost-efficiency and accuracy: the numerical scheme has to be sufficiently efficient to be used in an operational context but accurate enough to predict the phenomenon faithfully. To tackle this trade-off, we design a hybrid numerical scheme coupling a traditional fluid dynamic solver with a neural network approximating the chemical reactions. We rely on their power in terms of accuracy and dimension reduction when applied in a big data context and on their efficiency stemming from their matrix-vector structure to achieve important acceleration factors (×10 to ×18.6). This paper aims to explain how we design such cost-effective hybrid numerical schemes in practice. Above all, we describe methodologies to ensure accuracy guarantees, allowing us to go beyond traditional surrogate modeling and to use these schemes as references. • Deep Learning-based hybridization speeds up numerical schemes of atmospheric reentry while maintaining high accuracy. • Initializing a scheme with a hybrid code's prediction reduces the convergence time and keeps the exact same guarantees. • Uncertainty analysis provides statistical guarantees concerning approximation errors when using hybridization code. • Neural network approximation error is statistically lower than many other sources of error inherent to numerical simulations. [ABSTRACT FROM AUTHOR]

We present a numerical framework for learning unknown stochastic dynamical systems using measurement data. Termed stochastic flow map learning (sFML), the new framework is an extension of flow map learning (FML) that was developed for learning deterministic dynamical systems. For learning stochastic systems, we define a stochastic flow map that is a superposition of two sub-flow maps: a deterministic sub-map and a stochastic sub-map. The stochastic training data are used to construct the deterministic sub-map first, followed by the stochastic sub-map. The deterministic sub-map takes the form of residual network (ResNet), similar to the work of FML for deterministic systems. For the stochastic sub-map, we employ a generative model, particularly generative adversarial networks (GANs) in this paper. The final constructed stochastic flow map then defines a stochastic evolution model that is a weak approximation, in term of distribution, of the unknown stochastic system. A comprehensive set of numerical examples are presented to demonstrate the flexibility and effectiveness of the proposed sFML method for various types of stochastic systems. • Proposed a new numerical method for modeling SDEs with observation data. • Established a mathematical framework that utilizes stochastic flow map for effective modeling. • Developed novel stochastic flow map modeling and its DNN structure. [ABSTRACT FROM AUTHOR]

Recent work has focused on data-driven learning of the evolution of unknown systems via deep neural networks (DNNs), with the goal of conducting long time prediction of the evolution of the unknown system. Training a DNN with low generalization error is a particularly important task in this case as error is accumulated over time. Because of the inherent randomness in DNN training, chiefly in stochastic optimization, there is uncertainty in the resulting prediction, and therefore in the generalization error. Hence, the generalization error can be viewed as a random variable with some probability distribution. Well-trained DNNs, particularly those with many hyperparameters, typically result in probability distributions for generalization error with low bias but high variance. High variance causes variability and unpredictably in the results of a trained DNN. This paper presents a computational technique which decreases the variance of the generalization error, thereby improving the reliability of the DNN model to generalize consistently. In the proposed ensemble averaging method, multiple models are independently trained and model predictions are averaged at each time step. A mathematical foundation for the method is presented, including results regarding the distribution of the local truncation error. In addition, three time-dependent differential equation problems are considered as numerical examples, demonstrating the effectiveness of the method to decrease variance of DNN predictions generally. • Proposed ensemble averaged learning for deep neural network (DNN) modeling unknown systems. • Established mathematical foundation for variance reduction by the ensemble averaged learning. • Provided extensive numerical studies to verify the theoretical results and demonstrate the practical advantage of the method. [ABSTRACT FROM AUTHOR]

Generating quasirandom points with high uniformity is a fundamental task in many fields. Existing number-theoretic approaches produce evenly distributed points in [ 0 , 1 ] d in asymptotic sense but may not yield a good distribution for a given set size. It is also difficult to extend those techniques to other geometries like a disk or a manifold. In this paper, we present a novel physics-informed framework to transform a given set of points into a distribution with better uniformity. We model each point as a particle and assign the system with a potential energy. Upon minimizing the energy, the uniformity of distribution can be improved correspondingly. Two kinds of schemes are introduced: one based on molecular dynamics and another based on deep neural networks. The new physics-informed framework serves as a black-box transformer that is able to improve given distributions and can be easily extended to other geometries such as disks, spheres, complex manifolds, etc. Various experiments with different geometries are provided to demonstrate that the new framework is able to transform poorly distributed input into one with superior uniformity. • A novel physics-informed framework for improving the uniformity of a given distribution. • Molecular dynamics and deep neural networks are used for transforming distributions. • The new framework works for different geometries including general manifolds. • A new matrix-based metric is introduced to measure uniformity on general geometry. • Experiments show that the new approach is effective for various geometries. [ABSTRACT FROM AUTHOR]