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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]

Fourier analysis based optimization techniques have been developed for numerical schemes on structured grids and result in significant performance improvements. However, there lacks an effective optimization technique for numerical schemes on general unstructured grids, due to the mesh geometry complexity that makes Fourier transformation infeasible. To address this issue, an optimization framework based on machine learning is developed in this paper for numerical schemes on unstructured grids. The optimization of a compact high-order variational reconstruction on triangular grids is performed to illustrate the framework. An artificial neural network (ANN) is used to predict the optimal values of the derivative weights on cell interfaces that are the free parameters of the variational reconstruction, given the local geometry as input. The ANN is trained by minimizing the solution errors of a linear advection equation on a set of generated compact stencils, with a set of sampled sine waves as initial conditions. The developed optimization framework is applicable to general numerical schemes on unstructured grids as the training does not require explicit computation of dispersion or dissipation. Numerical results for inviscid flow problems show that the optimized variational finite volume scheme is remarkably more accurate and efficient than its non-optimized counterpart, demonstrating the effectiveness of the machine learning optimization. • A machine learning optimization framework for numerical schemes on unstructured grids. • A feedforward neural network to predict optimal parameter values of numerical schemes. • Training of the network by minimizing solution errors of a linear advection equation. • A machine learning optimized compact high-order variational reconstruction. • Generalization of the optimized variational reconstruction to nonlinear problems. [ABSTRACT FROM AUTHOR]

The Allen-Cahn equation is a well-known stiff semilinear parabolic equation used to describe the process of phase separation and transition in phase field modeling of multi-component physical systems, while the conservative Allen-Cahn equation is a modified version of the classic Allen-Cahn equation that can additionally conserve the mass. As neural networks and deep learning techniques have achieved significant successes in recent years in scientific and engineering applications, there has been growing interest in developing deep learning algorithms for numerical solutions of partial differential equations. In this paper, we propose a novel deep learning method for predicting the dynamics of the classic and conservative Allen-Cahn equations. Specifically, we design two special convolutional neural network models, one for each of the two equations, to learn the fully-discrete operators between two adjacent time steps. The loss functions of the two models are defined using the residual of the fully-discrete systems, which result from applying the central finite difference discretization in space and the Crank–Nicolson approximation in time. This approach enables us to train the models without requiring any ground-truth data. Moreover, we introduce an effective training strategy that automatically generates useful samples along the time evolution to facilitate training of the models. Finally, we conduct extensive experiments in two and three dimensions to demonstrate outstanding performance of our proposed method, including its dynamics prediction and generalization ability under different scenarios. • We propose two novel convolutional neural networks to learn the fully-discrete operators of the classic and conservative Allen-Chan equations. • The loss functions are defined using the residual of the fully-discrete systems, which enables us to train the models without ground-truth data. • We introduce a novel training strategy that automatically generates samples along the time to facilitate effective training of the two models. • Extensive experiments demonstrate outstanding performance of the proposed models for dynamics prediction under different scenarios. [ABSTRACT FROM AUTHOR]