Accelerating hypersonic reentry simulations using deep learning-based hybridization (with guarantees)

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 simulation code 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 simulation code 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 ($\times 10$ to $\times 18.6$). This paper aims to explain how we design such cost-effective hybrid simulation codes in practice. Above all, we describe methodologies to ensure accuracy guarantees, allowing us to go beyond traditional surrogate modeling and to use these codes as references.

Goal-Oriented Sensitivity Analysis of Hyperparameters in Deep Learning

Tackling new machine learning problems with neural networks always means optimizing numerous hyperparameters that define their structure and strongly impact their performances. In this work, we study the use of goal-oriented sensitivity analysis, based on the Hilbert-Schmidt Independence Criterion (HSIC), for hyperparameter analysis and optimization. Hyperparameters live in spaces that are often complex and awkward. They can be of different natures (categorical, discrete, boolean, continuous), interact, and have inter-dependencies. All this makes it non-trivial to perform classical sensitivity analysis. We alleviate these difficulties to obtain a robust analysis index that is able to quantify hyperparameters' relative impact on a neural network's final error. This valuable tool allows us to better understand hyperparameters and to make hyperparameter optimization more interpretable. We illustrate the benefits of this knowledge in the context of hyperparameter optimization and derive an HSIC-based optimization algorithm that we apply on MNIST and Cifar, classical machine learning data sets, but also on the approximation of Runge function and Bateman equations solution, of interest for scientific machine learning. This method yields neural networks that are both competitive and cost-effective.

Variance Based Samples Weighting for Supervised Deep Learning

In the context of supervised learning of a function by a Neural Network (NN), we claim and empirically justify that a NN yields better results when the distribution of the data set focuses on regions where the function to learn is steeper. We first traduce this assumption in a mathematically workable way using Taylor expansion. Then, theoretical derivations allow to construct a methodology that we call Variance Based Samples Weighting (VBSW). VBSW uses local variance of the labels to weight the training points. This methodology is general, scalable, cost effective, and significantly increases the performances of a large class of NNs for various classification and regression tasks on image, text and multivariate data. We highlight its benefits with experiments involving NNs from shallow linear NN to Resnet or Bert.

A Taylor Based Sampling Scheme for Machine Learning in Computational Physics

Machine Learning (ML) is increasingly used to construct surrogate models for physical simulations. We take advantage of the ability to generate data using numerical simulations programs to train ML models better and achieve accuracy gain with no performance cost. We elaborate a new data sampling scheme based on Taylor approximation to reduce the error of a Deep Neural Network (DNN) when learning the solution of an ordinary differential equations (ODE) system.