A machine learning framework for data driven acceleration of computations of differential equations

We propose a machine learning framework to accelerate numerical computations of time-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existing numerical methods as artificial neural networks, with a set of trainable parameters. These parameters are determined in an offline training process by (approximately) minimizing suitable (possibly non-convex) loss functions by (stochastic) gradient descent methods. The proposed algorithm is designed to be always consistent with the underlying differential equation. Numerical experiments involving both linear and non-linear ODE and PDE model problems demonstrate a significant gain in computational efficiency over standard numerical methods

Parallel Multi-Hypothesis Algorithm for Criticality Estimation in Traffic and Collision Avoidance

Due to the current developments towards autonomous driving and vehicle active safety, there is an increasing necessity for algorithms that are able to perform complex criticality predictions in real-time. Being able to process multi-object traffic scenarios aids the implementation of a variety of automotive applications such as driver assistance systems for collision prevention and mitigation as well as fall-back systems for autonomous vehicles. We present a fully model-based algorithm with a parallelizable architecture. The proposed algorithm can evaluate the criticality of complex, multi-modal (vehicles and pedestrians) traffic scenarios by simulating millions of trajectory combinations and detecting collisions between objects. The algorithm is able to estimate upcoming criticality at very early stages, demonstrating its potential for vehicle safety-systems and autonomous driving applications. An implementation on an embedded system in a test vehicle proves in a prototypical manner the compatibility of the algorithm with the hardware possibilities of modern cars. For a complex traffic scenario with 11 dynamic objects, more than 86 million pose combinations are evaluated in 21 ms on the GPU of a Drive PX~2

Fast Neural Network Predictions from Constrained Aerodynamics Datasets

Incorporating computational fluid dynamics in the design process of jets, spacecraft, or gas turbine engines is often challenged by the required computational resources and simulation time, which depend on the chosen physics-based computational models and grid resolutions. An ongoing problem in the field is how to simulate these systems faster but with sufficient accuracy. While many approaches involve simplified models of the underlying physics, others are model-free and make predictions based only on existing simulation data. We present a novel model-free approach in which we reformulate the simulation problem to effectively increase the size of constrained pre-computed datasets and introduce a novel neural network architecture (called a cluster network) with an inductive bias well-suited to highly nonlinear computational fluid dynamics solutions. Compared to the state-of-the-art in model-based approximations, we show that our approach is nearly as accurate, an order of magnitude faster, and easier to apply. Furthermore, we show that our method outperforms other model-free approaches

Data-driven acceleration of Photonic Simulations

Designing modern photonic devices often involves traversing a large parameter space via an optimization procedure, gradient based or otherwise, and typically results in the designer performing electromagnetic simulations of correlated devices. In this paper, we present an approach to accelerate the Generalized Minimal Residual (GMRES) algorithm for the solution of frequency-domain Maxwell's equations using two machine learning models (principal component analysis and a convolutional neural network) trained on simulations of correlated devices. These data-driven models are trained to predict a subspace within which the solution of the frequency-domain Maxwell's equations lie. This subspace can then be used for augmenting the Krylov subspace generated during the GMRES iterations. By training the proposed models on a dataset of grating wavelength-splitting devices, we show an order of magnitude reduction ($\sim 10 - 50$) in the number of GMRES iterations required for solving frequency-domain Maxwell's equations