Related papers: Neural network based limiter with transfer learnin…

Neural networks have recently become popular for a wide variety of uses, but have seen limited application in safety-critical domains such as robotics near and around humans. This is because it remains an open challenge to train a neural…

We implement a high-order numerical scheme for the entropy-based moment closure, the so-called M$_N$ model, for linear kinetic equations in slab geometry. A discontinuous Galerkin (DG) scheme in space along with a strong-stability…

We develop a novel cut discontinuous Galerkin (CutDG) method for stationary advection-reaction problems on surfaces embedded in $\mathbb{R}^d$. The CutDG method is based on embedding the surface into a full-dimensional background mesh and…

Training a neural network (NN) typically relies on some type of curve-following method, such as gradient descent (GD) (and stochastic gradient descent (SGD)), ADADELTA, ADAM or limited memory algorithms. Convergence for these algorithms…

We consider the problem of training input-output recurrent neural networks (RNN) for sequence labeling tasks. We propose a novel spectral approach for learning the network parameters. It is based on decomposition of the cross-moment tensor…

A novel class of Runge-Kutta discontinuous Galerkin schemes for coupled systems of conservation laws in multiple space dimensions that are separated by a fixed sharp interface is introduced. The schemes are derived from a relaxation…

Neural networks, a central tool in machine learning, have demonstrated remarkable, high fidelity performance on image recognition and classification tasks. These successes evince an ability to accurately represent high dimensional…

This work presents algorithms for the efficient implementation of discontinuous Galerkin methods with explicit time stepping for acoustic wave propagation on unstructured meshes of quadrilaterals or hexahedra. A crucial step towards…

Neural networks offer a computationally efficient approximation of model predictive control, but they lack guarantees on the resulting controlled system's properties. Formal certification of neural networks is crucial for ensuring safety,…

Modeling the localized intensive deformation in a damaged solid requires highly refined discretization for accurate prediction, which significantly increases the computational cost. Although adaptive model refinement can be employed for…

Training deep neural networks with stochastic gradient descent (SGD) can often achieve zero training loss on real-world tasks although the optimization landscape is known to be highly non-convex. To understand the success of SGD for…

Large-scale Graph Neural Networks (GNNs) are typically trained by sampling a vertex's neighbors to a fixed distance. Because large input graphs are distributed, training requires frequent irregular communication that stalls forward…

The paper proposes a scheme by combining the Runge-Kutta discontinuous Galerkin method with a {\delta}-mapping algorithm for solving hyperbolic conservation laws with discontinuous fluxes. This hybrid scheme is particularly applied to…

Linear Quadratic Regulator (LQR) is often combined with feedback linearization (FBL) for nonlinear systems that have the nonlinearity additive to the input. Conventional approaches estimate and cancel the nonlinearity based on the first…

In recent years, there has been an increasing interest in using deep learning and neural networks to tackle scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods, such…

Deep neural networks work well at approximating complicated functions when provided with data and trained by gradient descent methods. At the same time, there is a vast amount of existing functions that programmatically solve different…

How can local-search methods such as stochastic gradient descent (SGD) avoid bad local minima in training multi-layer neural networks? Why can they fit random labels even given non-convex and non-smooth architectures? Most existing theory…

Real-time accurate solutions of large-scale complex dynamical systems are critically needed for control, optimization, uncertainty quantification, and decision-making in practical engineering and science applications, particularly in…

In the previous article, we introduced a neural network framework based on symmetric differential equations. This novel framework exhibits complete symmetry, endowing it with perfect mathematical properties. While we have examined some of…

In this paper we develop a non-diffusive neural network (NDNN) algorithm for accurately solving weak solutions to hyperbolic conservation laws. The principle is to construct these weak solutions by computing smooth local solutions in…