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The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations
TLDR
This work represents the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error. Expand
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
Abstract We introduce physics-informed neural networks – neural networks that are trained to solve supervised learning tasks while respecting any given laws of physics described by general nonlinearExpand
Microflows and Nanoflows: Fundamentals and Simulation
Gas Flows.- Governing Equations and Slip Models.- Shear-Driven Flows.- Pressure-Driven Flows.- Thermal Effects in Microscales.- Prototype Applications of Gas Flows.- Basic Concepts and Technologies.-Expand
Spectral/hp Element Methods for Computational Fluid Dynamics
Introduction Fundamental concepts in one dimension Multi-dimensional expansion bases Multi-dimensional formulations Diffusion equation Advection and advection-diffusion Non-conforming elementsExpand
High-order splitting methods for the incompressible Navier-Stokes equations
Abstract A new pressure formulation for splitting methods is developed that results in high-order time-accurate schemes for the solution of the incompressible Navier-Stokes equations. In particular,Expand
Spectral/hp Element Methods for CFD
Introduction Fundamental concepts in one dimension Multi-dimensional expansion bases Multi-dimensional formulations Diffusion equation Advection and advection-diffusion Non-conforming elementsExpand
Modeling uncertainty in flow simulations via generalized polynomial chaos
We present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations. The stochastic input is represented spectrally by employing orthogonal polynomialExpand
REPORT: A MODEL FOR FLOWS IN CHANNELS, PIPES, AND DUCTS AT MICRO AND NANO SCALES
Rarefied gas flows in channels, pipes, and ducts with smooth surfaces are studied in a wide range of Knudsen number (Kn) at low Mach number (M) with the objective of developing simple, physics-basedExpand
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
We formulate a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with long-term integration and discontinuities in stochastic differential equations. We first present this method forExpand
Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
TLDR
This two part treatise introduces physics informed neural networks – neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations and demonstrates how these networks can be used to infer solutions topartial differential equations, and obtain physics-informed surrogate models that are fully differentiable with respect to all input coordinates and free parameters. Expand
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