The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations
- D. Xiu, G. Karniadakis
- MathematicsSIAM Journal on Scientific Computing
- 1 February 2002
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.
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
- M. Raissi, P. Perdikaris, G. Karniadakis
- Computer ScienceJournal of Computational Physics
- 1 February 2019
Microflows and Nanoflows: Fundamentals and Simulation
- G. Karniadakis, A. Beskok, N. Aluru, Chih-Ming Ho
- Physics, Engineering
- 16 November 2001
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.-…
Spectral/hp Element Methods for Computational Fluid Dynamics
- G. Karniadakis, S. Sherwin
- Mathematics
- 11 August 2005
Introduction Fundamental concepts in one dimension Multi-dimensional expansion bases Multi-dimensional formulations Diffusion equation Advection and advection-diffusion Non-conforming elements…
High-order splitting methods for the incompressible Navier-Stokes equations
- G. Karniadakis, M. Israeli, S. Orszag
- Computer Science
- 1 December 1991
Modeling uncertainty in flow simulations via generalized polynomial chaos
- D. Xiu, G. Karniadakis
- Mathematics
- 1 May 2003
Spectral/hp Element Methods for CFD
- G. Karniadakis, S. Sherwin
- Mathematics
- 15 April 1999
REPORT: A MODEL FOR FLOWS IN CHANNELS, PIPES, AND DUCTS AT MICRO AND NANO SCALES
- A. Beskok, G. Karniadakis
- Engineering, Physics
- 1 February 1999
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-based…
Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
- M. Raissi, P. Perdikaris, G. Karniadakis
- Computer ScienceArXiv
- 28 November 2017
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.
DeepXDE: A Deep Learning Library for Solving Differential Equations
- Lu Lu, Xuhui Meng, Zhiping Mao, G. Karniadakis
- Computer ScienceAAAI Spring Symposium: MLPS
- 10 July 2019
An overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation, and a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs.
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