A constrained integration (CINT) approach to solving partial differential equations using artificial neural networks

Rudd, Keith; Ferrari, Silvia

doi:10.1016/j.neucom.2014.11.058

Cited by 123 publications

(89 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…From this point of view, efforts are being undertaken to adapt solution algorithms to fit this new programming paradigm. This implies the use of technology based on training sets (determining weights of associated neural networks) computed outside critical time windows, that are successively applied to problems (including PDEs) described by approximated models constructed with the neural network weights (see, e.g., [77]). The weather and climate community should be aware of the factors driving hardware vendors and facilitate co-design of the underlying algorithms, in order to exploit new generation computing machines at their best.…”

Section: Discussion and Concluding Remarksmentioning

confidence: 99%

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

Mengaldo

Wyszogrodzki

Diamantakis

et al. 2018

Arch Computat Methods Eng

View full text Add to dashboard Cite

The continuous partial differential equations governing a given physical phenomenon, such as the Navier-Stokes equations describing the fluid motion, must be numerically discretized in space and time in order to obtain a solution otherwise not readily available in closed (i.e., analytic) form. While the overall numerical discretization plays an essential role in the algorithmic efficiency and physically-faithful representation of the solution, the time-integration strategy commonly is one of the main drivers in terms of cost-to-solution (e.g., time-or energy-to-solution), accuracy and numerical stability, thus constituting one of the key building blocks of the computational model. This is especially true in time-critical applications, including numerical weather prediction (NWP), climate simulations and engineering. This review provides a comprehensive overview of the existing and emerging time-integration (also referred to as time-stepping) practices used in the operational global NWP and climate industry, where global refers to weather and climate simulations performed on the entire globe. While there are many flavors of time-integration strategies, in this review we focus on the most widely adopted in NWP and climate centers and we emphasize the reasons why such numerical solutions were adopted. This allows us to make some considerations on future trends in the field such as the need to balance accuracy in time with substantially enhanced time-to-solution and associated implications on energy consumption and running costs. In addition, the potential for the co-design of time-stepping algorithms and underlying high performance computing hardware, a keystone to accelerate the computational performance of future NWP and climate services, is also discussed in the context of the demanding operational requirements of the weather and climate industry.

show abstract

Section: Discussion and Concluding Remarksmentioning

confidence: 99%

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

Mengaldo

Wyszogrodzki

Diamantakis

et al. 2018

Arch Computat Methods Eng

View full text Add to dashboard Cite

show abstract

“…In [20,21], the solution of ordinary differential equations (ODE) is approximated by the combination of splines, where the combination parameters are determined by training a neural network with piecewise linear activation functions. A constrained integration method called GINT is proposed to solving initial boundary value PDEs in [1], where neural networks are combined with the classical Galerkin method. In [22], convolutional neural networks (CNN) is used to solve the large linear system derived from the discretization of incompressible Euler equations.…”

Section: Related Workmentioning

confidence: 99%

Weak adversarial networks for high-dimensional partial differential equations

Zang

Bao

et al. 2020

Journal of Computational Physics

283

171

View full text Add to dashboard Cite

Solving general high-dimensional partial differential equations (PDE) is a long-standing challenge in numerical mathematics. In this paper, we propose a novel approach to solve high-dimensional linear and nonlinear PDEs defined on arbitrary domains by leveraging their weak formulations. We convert the problem of finding the weak solution of PDEs into an operator norm minimization problem induced from the weak formulation. The weak solution and the test function in the weak formulation are then parameterized as the primal and adversarial networks respectively, which are alternately updated to approximate the optimal network parameter setting. Our approach, termed as the weak adversarial network (WAN), is fast, stable, and completely mesh-free, which is particularly suitable for high-dimensional PDEs defined on irregular domains where the classical numerical methods based on finite differences and finite elements suffer the issues of slow computation, instability and the curse of dimensionality. We apply our method to a variety of test problems with high-dimensional PDEs to demonstrate its promising performance.

show abstract

“…For this reason, ANNs are popular in applications such as image identification [19] and speech recognition [20], as a substitute for complex rule-based algorithms which are often difficult to program. ANNs have also been used to solve certain classes of ordinary and partial differential equations [21,22,23].…”

Section: Introductionmentioning

confidence: 99%

An artificial neural network as a troubled-cell indicator

Ray

Hesthaven

2018

Journal of Computational Physics

129

View full text Add to dashboard Cite

High-resolution schemes for conservation laws need to suitably limit the numerical solution near discontinuities, in order to avoid Gibbs oscillations. The solution quality and the computational cost of such schemes strongly depend on their ability to correctly identify troubled-cells, namely, cells where the solution loses regularity. Motivated by the objective to construct a universal troubled-cell indicator that can be used for general conservation laws, we propose a new approach to detect discontinuities using artificial neural networks (ANNs). In particular, we construct a multilayer perceptron (MLP), which is trained offline using a supervised learning strategy, and thereafter used as a black-box to identify troubled-cells. The proposed MLP indicator can accurately identify smooth extrema and is independent of problem-dependent parameters, which gives it an advantage over traditional limiter-based indicators. Several numerical results are presented to demonstrate the robustness of the MLP indicator in the framework of Runge-Kutta discontinuous Galerkin schemes, and its performance is compared with the minmod limiter and the minmod-based TVB limiter.

show abstract

A constrained integration (CINT) approach to solving partial differential equations using artificial neural networks

Cited by 123 publications

References 27 publications

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

Weak adversarial networks for high-dimensional partial differential equations

An artificial neural network as a troubled-cell indicator

Contact Info

Product

Resources

About