Conditional Karhunen-Loève expansion for uncertainty quantification and active learning in partial differential equation models

Tipireddy, Ramakrishna; Barajas-Solano, David A.; Tartakovsky, Alexandre M.

doi:10.1016/j.jcp.2020.109604

Cited by 22 publications

(19 citation statements)

References 23 publications

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“…Tartakovsky et al (2020) for solving inverse diffusion equations with unknown diffusion coefficients. In PICKLE, the unknown parameter field y(x) as well as the hydraulic head u(x) are represented with the so-called conditional Karhunen-Loéve expansion (CKLEs) (Tipireddy et al, 2020b) as…”

Section: Methods Formulationmentioning

confidence: 99%

“…Also, there are several computationally efficient alternatives to MC methods, including the moment equation method (e.g., (Neuman, 1993;D. M. Tartakovsky et al, 2003;Jarman & Tartakovsky, 2013)) and polynomial-chaosbased approaches (Lin & Tartakovsky, 2010;Tipireddy et al, 2020a;Li & Tartakovsky, 2020), surrogate models (X. Yang, Li, & Tartakovsky, 2018), and generative physicsinformed machine learning methods (L. .…”

Section: Computing Covariance Functionsmentioning

confidence: 99%

“…Conditioning models on direct measurements of space-varying parameters (e.g., conductivity) is relatively straightforward and can be achieved using Kriging or Gaussian process regression (GPR) (Neuman, 1993;Tipireddy et al, 2020a). In Kriging, an estimate of the conductivity field is obtained by means of a nonlinear interpolation between its measured values with the interpolation coefficients given in terms of the covariance function of the conductivity field learned from data (Matheron, 1963;Rasmussen, 2003).…”

Section: Introductionmentioning

confidence: 99%

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Physics-Informed Machine Learning Method for Large-Scale Data Assimilation Problems

Yeung,

Barajas-Solano,

Tartakovsky

2021

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Section: Methods Formulationmentioning

confidence: 99%

Section: Computing Covariance Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Physics-Informed Machine Learning Method for Large-Scale Data Assimilation Problems

Yeung,

Barajas-Solano,

Tartakovsky

2021

Preprint

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“…[1], which provide a suite of disturbance models). Physics-based ML methods such as neural ODEs, graph neural networks, physics-informed neural networks (PINNs) [6,10], physics-informed coKriging [11], approximate Bayesian methods [10], and other methods using a functional representation of unknown physics, parameters, and quantities of interest [4,5,[7][8][9] are used to develop the AI models. Note that each functional representation of extremes introduces bias.…”

Section: Earth System Models a Key Question To Address Is "How Can Physics-informed Ai Dynamically Guide The Real-time Collection And Intmentioning

confidence: 99%

Emergent Concepts from a Community Ideation on AI4ESP

Stegen

2021

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“…If trained accurately, PINNs can work faster and more accurately than numerical simulators of complex real-world phenomena. PINNs may also be used to assimilate data and observations into numerical models, or be used in parameter identification (the inverse problem) [14,21] and uncertainty quantification [22,23,24,25].…”

Section: Introductionmentioning

confidence: 99%

Physics-informed attention-based neural network for solving non-linear partial differential equations

Rodriguez-Torrado,

Ruiz,

Cueto-Felgueroso

et al. 2021

Preprint

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Physics-Informed Neural Networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs). PINNs are based on simple architectures, and learn the behavior of complex physical systems by optimizing the network parameters to minimize the residual of the underlying PDE. Current network architectures share some of the limitations of classical numerical discretization schemes when applied to non-linear differential equations in continuum mechanics. A paradigmatic example is the solution of hyperbolic conservation laws that develop highly localized nonlinear shock waves. Learning solutions of PDEs with dominant hyperbolic character is a challenge for current PINN approaches, which rely, like most grid-based numerical schemes, on adding artificial dissipation. Here, we address the fundamental question of which network architectures are best suited to learn the complex behavior of non-linear PDEs. We focus on network architecture rather than on residual regularization. Our new methodology, called Physics-Informed Attention-based Neural Networks, (PIANNs), is a combination of recurrent neural networks and attention mechanisms. The attention mechanism adapts the behavior of the deep neural network to the non-linear features of the solution, and break the current limitations of PINNs. We find that PIANNs effectively capture the shock front in a hyperbolic model problem, and are capable of providing high-quality solutions inside and beyond the training set.

show abstract

Conditional Karhunen-Loève expansion for uncertainty quantification and active learning in partial differential equation models

Cited by 22 publications

References 23 publications

Physics-Informed Machine Learning Method for Large-Scale Data Assimilation Problems

Physics-Informed Machine Learning Method for Large-Scale Data Assimilation Problems

Emergent Concepts from a Community Ideation on AI4ESP

Physics-informed attention-based neural network for solving non-linear partial differential equations

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