Mathematical methods for uncertainty quanti cation in nonlin-ear multi-physics systems and their numerical simulations

Wang, Ruili; Jiang, Song

doi:10.1360/n012014-00115

Cited by 8 publications

(7 citation statements)

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“…UQ methods have been applied in widespread fields like fluid dynamics [19], weather forecasting [20], etc. At present, UQ methods are shown as follows [21]:…”

Section: A Review Of Resilience Methodsmentioning

confidence: 99%

Uncertainty of Resilience in Complex Networks With Nonlinear Dynamics

Moutsinas

Zou

Guo

2021

IEEE Systems Journal

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Resilience is a system's ability to maintain its function when perturbations and errors occur. Whilst we understand low-dimensional networked systems' behaviour well, our understanding of systems consisting of a large number of components is limited. Recent research in predicting the network level resilience pattern has advanced our understanding of the coupling relationship between global network topology and local nonlinear component dynamics. However, when there is uncertainty in the model parameters, our understanding of how this translates to uncertainty in resilience is unclear for a large-scale networked system. Here we develop a polynomial chaos expansion method to estimate the resilience for a wide range of uncertainty distributions. By applying this method to case studies, we not only reveal the general resilience distribution with respect to the topology and dynamics sub-models, but also identify critical aspects to inform better monitoring to reduce uncertainty.

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“…UQ methods have been applied in widespread fields like fluid dynamics [19], weather forecasting [20], etc. At present, UQ methods are shown as follows [21]:…”

Section: A Review Of Resilience Methodsmentioning

confidence: 99%

Uncertainty of Resilience in Complex Networks With Nonlinear Dynamics

Moutsinas

Zou

Guo

2021

IEEE Systems Journal

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“…It is the process of determining the degree to which computational simulation results agree with experimental data and real world. The fundamental method of validation involves identifying and quantifying the error and uncertainty in the physical and computational models [6,7] , quantifying the numerical error in the computational solution, estimating the experimental uncertainty, and then comparing the computational results with experimental data. Cylinder test is to put the detonator into a copper tube of equal thickness, which is detonated from one side and measured by a high-speed rotating camera.…”

Section: Validation Of the Lad2d Softwarementioning

confidence: 99%

Verification and Validation of LAD2D Hydrocodes for Multi-material Compressible Flows

Wang¹,

Liang²

2018

dtetr

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Lagrangian Hydrocodes is the main simulation tool in multi-material compressible flows of high temperature and high pressure. The important problems we are up against are how be should confidence in modeling and simulation be critically assessed. The verification and validation (V&V) of computational simulations are the primary methods for building and quantifying this confidence. Verification is the assessment of the accuracy of the solution to a computational model by comparison with known solutions. Validation is the assessment of the accuracy of a computational simulation by comparison with experimental data. In this paper, The verification and validation (V&V) of Lagrangian Hydrocodes of multi-material compressible flows was present. Verification of the LAD2D Software was implemented by the Sod problem. Validation of the LAD2D Software was implemented by the cylinder test.

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“…These uncertain factors may have an important impact on the system evolution, especially in long-term forecasting, and hence quantifying uncertainty is very important. Following a probability framework, the uncertainty is usually modeled as a random field [6], and therefore modeling nonlocal interactions with uncertainty requires the formulation of fractional partial differential equation with random inputs. Although there have been some achievements in the numerical solution of stochastic fractional partial differential equations (SFPDEs) [7][8][9], the design of reliable algorithms that can tackle high-dimensions and long-term integration is still an open challenge in the context of solving efficiently time-dependent stochastic fractional partial differential equations (SFPDEs).…”

Section: Introductionmentioning

confidence: 99%

“…The polynomial expansion coefficients are solved by the stochastic Galerkin method or the stochastic collocation method, and then the statistics of interest can be readily computed [12,13]. For more applications and introduction of the PC method, please refer to [6,14]. Although the PC method is very effective, with the increase of the random input dimension, the number of basis functions increases exponentially leading to the well known problem of the curse-of-dimensionality.…”

Section: Introductionmentioning

confidence: 99%

Bi-orthogonal fPINN: A physics-informed neural network method for solving time-dependent stochastic fractional PDEs

Ma¹,

Li²,

Zeng³

et al. 2023

Preprint

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Fractional partial differential equations (FPDEs) can effectively represent anomalous transport and nonlocal interactions. However, inherent uncertainties arise naturally in real applications due to random forcing or unknown material properties. Mathematical models considering nonlocal interactions with uncertainty quantification can be formulated as stochastic fractional partial differential equations (SFPDEs). There are many challenges in solving SFPDEs numerically, especially for long-time integration since such problems are high-dimensional and nonlocal. Here, we combine the bi-orthogonal (BO) method for representing stochastic processes with physics-informed neural networks (PINNs) for solving partial differential equations to formulate the bi-orthogonal PINN method (BO-fPINN) for solving time-dependent SFPDEs. Specifically, we introduce a deep neural network for the stochastic solution of the time-dependent SFPDEs, and include the BO constraints in the loss function following a weak formulation. Since automatic differentiation is not currently applicable to fractional derivatives, we employ discretization on a grid to to compute the fractional derivatives of the neural network output. The weak formulation loss function of the BO-fPINN method can overcome some drawbacks of the BO methods and thus can be used to solve SFPDEs with eigenvalue crossings. Moreover, the BO-fPINN method can be used for inverse SFPDEs with the same framework and same computational complexity as for forward problems. We demonstrate the effectiveness of the BO-fPINN method for different benchmark problems. Specifically, we first consider an SFPDE with eigenvalue crossing and obtain good results while the original BO method fails. We then solve several forward and inverse problems governed by SFPDEs, including problems with noisy initial conditions. We study the effect of the fractional order as well as the number of the BO modes on the accuracy of the BO-fPINN method. The results demonstrate the flexibility and efficiency of the proposed method, especially for inverse problems. We also present a simple example of transfer learning (for the fractional order) that can help in accelerating the training of BO-fPINN for SFPDEs. Taken together, the simulation results show that the BO-fPINN method can be employed to effectively solve time-dependent SFPDEs and may provide a reliable computational strategy for real applications exhibiting anomalous transport.

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Mathematical methods for uncertainty quanti cation in nonlin-ear multi-physics systems and their numerical simulations

Cited by 8 publications

References 0 publications

Uncertainty of Resilience in Complex Networks With Nonlinear Dynamics

Uncertainty of Resilience in Complex Networks With Nonlinear Dynamics

Verification and Validation of LAD2D Hydrocodes for Multi-material Compressible Flows

Bi-orthogonal fPINN: A physics-informed neural network method for solving time-dependent stochastic fractional PDEs

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