Reconstruction of two time independent coefficients in an inverse problem for a phase field system

Baranibalan, N.; Sakthivel, K.M.; Balachandran, K.; Kim, J.-H.

doi:10.1016/j.na.2009.11.027

Cited by 6 publications

(5 citation statements)

References 26 publications

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“…Wu et al [28] proved the Lipschitz stability and uniqueness of the inverse problem of phase field models with an inertial term. Stability results concerning the inverse problem of determining two time-independent coefficients with the observation data were presented in [29]. For other examples, please see references [30,31].…”

Section: Introductionmentioning

confidence: 99%

Fractional physics-informed neural networks for time-fractional phase field models

2022

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In this paper, a new fractional physics-informed neural networks (fPINNs) is proposed, which combines f-PINNs with spectral collocation method to solve the timefractional phase field models. Compared to fPINNs, it has large representation capacity due to the property of spectral collocation method, which reduces the number of approximate points of discrete fractional operators, improves the training efficiency and has higher error accuracy. Unlike traditional numerical method, it directly optimizes the spectral collocation coefficient, saves the time of matrix calculation, is easy to deal with the high-dimensional model, and also has higher error accuracy. First, fPINNs based on a spectral collocation method is used to obtain the numerical solutions of the models under consideration. The spectral collocation method is used to discretize the space direction, and the fractional backward difference formula is used to approximate the time-fractional derivative. The error accuracy in different cases is discussed, and it is observed that the point-wise error is 10 −5 to 10 −7 . Next, fPINNs is employed to solve several inverse problems in time-fractional phase field models to identify the order of fractional derivative, mobility constant, and other coefficients. The results of numerical experiments are presented to prove the effectiveness of fPINNs in solving time-fractional phase field models and their inverse problems.

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Section: Introductionmentioning

confidence: 99%

Fractional physics-informed neural networks for time-fractional phase field models

2022

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“…(1.5) It is noted that in our inverse problem we need the observation of only one component of the order parameter v in ω × (0, T ) and at some suitable time t 0 over the whole Ω. This is different from those papers aimed at discussing the coefficient inverse problem for the coupling system [3][4][5][6], where there are all components' measurements in the system to derive the stability result for the coefficient inverse problem.…”

Section: Introductionmentioning

confidence: 99%

“…In comparison with [3,4], there are two different aspects with our work. The first one is the type of system.…”

Section: Introductionmentioning

confidence: 99%

“…In that paper, they still used the measurements of all two components. In comparison with [4], the main difference about assumptions is the regularity of the solutions (see A5 below). Our regularity assumption is stronger than the one in [4].…”

Section: Introductionmentioning

confidence: 99%

“…In comparison with [4], the main difference about assumptions is the regularity of the solutions (see A5 below). Our regularity assumption is stronger than the one in [4]. The reason is that in order to deal with the Carleman estimate by only one component, we need to replace the measurement data of u by the data of v with higher regularity.…”

Section: Introductionmentioning

confidence: 99%

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Uniqueness and stability of an inverse problem for a phase field model using data from one component

Chen

Wang

2013

Computers & Mathematics with Applications

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h i g h l i g h t s• A Carleman estimate for the parabolic-hyperbolic phase field system is proved. • Lipschitz stability and uniqueness for a coefficient inverse problem for this phase field system using data from one component are established.• Lipschitz stability provides theoretical support for numerical methods. a b s t r a c tWe study an inverse problem of determining a spatial varying coefficient in a parabolichyperbolic phase field model with the following observation data of only one component: the order parameter in a subdomain ω satisfying ∂ω ⊃ ∂Ω along a sufficiently large time interval and at a suitable time over the whole spatial domain. Based on a Carleman estimate for the parabolic-hyperbolic phase field system, we prove the Lipschitz stability and uniqueness for this inverse problem.

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Simultaneous Identification of Two Time Independent Coefficients in a Nonlinear Phase Field System

Gnanavel

Balan

Balachandran

2013

J Optim Theory Appl

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Reconstruction of two time independent coefficients in an inverse problem for a phase field system

Cited by 6 publications

References 26 publications

Fractional physics-informed neural networks for time-fractional phase field models

Fractional physics-informed neural networks for time-fractional phase field models

Uniqueness and stability of an inverse problem for a phase field model using data from one component

Simultaneous Identification of Two Time Independent Coefficients in a Nonlinear Phase Field System

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