Neural network augmented inverse problems for PDEs

Berg, Jens; Nyström, Kaj

doi:10.48550/arxiv.1712.09685

Cited by 10 publications

(25 citation statements)

References 17 publications

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“…The dependence of C rand on the basis. Lemma 1 shows that the ONB used to randomise our input plays a role, as it appears explicitly in the formula (9). The following proposition underscores that point.…”

Section: 22mentioning

confidence: 66%

On the randomised stability constant for inverse problems

Alberti¹,

Capdeboscq²,

Privat³

2020

Mathematics in Engineering

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In this paper we introduce the randomised stability constant for abstract inverse problems, as a generalisation of the randomised observability constant, which was studied in the context of observability inequalities for the linear wave equation. We study the main properties of the randomised stability constant and discuss the implications for the practical inversion, which are not straightforward.2010 Mathematics Subject Classification. 65J22, 35R30.

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Section: 22mentioning

confidence: 66%

On the randomised stability constant for inverse problems

Alberti¹,

Capdeboscq²,

Privat³

2020

Mathematics in Engineering

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“…Being universal approximators, neural networks have been widely used in nonlinear system identification: depending on the architecture and on the properties of the loss function, they can be used as sparse regression models, they can act as priors on unknown coefficients or completely determine an unknown differential operator. Many kind of architectures have been used for system identification, among which multi-layer feed forward networks (see for instance [16], [13], [3], [22], [4]) and recurrent networks and its variants which have been used in dynamic identification of nonlinear systems because of their ability to retain information in time across layers (see for instance [30], [8], [19], [21]).…”

Section: Introductionmentioning

confidence: 99%

System Identification Through Lipschitz Regularized Deep Neural Networks

Negrini,

Citti,

Capogna

2020

Preprint

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In this paper we use neural networks to learn governing equations from data. Specifically we reconstruct the right-hand side of a system of ODEs ẋ(t) = f (t, x(t)) directly from observed uniformly time-sampled data using a neural network. In contrast with other neural network based approaches to this problem, we add a Lipschitz regularization term to our loss function. In the synthetic examples we observed empirically that this regularization results in a smoother approximating function and better generalization properties when compared with non-regularized models, both on trajectory and non-trajectory data, especially in presence of noise. In contrast with sparse regression approaches, since neural networks are universal approximators, we don't need any prior knowledge on the ODE system. Since the model is applied component wise, it can handle systems of any dimension, making it usable for real-world data.

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“…Jianyu et al [12] used ANN with a radial basis function as an activation function for Poisson equations. More recently, Berg et al in [4] used a DNN to solve steady (time-independent) problems for a domain with complex geometry in one and two space dimensions, and later in [3] they studied DNN architectures to solve augmented Poisson equations (inverse problems) including three space dimensions.…”

Section: Introductionmentioning

confidence: 99%

Deep Neural Network Approach to Forward-Inverse Problems

Jo¹,

Son²,

Hwang³

et al. 2019

Preprint

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In this paper, we construct approximated solutions of Differential Equations (DEs) using the Deep Neural Network (DNN). Furthermore, we present an architecture that includes the process of finding model parameters through experimental data, the inverse problem. That is, we provide a unified framework of DNN architecture that approximates an analytic solution and its model parameters simultaneously. The architecture consists of a feed forward DNN with non-linear activation functions depending on DEs, automatic differentiation [2], reduction of order, and gradient based optimization method. We also prove theoretically that the proposed DNN solution converges to an analytic solution in a suitable function space for fundamental DEs. Finally, we perform numerical experiments to validate the robustness of our simplistic DNN architecture for 1D transport equation, 2D heat equation, 2D wave equation, and the Lotka-Volterra system.

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Neural network augmented inverse problems for PDEs

Cited by 10 publications

References 17 publications

On the randomised stability constant for inverse problems

On the randomised stability constant for inverse problems

System Identification Through Lipschitz Regularized Deep Neural Networks

Deep Neural Network Approach to Forward-Inverse Problems

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