Deep Neural Network Approach to Forward-Inverse Problems

Jo, Hyeontae; Son, Hwijae; Hwang, Hyung Ju; Kim, Eunheui

doi:10.48550/arxiv.1907.12925

Cited by 3 publications

(4 citation statements)

References 23 publications

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“…Some empirical study of PINN is also conducted in [14]. Solutions to IPs based on PINN with data given in problem domain are also considered in [34], and refinement of solutions using adaptively sampled collocation points is proposed in [4]. In [59], the weak formulation of the PDE is leveraged as the objective function, where the solution of the PDE and the test function are both parameterized as deep neural networks trying to minimize and maximize the objective function, respectively.…”

Section: Related Workmentioning

confidence: 99%

Numerical solution of inverse problems by weak adversarial networks

Bao

Zang

et al. 2020

Inverse Problems

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In this paper, a weak adversarial network approach is developed to numerically solve a class of inverse problems, including electrical impedance tomography and dynamic electrical impedance tomography problems. The weak formulation of the partial differential equation for the given inverse problem is leveraged, where the solution and the test function are parameterized as deep neural networks. Then, the weak formulation and the boundary conditions induce a minimax problem of a saddle function of the network parameters. As the parameters are alternatively updated, the network gradually approximates the solution of the inverse problem. Theoretical justifications are provided on the convergence of the proposed algorithm. The proposed method is completely mesh-free without any spatial discretization, and is particularly suitable for problems with high dimensionality and low regularity on solutions. Numerical experiments on a variety of test inverse problems demonstrate the promising accuracy and efficiency of this approach.

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Section: Related Workmentioning

confidence: 99%

Numerical solution of inverse problems by weak adversarial networks

Bao

Zang

et al. 2020

Inverse Problems

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show abstract

“…The past few years have witnessed an emerging trend of using deep learning based methods to solve forward and inverse problems [3,9,14,24,25,27,36,39,42,45,50,51,54,56,59,61,64,73]. These methods can be roughly classified into two categories.…”

Section: Related Workmentioning

confidence: 99%

“…The second category features unsupervised learning methods that directly solve the forward or inverse problem based on the problem formulation rather than additional training data, which can be more advantageous than those in the first category in practice [9,24,25,36,39,42,50,51,54,59,73]. For example, feed-forward neural networks are used to parameterize the coefficient functions and trained by minimizing the performance function in [20].…”

Section: Related Workmentioning

confidence: 99%

“…A mesh-free framework, called physics-informed neural networks (PINN), for solving both the forward and inverse problems using deep neural networks based on the strong formulation of PDEs is proposed in [59], where a constant coefficient function is considered for the inverse problem part. Solutions to IPs based on PINN with data given in problem domain are also considered in [42], and refinement of solutions using adaptively sampled collocation points is proposed in [3]. In [54], three neural networks, one for low-fidelity data and the other two for the linear and nonlinear functions for high-fidelity data, are used by following the PINN approach.…”

Section: Related Workmentioning

confidence: 99%

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Numerical Solution of Inverse Problems by Weak Adversarial Networks

Bao,

Ye,

Zang

et al. 2020

Preprint

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We consider a weak adversarial network approach to numerically solve a class of inverse problems, including electrical impedance tomography and dynamic electrical impedance tomography problems. We leverage the weak formulation of PDE in the given inverse problem, and parameterize the solution and the test function as deep neural networks. The weak formulation and the boundary conditions induce a minimax problem of a saddle function of the network parameters. As the parameters are alternatively updated, the network gradually approximates the solution of the inverse problem. We provide theoretical justifications on the convergence of the proposed algorithm. Our method is completely mesh-free without any spatial discretization, and is particularly suitable for problems with high dimensionality and low regularity on solutions. Numerical experiments on a variety of test inverse problems demonstrate the promising accuracy and efficiency of our approach.

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Traveling Wave Solutions of Partial Differential Equations via Neural Networks

Cho¹,

Hwang²,

Son³

2021

Preprint

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This paper focuses on how to approximate traveling wave solutions for various kinds of partial differential equations via artificial neural networks. A traveling wave solution is hard to obtain with traditional numerical methods when the corresponding wave speed is unknown in advance. We propose a novel method to approximate both the traveling wave solution and the unknown wave speed via a neural network and an additional free parameter. We proved that under a mild assumption, the neural network solution converges to the analytic solution and the free parameter accurately approximates the wave speed as the corresponding loss tends to zero for the Keller-Segel equation. We also demonstrate in the experiments that reducing loss through training assures an accurate approximation of the traveling wave solution and the wave speed for the Keller-Segel equation, the Allen-Cahn model with relaxation, and the Lotka-Volterra competition model.

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Deep Neural Network Approach to Forward-Inverse Problems

Cited by 3 publications

References 23 publications

Numerical solution of inverse problems by weak adversarial networks

Numerical solution of inverse problems by weak adversarial networks

Numerical Solution of Inverse Problems by Weak Adversarial Networks

Traveling Wave Solutions of Partial Differential Equations via Neural Networks

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