Deep neural network approach to forward-inverse problems

Jo, Hyeontae; Son, Hwijae; Hwang, Hyung Ju; Kim, Eun Heui

doi:10.3934/nhm.2020011

Cited by 23 publications

(18 citation statements)

References 20 publications

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“…The data-driven method to solve the high-dimensional PDEs with a DNN is proposed in [74]. The second problem, called the forward-inverse problem, is also considered in [57] with a theoretical analysis of the convergence of the DNN solutions to the classical solutions. In [2], they present a method for approximating the solution of PDEs using an adaptive collocation strategy.…”

Section: Figure 2 Illustration Of Ap Schemesmentioning

confidence: 99%

The model reduction of the Vlasov–Poisson–Fokker–Planck system to the Poisson–Nernst–Planck system via the Deep Neural Network Approach

2021

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The model reduction of a mesoscopic kinetic dynamics to a macroscopic continuum dynamics has been one of the fundamental questions in mathematical physics since Hilbert's time. In this paper, we consider a diagram of the diffusion limit from the Vlasov-Poisson-Fokker-Planck (VPFP) system on a bounded interval with the specular reflection boundary condition to the Poisson-Nernst–Planck (PNP) system with the no-flux boundary condition. We provide a Deep Learning algorithm to simulate the VPFP system and the PNP system by computing the time-asymptotic behaviors of the solution and the physical quantities. We analyze the convergence of the neural network solution of the VPFP system to that of the PNP system via the Asymptotic-Preserving (AP) scheme. Also, we provide several theoretical evidence that the Deep Neural Network (DNN) solutions to the VPFP and the PNP systems converge to the a priori classical solutions of each system if the total loss function vanishes.

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Section: Figure 2 Illustration Of Ap Schemesmentioning

confidence: 99%

The model reduction of the Vlasov–Poisson–Fokker–Planck system to the Poisson–Nernst–Planck system via the Deep Neural Network Approach

2021

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“…Furthermore, the universal approximation property of neural networks suggests the possibility of approximating solutions of the partial differential equations. By penalizing a neural network to satisfy given PDEs, one can guarantee the convergence of the neural network to an actual solution using the energy estimate method (see, [9,10,30]).…”

Section: Motivationmentioning

confidence: 99%

Traveling Wave Solutions of Partial Differential Equations Via Neural Networks

2021

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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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“…The concrete model structures are presented in Figure 6. We applied similar training methods as introduced in [8] and [7].…”

Section: Deep Learning: Dnnsmentioning

confidence: 99%

“…Moreover, [5, 6] considered the parameters as functions of time and proposed methods to approximate the time-varying parameters. Additionally, [7, 8] have shown that neural networks (NNs) with proper loss functions represent a powerful tool for solving forward-inverse problems. Most previous studies considered the parameters constants because of the complexity of the model.…”

Section: Introductionmentioning

confidence: 99%

Analysis of COVID-19 spread in South Korea using the SIR model with time-dependent parameters and deep learning

Jo¹,

Son²,

Hwang³

et al. 2020

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Mathematical modeling is a process aimed at finding a mathematical description of a system and translating it into a relational expression. When a system is continuously changing over time (e.g., infectious diseases) differential equations, which may include parameters, are used for modeling the system. The process of finding those parameters that best fit the given data from the system is called an inverse problem. This study aims at analyzing the novel coronavirus infection (COVID-19) spread in South Korea using the susceptible-infectedrecovered (SIR) model. We collect the data from Korea Centers for Disease Control & Prevention (KCDC). We assume that each parameter in the SIR model is a function of time so that we can compute important parameters, such as the basic reproduction number (R0), more delicately. Using neural networks, we propose a method to find the best time-varying parameters and the solution for the model simultaneously. Moreover, using time-dependent parameters, we find that traditional numerical algorithms, such as the Runge-Kutta methods, can successfully approximate the SIR model while fitting the COVID-19 data, thus modeling the propagation patterns of COVID-19 more precisely.

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Deep neural network approach to forward-inverse problems

Cited by 23 publications

References 20 publications

The model reduction of the Vlasov–Poisson–Fokker–Planck system to the Poisson–Nernst–Planck system via the Deep Neural Network Approach

The model reduction of the Vlasov–Poisson–Fokker–Planck system to the Poisson–Nernst–Planck system via the Deep Neural Network Approach

Traveling Wave Solutions of Partial Differential Equations Via Neural Networks

Analysis of COVID-19 spread in South Korea using the SIR model with time-dependent parameters and deep learning

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