A-PINN: Auxiliary Physics Informed Neural Networks for Forward and Inverse Problems of Nonlinear Integro-Differential Equations

Yuan, Lujiang; Ni, Yi-Qing; Deng, Xiang-Yun; Hao, Shuo

doi:10.2139/ssrn.4000235

Cited by 6 publications

(8 citation statements)

References 27 publications

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“…PINNs can be seen as a universal function approximator based on compositional functions, named deep neural networks (DNNs). During the training of neural networks, the DNNs are optimized to satisfy a given PDE problem (Yuan et al 2022).…”

Section: Pinns As Methods To Solve Forward and Inverse Pde System Pro...mentioning

confidence: 99%

“…The necessary numerical differentiation is realized with automatic differentiation (Lu et al 2021) which does not require meshes and, hence, obliterates common issues related to mesh-based differentiation in, for example, finite element methods (FEM). For in-depth descriptions of the PINN method we refer the reader to Raissi et al 2017, Yuan et al (2022, Bischof and Kraus (2021), Lu et al (2021) and Pakravan and A. Mistani P, Aragon-Calvo MA, (2021).…”

Section: Introductionmentioning

confidence: 99%

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Transport of Organic Volatiles through Paper: Physics-Informed Neural Networks for Solving Inverse and Forward Problems

et al. 2022

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Transport of volatile organic compounds (VOCs) through porous media with active surfaces takes place in many important applications, such as in cellulose-based materials for packaging. Generally, it is a complex process that combines diffusion with sorption at any time. To date, the data needed to use and validate the mathematical models proposed in literature to describe the mentioned processes are scarce and have not been systematically compiled. As an extension of the model of Ramarao et al. (Dry Technol 21(10):2007–2056, 2003) for the water vapor transport through paper, we propose to describe the transport of VOCs by a nonlinear Fisher–Kolmogorov–Petrovsky–Piskunov equation coupled to a partial differential equation (PDE) for the sorption process. The proposed PDE system contains specific material parameters such as diffusion coefficients and adsorption rates as multiplication factors. Although these parameters are essential for solving the PDEs at a given time scale, not all of the required parameters can be directly deduced from experiments, particularly diffusion coefficients and sorption constants. Therefore, we propose to use experimental concentration data, obtained for the migration of dimethyl sulfoxide (DMSO) through a stack of paper sheets, to infer the sorption constant. These concentrations are considered as the outcome of a model prediction and are inserted into an inverse boundary problem. We employ Physics-Informed Neural Networks (PINNs) to find the underlying sorption constant of DMSO on paper from this inverse problem. We illustrate how to practically combine PINN-based calculations with experimental data to obtain trustworthy transport-related material parameters. Finally we verify the obtained parameter by solving the forward migration problem via PINNs and finite element methods on the relevant time scale and show the satisfactory correspondence between the simulation and experimental results.

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Section: Pinns As Methods To Solve Forward and Inverse Pde System Pro...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Transport of Organic Volatiles through Paper: Physics-Informed Neural Networks for Solving Inverse and Forward Problems

et al. 2022

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“…Physics-informed neural network. PINN (Raissi et al, 2019) is a representative approach that employs a neural network to solve PDEs and operates with few or even without data (Yuan et al, 2022). The main characteristic of PINN is to learn to minimize the PDE residual loss by enforcing physical constraints.…”

Section: Related Workmentioning

confidence: 99%

PIXEL: Physics-Informed Cell Representations for Fast and Accurate PDE Solvers

Kang¹,

Byeonghyeon²,

Hong³

et al. 2022

Preprint

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With the increases in computational power and advances in machine learning, data-driven learning-based methods have gained significant attention in solving PDEs. Physics-informed neural networks (PINNs) have recently emerged and succeeded in various forward and inverse PDEs problems thanks to their excellent properties, such as flexibility, mesh-free solutions, and unsupervised training. However, their slower convergence speed and relatively inaccurate solutions often limit their broader applicability in many science and engineering domains. This paper proposes a new kind of data-driven PDEs solver, physics-informed cell representations (PIXEL), elegantly combining classical numerical methods and learning-based approaches. We adopt a grid structure from the numerical methods to improve accuracy and convergence speed and overcome the spectral bias presented in PINNs. Moreover, the proposed method enjoys the same benefits in PINNs, e.g., using the same optimization frameworks to solve both forward and inverse PDE problems and readily enforcing PDE constraints with modern automatic differentiation techniques. We provide experimental results on various challenging PDEs that the original PINNs have struggled with and show that PIXEL achieves fast convergence speed and high accuracy.

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“…Several machine-learning based schemes for solving non-local PDEs can also be found in the literature. In particular, the physics-informed neural network and deep Galerkin approaches [84,87], based on representing an approximation of the whole solution of the PDE as a neural network and using automatic differentiation to do a least-squares minimization of the residual of the PDE, have been extended to fractional PDEs and other non-local PDEs [81,72,47,2,94]. While some of these approaches use classical methods susceptible to the curse of dimensionality for the non-local part [81,72], mesh-free methods suitable for high-dimensional problems have also been investigated [47,2,94].…”

Section: Introductionmentioning

confidence: 99%

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

Boussange¹,

Becker²,

Jentzen³

et al. 2022

Preprint

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Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately account for certain non-local phenomena such as, e.g., interactions at a distance. In order to properly capture these phenomena non-local nonlinear PDE models are frequently employed in the literature. In this article we propose two numerical methods based on machine learning and on Picard iterations, respectively, to approximately solve non-local nonlinear PDEs. The proposed machine learning-based method is an extended variant of a deep learning-based splitting-up type approximation method previously introduced in the literature and utilizes neural networks to provide approximate solutions on a subset of the spatial domain of the solution. The Picard iterations-based method is an extended variant of the so-called full history recursive multilevel Picard approximation scheme previously introduced in the literature and provides an approximate solution for a single point of the domain. Both methods are mesh-free and allow non-local nonlinear PDEs with Neumann boundary conditions to be solved in high dimensions.In the two methods, the numerical difficulties arising due to the dimensionality of the PDEs are avoided by (i) using the correspondence between the expected trajectory of reflected stochastic processes and the solution of PDEs (given by the Feynman-Kac formula) and by (ii) using a plain vanilla Monte Carlo integration to handle the nonlocal term. We evaluate the performance of the two methods on five different PDEs arising in physics and biology. In all cases, the methods yield good results in up to 10 dimensions with short run times. Our work extends recently developed methods to overcome the curse of dimensionality in solving PDEs.

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A-PINN: Auxiliary Physics Informed Neural Networks for Forward and Inverse Problems of Nonlinear Integro-Differential Equations

Cited by 6 publications

References 27 publications

Transport of Organic Volatiles through Paper: Physics-Informed Neural Networks for Solving Inverse and Forward Problems

Transport of Organic Volatiles through Paper: Physics-Informed Neural Networks for Solving Inverse and Forward Problems

PIXEL: Physics-Informed Cell Representations for Fast and Accurate PDE Solvers

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

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