Physics-informed neural networks for solving nonlinear diffusivity and Biot’s equations

Abstract

This paper presents the potential of applying physics-informed neural networks for solving nonlinear multiphysics problems, which are essential to many fields such as biomedical engineering, earthquake prediction, and underground energy harvesting. Specifically, we investigate how to extend the methodology of physics-informed neural networks to solve both the forward and inverse problems in relation to the nonlinear diffusivity and Biot's equations. We explore the accuracy of the physics-informed neural networks with different training example sizes and choices of hyperparameters. The impacts of the stochastic variations between various training realizations are also investigated. In the inverse case, we also study the effects of noisy measurements. Furthermore, we address the challenge of selecting the hyperparameters of the inverse model and illustrate how this challenge is linked to the hyperparameters selection performed for the forward one.

Data-Driven Solutions and Discoveries in Mechanics Using Physics Informed Neural Network

“…In the work done by Raissi et al [30,31,32], they named such strong form approach for differential equation as the physicsinformed neural network (PINN) for the first time. Today, the PINN becomes more and more popular in many engineering fields such as hydrogeology [7,18,35], geomechanics [19], cardiovascular system [21] and so on. One attractive feature of PINN is that it is fully differentiable with respect to all the input coordinates and free parameters, in other words, the trial solution (via the trained PINN) represents a smooth approximation that can be evaluated and differentiated continuously on the domain [9].…”

“…Solving these differential equations and obtaining parameter estimations from given observations are always intriguing topics, and significant research has been done to develop many advanced (semi-)analytical or numerical algorithms. While in the last decade, machine learning especially neural networks have yielded revolutionary results across diverse disciplines, including image and pattern recognition, natural language processing, genomics, and material constitutive modeling [15,23,26,34], among which a fair amount of research has also been done related to differential equations [1,9,12,13,17,18,19,22,24,25,27,28,29,30,31,32,33,35,36,38], owing to the dramatic increase in the computing resources. Therefore, it will be interesting to adopt neural networks as an important alternative to traditional mathematical methods to approximate the solutions to differential equations through iterative update of the network weights and biases.…”

Physics-informed neural networks for estimating stress transfer mechanics in single lap joints

“…Meanwhile, physics-informed neural networks (PINNs) are neural networks trained to solve supervised learning tasks while satisfying the applicable laws of physics described by general nonlinear PDEs (Fang and Zhan, 2020). These PINNs can replace the traditional discretization methods with a neural network that approximates the solutions of differential equations (Raissi et al, 2019;Kadeethum et al, 2020). A PINN algorithm for solving brittle fracture problems by minimizing the variational energy of the system is presented by Goswami et al (2020), where the boundary conditions are entirely satisfied via appropriate modification of the neural network output.…”

Solving Partial Differential Equations Using Deep Learning and Physical Constraints

“…This neural network model takes into account the physical laws contained in PDEs and encodes them into the neural network as regularization terms, which improves performance of the neural network model. Nowadays, physics-informed neural networks are gaining more and more attention from researchers and they are gradually being applied to various fields of research [32][33][34][35][36]. Jagtap et al [37] introduce adaptive activation functions into deep and physics-informed neural networks (PINNs) to better approximate complex functions and the solutions of partial differential equations.…”