New efficient numerical methods for interface problems using the deep learning approach

Dr Zhang, Zhiwen   (Principal Investigator (PI))

30

2019-04-01

2021-09-30

93200

New efficient numerical methods for interface problems using the deep learning approach

deep learning, deep neural network, interface problems, partial differential equations, stochastic gradient descent, variational problems

Applied Mathematics,Others - Mathematics

Physical Sciences (P)

201811159110

Seed Fund for PI Research – Basic Research

2018

Completed

In recent years, deep learning methods have achieved unprecedented successes in various application fields, including computer vision, speech recognition, natural language processing, audio recognition, social network filtering, and bioinformatics, where they have produced results comparable to and in some cases superior to human experts [4,8]. Motivated by these exciting progress, there are increased new research interests in the literature for the application of deep learning methods for scientific computation, including approximating multivariate functions and solving differential equations using the deep neural network; see [5,6,11,13-16,18] and references therein. To the best of the PI's knowledge, this research area of solving differential equations using the deep learning approach is quite new and there are many important and open problems. In this project, the PI aims to develop new efficient numerical methods for interface problems using the deep learning approach. Here the interface problems refer to partial differential equations (PDEs) with discontinuous coefficients, which have many applications in the physical and engineering sciences. For example, to model the heterogeneous porous medium in the reservoir simulation, the permeability field is often assumed to be a multiscale function with high-contrast and discontinuous features. Another example is to study the evolution of the shape and location of fibroblast cells under stress [17]. In this model, the stress tensor has discontinuity across the cell surface due to the transformation in the strain tensor caused by a contraction in the cell. Due to its importance in applications, there has been a great deal of effort in developing accurate and efficient numerical methods for interface problems. Some methods are designed based on the finite element method. In [10], Li et.al. developed the immersed-interface finite element method to solve elliptic interface problems with non-homogeneous jump conditions. In [3], Hou et.al. developed a new multiscale finite element method which was able to accurately capture solutions of elliptic interface problems with high-contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. In [2], Chen and Zou approximated the smooth interface by a polygon and used classical finite element methods to solve both elliptic and parabolic interface equations, where the mesh must align with the interface. Alternatively, some efficient finite difference methods were proposed to solve interface problems. In [12], Peskin developed the immersed boundary method (IBM) to study the motion of one or more massless, elastic surfaces immersed in an incompressible, viscous fluid, particularly in bio-fluid dynamics problems where complex geometries and immersed elastic membranes are present. Another related work is the immersed interface method (IIM) for elliptic interface problems developed by LeVeque and Li [9]. By incorporating the jump condition across the interface to modify the finite difference approximation near the interface, a second order accuracy was maintained. Those above mentioned numerical methods are efficient in solving interface problems. However, their implementations are not easy in general. Namely, one needs to choose an adaptive mesh or specially designed finite element basis functions or finite difference schemes to compute the PDE solutions. In this project, the PI aims to develop efficient and stable computational methods to solve interface problems (arising from PDEs with discontinuous coefficients) by using the deep learning approach. First of all, the PI plans to investigate the expressibility of the deep neural network in solving PDEs with discontinuous coefficients. Here the expressibility means whether the deep neural network can be used to represent the numerical solution of the PDEs. Currently, deep learning has many successful applications in classification and regression problems. For scientific computing problems, however, there are not so many results. The PI believes that urgent actions are required to seize this research opportunity and to make substantial contributions to the fast developing deep learning research area. Secondly, the PI intends to study the performance of optimization methods in deep learning problems. Although neural networks with deep layers can represent the solutions of the PDEs, to find the correct parameters (weights and biases) in the neural networks may pose a challenge. In this project, the PI will reformulate the PDE problems into optimization problems and apply the stochastic gradient descent to solve these optimization problems. Those optimization problems are non-convex (with respect to the parameters) in general, which brings challenges to existing optimization solvers. In addition, the PI will consider how to design adaptive neural networks based on the property of the PDE solutions. Finally, the PI will study the numerical accuracy and stability of the proposed numerical methods and carry out many numerical simulations to test the accuracy and efficiency of the proposed numerical methods, including the model problems from reservoir simulation and wave propagation in layered media. To the best of the PI's knowledge, it is hard to obtain a convergence order for the numerical methods based on the deep learning approach. The PI is interested in investigating the convergence order of the numerical methods developed in this project.