A Mesh-free Method Using Piecewise Deep Neural Network for Elliptic Interface Problems

Abstract

In this paper, we propose a novel mesh-free numerical method for solving the elliptic interface problems based on deep learning. We approximate the solution by the neural networks and, since the solution may change dramatically across the interface, we employ different neural networks in different sub-domains. By reformulating the interface problem as a least-squares problem, we discretize the objective function using mean squared error via sampling and solve the proposed deep least-squares method by standard training algorithms such as stochastic gradient descent. The discretized objective function utilizes only the point-wise information on the sampling points and thus no underlying mesh is required. Doing this circumvents the challenging meshing procedure as well as the numerical integration on the complex interface. To improve the computational efficiency for more challenging problems, we further design an adaptive sampling strategy based on the residual of the least-squares function and propose an adaptive algorithm. Finally, we present several numerical experiments in both 2D and 3D to show the flexibility, effectiveness, and accuracy of the proposed deep least-square method for solving interface problems.

A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems

“…Example 2. As the second example, we compare the results of DCSNN with the piecewise deep neural network proposed in [14], in which two (or multiple) individual neural nets are trained to approximate the function in each subdomain. Here, the regular domain Ω = [−1, 1] 2 is separated by the interface that is given by a polar curve r(θ) = 1/2 + sin(5θ)/7.…”

“…Moreover, as shown in some examples in next section, the present DCSNN not only achieves better accuracy than the traditional finite difference method such as immersed interface method [15,18,19] in solving Eq. ( 8) but also outperforms other piecewise DNN [14] in terms of accuracy and network complexity.…”

“…To the best of our knowledge, so far the most successful methodology for solving elliptic problems with discontinuous solution and derivatives across the interface is to use the piecewise deep neural network proposed by He et al [14]. They approximate the solution by two neural networks corresponding to two disjoint subdomains so that in each subdomain, the solution remains smooth.…”

VPVnet: a velocity-pressure-vorticity neural network method for the Stokes' equations under reduced regularity

“…7(a)), respectively, on the boundary and inside of the computation domain. As for the 3D test case (5.1b), the domain boundary is described by the zero iso-value of the function φ [24]:…”

Towards fast weak adversarial training to solve high dimensional parabolic partial differential equations using XNODE-WAN

“…The application of neural networks on high dimensional data, including image classification [3] and natural language processing [4], has empirically proven to be successful. Recently, applying deep learning models to tackle high dimensional PDE problems, which achieved superior empirical results [1,5,6,7,8], has been a growing area of research. Relevant numerical analysis that takes into account of approximation rates, generality and optimization theory can be also found in many works, see [9,10,2,11].…”