Wan Chen

Wan Chen, Brian Wetton

This paper describes a technique to determine the linear well-posedness of a general class of vector elliptic problems that include a steady interface, to be determined as part of the problem, that separates two subdomains. The interface satisfies mixed Dirichlet and Neumann conditions. We consider ``2+2'' models, meaning two independent variables respectively on each subdomain. The governing equations are taken to be vector Laplacian, to be able to make analytic progress. The interface conditions can be classified into four large categories, and we concentrate on the one with most physical interest. The well-posedness criteria in this case are particularly clear. In many physical cases, the movement of the interface in time-dependent situations can be reduced to a normal motion proportional to the residual in one of the steady state interface conditions (the elliptic interior problems and the other interface conditions are satisfied at each time). If only the steady state is of interest, one can consider using other residuals for the normal velocity. Our analysis can be extended to give insight into choosing residual velocities that have superior numerical properties. Hence, in the second part, we discuss an iterative method to solve free boundary problems. The advantages of the correctly chosen, non-physical residual velocities are demonstrated in a numerical example, based on a simplified model of two-phase flow with phase change in porous media.

Chen Wan, Fangjun Huang

Deep neural networks (DNNs) are vulnerable to adversarial examples obtained by adding small perturbations to original examples. The added perturbations in existing attacks are mainly determined by the gradient of the loss function with respect to the inputs. In this paper, the close relationship between gradient-based attacks and the numerical methods for solving ordinary differential equation (ODE) is studied for the first time. Inspired by the numerical solution of ODE, a new prediction-correction (PC) based adversarial attack is proposed. In our proposed PC-based attack, some existing attack can be selected to produce a predicted example first, and then the predicted example and the current example are combined together to determine the added perturbations. The proposed method possesses good extensibility and can be applied to all available gradient-based attacks easily. Extensive experiments demonstrate that compared with the state-of-the-art gradient-based adversarial attacks, our proposed PC-based attacks have higher attack success rates, and exhibit better transferability.