In this article, we develop a higher order approximation for the generalized fractional derivative that includes a scale function z(t) and a weight function w(t). This is then used to solve a generalized fractional diffusion problem numerically. The stability and convergence analysis of the numerical scheme are conducted by the energy method. It is proven that the temporal convergence order is 3 and this is the best result to date. Finally, we present four examples to confirm the theoretical results.

In this paper, we shall solve a time-fractional nonlinear Schrödinger equation by using the quintic non-polynomial spline and the L1 formula. The unconditional stability, unique solvability and convergence of our numerical scheme are proved by the Fourier method. It is shown that our method is sixth order accurate in the spatial dimension and $(2-\gamma )$ ( 2 − γ ) th order accurate in the temporal dimension, where γ is the fractional order. The efficiency of the proposed numerical scheme is further illustrated by numerical experiments, meanwhile the simulation results indicate better performance over previous work in the literature.

Counterfactual explanations are minimum changes of a given input to alter the original prediction by a machine learning model, usually from an undesirable prediction to a desirable one. Previous works frame this problem as a constrained cost minimization, where the cost is defined as 𝐿 1 /𝐿 2 distance (or variants) over multiple features to measure the change. In real-life applications, features of different types are hardly comparable and it is difficult to measure the changes of heterogeneous features by a single cost function. Moreover, existing approaches do not support interactive exploration of counterfactual explanations. To address above issues, we propose the skyline counterfactual explanations that define the skyline of counterfactual explanations as all non-dominated changes. We solve this problem as multi-objective optimization over actionable features. This approach does not require any cost function over heterogeneous features. With the skyline, the user can interactively and incrementally refine their goals on the features and magnitudes to be changed, especially when lacking prior knowledge to express their needs precisely. Intensive experiment results on three real-life datasets demonstrate that the skyline method provides a friendly way for finding interesting counterfactual explanations, and achieves superior results compared to the state-of-the-art methods. CCS CONCEPTS• Computing methodologies → Philosophical/theoretical foundations of artificial intelligence; Machine learning.

Deep neural networks (DNNs) are well-known to be vulnerable to adversarial attacks, where malicious human-imperceptible perturbations are included in the input to the deep network to fool it into making a wrong classification. Recent studies have demonstrated that neural Ordinary Differential Equations (ODEs) are intrinsically more robust against adversarial attacks compared to vanilla DNNs. In this work, we propose a stable neural ODE with Lyapunov-stable equilibrium points for defending against adversarial attacks (SODEF). By ensuring that the equilibrium points of the ODE solution used as part of SODEF is Lyapunov-stable, the ODE solution for an input with a small perturbation converges to the same solution as the unperturbed input. We provide theoretical results that give insights into the stability of SODEF as well as the choice of regularizers to ensure its stability. Our analysis suggests that our proposed regularizers force the extracted feature points to be within a neighborhood of the Lyapunov-stable equilibrium points of the ODE. SODEF is compatible with many defense methods and can be applied to any neural network's final regressor layer to enhance its stability against adversarial attacks.

In this paper, a mid-knot cubic non-polynomial spline is applied to obtain the numerical solution of a system of second-order boundary value problems. The numerical method is proved to be uniquely solvable and it is of second-order accuracy. We further include three examples to illustrate the accuracy of our method and to compare with other methods in the literature.