Does a Neural Network Really Encode Symbolic Concepts?

Recently, a series of studies have tried to extract interactions between input variables modeled by a DNN and define such interactions as concepts encoded by the DNN. However, strictly speaking, there still lacks a solid guarantee whether such interactions indeed represent meaningful concepts. Therefore, in this paper, we examine the trustworthiness of interaction concepts from four perspectives. Extensive empirical studies have verified that a well-trained DNN usually encodes sparse, transferable, and discriminative concepts, which is partially aligned with human intuition

Generic regularity of conservative solutions to Camassa-Holm type equations

This paper mainly proves the generic properties of the Camassa-Holm equation and the two-component Camassa-Holm equation by Thom's transversality Lemma. We reveal their differences in generic regularity and singular behavior

Technical Note: Defining and Quantifying AND-OR Interactions for Faithful and Concise Explanation of DNNs

In this technical note, we aim to explain a deep neural network (DNN) by quantifying the encoded interactions between input variables, which reflects the DNN's inference logic. Specifically, we first rethink the definition of interactions, and then formally define faithfulness and conciseness for interaction-based explanation. To this end, we propose two kinds of interactions, i.e., the AND interaction and the OR interaction. For faithfulness, we prove the uniqueness of the AND (OR) interaction in quantifying the effect of the AND (OR) relationship between input variables. Besides, based on AND-OR interactions, we design techniques to boost the conciseness of the explanation, while not hurting the faithfulness. In this way, the inference logic of a DNN can be faithfully and concisely explained by a set of symbolic concepts.Comment: arXiv admin note: text overlap with arXiv:2111.0620

Stability of Stationary Solutions to the Nonisentropic Euler-Poisson System in a Perturbed Half Space

The main concern of this paper is to mathematically investigate the formation of a plasma sheath near the surface of nonplanar walls. We study the existence and asymptotic stability of stationary solutions for the nonisentropic Euler-Poisson equations in a domain of which boundary is drawn by a graph, by employing a space weighted energy method. Moreover, the convergence rate of the solution toward the stationary solution is obtained, provided that the initial perturbation belongs to the weighted Sobolev space. Because the domain is the perturbed half space, we first show the time-global solvability of the nonisentropic Euler-Poisson equations, then construct stationary solutions by using the time-global solutions.Comment: arXiv admin note: substantial text overlap with arXiv:1910.0321