In this paper, we propose a discontinuity-capturing shallow neural network with categorical embedding to represent piecewise smooth functions. The network comprises three hidden layers, a discontinuity-capturing layer, a categorical embedding layer, and a fully-connected layer. Under such a design, we show that a piecewise smooth function, even with a large number of pieces, can be approximated by a single neural network with high prediction accuracy. We then leverage the proposed network model to solve anisotropic elliptic interface problems. The network is trained by minimizing the mean squared error loss of the system. Our results show that, despite its simple and shallow structure, the proposed neural network model exhibits comparable efficiency and accuracy to traditional grid-based numerical methods.

Detecting discontinuity interfaces of discontinuous functions is a challenging task with significant implications across various scientific and engineering applications. Identifying these interfaces is particularly critical for functions with a high-dimensional domain, as their discontinuities can significantly influence the behavior of numerical methods and simulations; for example, within the realm of uncertainty quantification, where the smoothness of the target function plays a fundamental role in the use of stochastic collocation methods. Specifically, the knowledge of discontinuity interfaces enables the partitioning of the function domain into regions of smoothness, a crucial factor in improving the performance of numerical methods (e.g., see [17]). Other examples of discontinuity detection applications include signal processing, nonlinear partial differential equation (PDE) simulations, investigations of phase transitions in physical systems [14], and change-point analyses in geology or biology, to name a few [30]. The central objective of most discontinuity detection methods is to identify the position of discontinuities in the function domain using function evaluations on sets of points. Over the last few decades, progresses has been made in discontinuity detection, leading to the development of various algorithms. Notable works, such as [3, 2, 16, 35], have introduced significant contributions in this field. In particular, [3] introduced a polynomial annihilation edge detection method designed for piece-wise smooth functions with low-dimensional domains (n 2). This method identifies discontinuous interfaces by reconstructing jump functions based on a set of function evaluations.

Surrogate modeling and active subspaces have emerged as powerful paradigms in computational science and engineering. Porting such techniques to computational models in the social sciences brings into sharp relief their limitations in dealing with discontinuous simulators, such as Agent-Based Models, which have discrete outputs. Nevertheless, prior applied work has shown that surrogate estimates of active subspaces for such estimators can yield interesting results. But given that active subspaces are defined by way of gradients, it is not clear what quantity is being estimated when this methodology is applied to a discontinuous simulator. We begin this article by showing some pathologies that can arise when conducting such an analysis. This motivates an extension of active subspaces to discontinuous functions, clarifying what is actually being estimated in such analyses. We also conduct numerical experiments on synthetic test functions to compare Gaussian process estimates of active subspaces on continuous and discontinuous functions. Finally, we deploy our methodology on Flee, an agent-based model of refugee movement, yielding novel insights into which parameters of the simulation are most important across 8 displacement crises in Africa and the Middle East.

Adversarial examples have been found for various deep as well as shallow learning models, and have at various times been suggested to be either fixable model-specific bugs, or else inherent dataset feature, or both. We present theoretical and empirical results to show that adversarial examples are approximate discontinuities resulting from models that specify approximately bijective maps $f: \Bbb R^n \to \Bbb R^m; n \neq m$ over their inputs, and this discontinuity follows from the topological invariance of dimension.

Computation of confidence sets is central to data science and machine learning, serving as the workhorse of A/B testing and underpinning the operation and analysis of reinforcement learning algorithms. This paper studies the geometry of the minimum-volume confidence sets for the multinomial parameter. When used in place of more standard confidence sets and intervals based on bounds and asymptotic approximation, learning algorithms can exhibit improved sample complexity. Prior work showed the minimum-volume confidence sets are the level-sets of a discontinuous function defined by an exact p-value. While the confidence sets are optimal in that they have minimum average volume, computation of membership of a single point in the set is challenging for problems of modest size. Since the confidence sets are level-sets of discontinuous functions, little is apparent about their geometry. This paper studies the geometry of the minimum volume confidence sets by enumerating and covering the continuous regions of the exact p-value function. This addresses a fundamental question in A/B testing: given two multinomial outcomes, how can one determine if their corresponding minimum volume confidence sets are disjoint? We answer this question in a restricted setting.

In 1987, Hecht-Nielsen showed that any continuous multivariate function could be implemented by a certain type three-layer neural network. This result was very much discussed in neural network literature. In this paper we prove that not only continuous functions but also all discontinuous functions can be implemented by such neural networks.

This paper presents a method for solving the supervised learning problem in which the output is highly nonlinear and discontinuous. It is proposed to solve this problem in three stages: (i) cluster the pairs of input-output data points, resulting in a label for each point; (ii) classify the data, where the corresponding label is the output; and finally (iii) perform one separate regression for each class, where the training data corresponds to the subset of the original input-output pairs which have that label according to the classifier. It has not yet been proposed to combine these 3 fundamental building blocks of machine learning in this simple and powerful fashion. This can be viewed as a form of deep learning, where any of the intermediate layers can itself be deep. The utility and robustness of the methodology is illustrated on some toy problems, including one example problem arising from simulation of plasma fusion in a tokamak.

We consider the problem of minimizing the regret in stochastic multi-armed bandit, when the measure of goodness of an arm is not the mean return, but some general function of the mean and the variance. We characterize the conditions under which learning is possible and present examples for which no natural algorithm can achieve sublinear regret.