Shiping Zhou, a doctoral student in mathematics with computational and applied math emphasis, will present their doctoral dissertation titled “Nonlocal Modeling and Computation: Numerical Methods and Data-Driven Approaches.” Shiping’s advisor, Dr. Yanzhi Zhang, is a professor of mathematics and statistics.

Abstract: In recent decades, nonlocal models have garnered significant attention for describing complex systems. These models excel in capturing long-range interactions, multiscale properties, and memory effects more accurately than local models. Among these nonlocal operators, the fractional Laplacian $(-\Delta)^{\alpha/2}$ stands out as one of the most popular for describing anomalous phenomena in homogeneous media, and its variable-order generalization $(-\Delta)^{\alpha({\bf x})/2}$ for heterogeneous media. While nonlocal operators enable more accurate modeling, they also introduce new numerical challenges. Currently, numerical studies for these nonlocal models still remain limited, especially for the variable-order cases.

The objective of this dissertation is to develop accurate and efficient methods for solving nonlocal problems, with the application in seismic wave propagation in heterogeneous media. In the first part, we present a spectral method for the fractional Laplacian, which can be viewed as an exact discrete analogue of the fractional Laplacian. Furthermore, we introduce the first finite difference and pseudospectral methods for the variable-order fractional Laplacian, addressing the challenges brought by the spatial dependency in the power $\alpha({\bf x})$. In the second part, we propose a data-driven surrogate solver for time-dependent nonlocal problems. This surrogate solver utilizes the convolutional neural network (CNN) and works independently of the nonlocal PDE. Leveraging limited time-series observed data, it can learn the underlying dynamics and predict long-term future solutions accurately. Furthermore, the trained surrogate solver demonstrates robustness to initial conditions beyond the observed data.

The proposed finite difference and pseudospectral methods as well as the CNN surrogate solver offer accurate and efficient approaches for solving nonlocal problems. These methods could effectively address the numerical challenges arising from the nonlocality and spatial dependency inherent in fractional Laplacian, so as to advance the applications of nonlocal models in other fields.