Biraj Dahal

Biraj Dahal, Alex Havrilla, Minshuo Chen et al.

Generative networks have experienced great empirical successes in distribution learning. Many existing experiments have demonstrated that generative networks can generate high-dimensional complex data from a low-dimensional easy-to-sample distribution. However, this phenomenon can not be justified by existing theories. The widely held manifold hypothesis speculates that real-world data sets, such as natural images and signals, exhibit low-dimensional geometric structures. In this paper, we take such low-dimensional data structures into consideration by assuming that data distributions are supported on a low-dimensional manifold. We prove statistical guarantees of generative networks under the Wasserstein-1 loss. We show that the Wasserstein-1 loss converges to zero at a fast rate depending on the intrinsic dimension instead of the ambient data dimension. Our theory leverages the low-dimensional geometric structures in data sets and justifies the practical power of generative networks. We require no smoothness assumptions on the data distribution which is desirable in practice.

Biraj Dahal, Pablo Seleson

Peridynamics is a fast growing field of continuum mechanics, especially developed for the modeling and simulation of fracture problems, initiated a quarter century ago. In this study, we analyze the evolution of the peridynamics community since its inception in terms of publication co-authorship. For this purpose, we construct a peridynamics co-authorship network for each year from 2000 to 2024 and perform network analysis based on selected metrics. Nodes represent scientists, and links connect co-authoring scientists with link weights representing the number of co-authorships (based on the total number of co-authors per publication). Network-level metrics are used to quantify the evolution of the field, and node-level metrics are used to identify trends in the most collaborative scientists in peridynamics. We noticed a deviation in network trends that occurred in the years since 2019, and we subsequently performed a country-based analysis with insights about the impact of the COVID-19 pandemic on the evolution of the peridynamics co-authorship network.

Hao Liu, Biraj Dahal, Rongjie Lai et al.

Many physical processes in science and engineering are naturally represented by operators between infinite-dimensional function spaces. The problem of operator learning, in this context, seeks to extract these physical processes from empirical data, which is challenging due to the infinite or high dimensionality of data. An integral component in addressing this challenge is model reduction, which reduces both the data dimensionality and problem size. In this paper, we utilize low-dimensional nonlinear structures in model reduction by investigating Auto-Encoder-based Neural Network (AENet). AENet first learns the latent variables of the input data and then learns the transformation from these latent variables to corresponding output data. Our numerical experiments validate the ability of AENet to accurately learn the solution operator of nonlinear partial differential equations. Furthermore, we establish a mathematical and statistical estimation theory that analyzes the generalization error of AENet. Our theoretical framework shows that the sample complexity of training AENet is intricately tied to the intrinsic dimension of the modeled process, while also demonstrating the remarkable resilience of AENet to noise.