Related papers: Neural active manifolds: nonlinear dimensionality …

High-dimensional data analysis has been an active area, and the main focuses have been variable selection and dimension reduction. In practice, it occurs often that the variables are located on an unknown, lower-dimensional nonlinear…

Uncertainty quantification in automated image analysis is highly desired in many applications. Typically, machine learning models in classification or segmentation are only developed to provide binary answers; however, quantifying the…

Active learning is a unique abstraction of machine learning techniques where the model/algorithm could guide users for annotation of a set of data points that would be beneficial to the model, unlike passive machine learning. The primary…

Large scale dynamical systems (e.g. many nonlinear coupled differential equations) can often be summarized in terms of only a few state variables (a few equations), a trait that reduces complexity and facilitates exploration of behavioral…

We present an approach to analyze $C^1(\mathbb{R}^m)$ functions that addresses limitations present in the Active Subspaces (AS) method of Constantine et al.(2015; 2014). Under appropriate hypotheses, our Active Manifolds (AM) method…

Operator learning frameworks, because of their ability to learn nonlinear maps between two infinite dimensional functional spaces and utilization of neural networks in doing so, have recently emerged as one of the more pertinent areas in…

Uniform Manifold Approximation and Projection (UMAP) is a widely used manifold learning technique for dimensionality reduction. This paper studies UMAP, supervised UMAP, and several competing dimensionality reduction methods, including…

Neural ordinary differential equations (NODE) have garnered significant attention for their design of continuous-depth neural networks and the ability to learn data/feature dynamics. However, for high-dimensional systems, estimating…

We introduce a generative smoothness regularization on manifolds (SToRM) model for the recovery of dynamic image data from highly undersampled measurements. The model assumes that the images in the dataset are non-linear mappings of…

Neural Linear Models (NLM) are deep Bayesian models that produce predictive uncertainty by learning features from the data and then performing Bayesian linear regression over these features. Despite their popularity, few works have focused…

Uncertainty quantification based on stochastic spectral methods suffers from the curse of dimensionality. This issue was mitigated recently by low-rank tensor methods. However, there exist two fundamental challenges in low-rank tensor-based…

Recent advances in reconstruction methods for inverse problems leverage powerful data-driven models, e.g., deep neural networks. These techniques have demonstrated state-of-the-art performances for several imaging tasks, but they often do…

Deploying deep learning models, comprising of non-linear combination of millions, even billions, of parameters is challenging given the memory, power and compute constraints of the real world. This situation has led to research into model…

Deep neural networks are often ignorant about what they do not know and overconfident when they make uninformed predictions. Some recent approaches quantify classification uncertainty directly by training the model to output high…

System identification has greatly benefited from deep learning techniques, particularly for modeling complex, nonlinear dynamical systems with partially unknown physics where traditional approaches may not be feasible. However, deep…

In many areas of engineering, nonlinear numerical analysis is playing an increasingly important role in supporting the design and monitoring of structures. Whilst increasing computer resources have made such formerly prohibitive analyses…

In chemometrics, data from infrared or near-infrared (NIR) spectroscopy are often used to identify a compound or to analyze the composition of amaterial. This involves the calibration of models that predict the concentration ofmaterial…

We present a fast method for nonlinear data-driven model reduction of dynamical systems onto their slowest nonresonant spectral submanifolds (SSMs). We use observed data to locate a low-dimensional, attracting slow SSM and compute a…

Statistical inference from high-dimensional data with low-dimensional structures has recently attracted lots of attention. In machine learning, deep generative modeling approaches implicitly estimate distributions of complex objects by…

Inverse design in science and engineering involves determining optimal design parameters that achieve desired performance outcomes, a process often hindered by the complexity and high dimensionality of design spaces, leading to significant…