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Symmetry-aware architectures are central to geometric deep learning. We present a systematic approach for constructing continuous rotationally invariant and equivariant functions using symmetric tensor networks. The proposed framework…

Accurately following a geometric desired path in a two-dimensional space is a fundamental task for many engineering systems, in particular mobile robots. When the desired path is occluded by obstacles, it is necessary and crucial to…

Dynamic Connectivity is a fundamental algorithmic graph problem, motivated by a wide range of applications to social and communication networks and used as a building block in various other algorithms, such as the bi-connectivity and the…

A computational framework that leverages data from self-consistent field theory simulations with deep learning to accelerate the exploration of parameter space for block copolymers is presented. This is a substantial two-dimensional…

The 2x2 space-filling curve is a type of generalized space-filling curve characterized by a basic unit is in a "U-shape" that traverses a 2x2 grid. In this work, we propose a universal framework for constructing general 2x2 curves where…

We develop a rigorous theoretical framework for principal manifold estimation that recovers a latent low-dimensional manifold from a point cloud observed in a high-dimensional ambient space. Our framework accommodates manifolds with…

X-ray Computed Tomography (XCT) techniques have evolved to a point that high-resolution data can be acquired so fast that classic segmentation methods are prohibitively cumbersome, demanding automated data pipelines capable of dealing with…

We introduce the Gaussian Ensemble Topology (GET) method, a new explicit and manufacture-ready framework for topology optimization in which design geometries are represented as superpositions of anisotropic Gaussian functions. By combining…

This paper develops a new mathematical framework that enables nonparametric joint semantic and geometric representation of continuous functions using data. The joint embedding is modeled by representing the processes in a reproducing kernel…

We propose a manifold matching approach to generative models which includes a distribution generator (or data generator) and a metric generator. In our framework, we view the real data set as some manifold embedded in a high-dimensional…

Invariant manifolds are one of the key features that organize the dynamics of a differential equation. We introduce a novel approach to visualizing and studying invariant manifolds by using 3D printing technology, combining advanced…

We prove the existence of nonradial classical solutions to the 2D incompressible Euler equations with compact support. More precisely, for any positive integer $k$, we construct compactly supported stationary Euler flows of class…

Given any smooth solenoidal vector field $v_0$ on $\mathbf T^3$, we show the existence of infinitely many H\"older-continuous steady Euler flows $v$ with the same topology as $v_0$, in certain weak sense. In particular, we show that $v$…

Fully convolutional neural networks like U-Net have been the state-of-the-art methods in medical image segmentation. Practically, a network is highly specialized and trained separately for each segmentation task. Instead of a collection of…

The rise of machine learning has fueled the discovery of new materials and, especially, metamaterials--truss lattices being their most prominent class. While their tailorable properties have been explored extensively, the design of…

Implicit generative models have been widely employed to model 3D data and have recently proven to be successful in encoding and generating high-quality 3D shapes. This work builds upon these models and alleviates current limitations by…

High-quality quadrilateral mesh generation is a fundamental challenge in computer graphics. Traditional optimization-based methods are often constrained by the topological quality of input meshes and suffer from severe efficiency…

Additive manufacturing (AM), also known as 3D printing, is one of the most promising digital manufacturing technologies, thanks to its potential to produce highly complex geometries rapidly. AM has been promoted from a prototyping…

In this paper, we are concerned with the three dimensional Euler equations driven by an additive stochastic forcing. First, we construct global H\"{o}lder continuous (stationary) solutions in $C(\mathbb{R};C^{\vartheta})$ space for some…

This paper consists of a formal analysis and one solid solution to the knot finding problem given a source polyline and a parametric curve (e.g. circular arc, ellipse or biarc). We solve the problem using both a greedy algorithm to collect…