Related papers: Geometric Numerical Integration of the Assignment …
Manifold learning is a popular and quickly-growing subfield of machine learning based on the assumption that one's observed data lie on a low-dimensional manifold embedded in a higher-dimensional space. This thesis presents a mathematical…
Computational fluid dynamics models based on Reynolds-averaged Navier--Stokes equations with turbulence closures still play important roles in engineering design and analysis. However, the development of turbulence models has been stagnant…
The study of high dimensional data sets often rely on their low dimensional projections that preserve the local geometry of the original space. While numerous methods have been developed to summarize this space as variations of tree-like…
Sampling a target probability distribution with an unknown normalization constant is a fundamental challenge in computational science and engineering. Recent work shows that algorithms derived by considering gradient flows in the space of…
In this work, we investigate the use of data-driven equation discovery for dynamical systems to model and forecast continuous-time dynamics of unconstrained optimization problems. To avoid expensive evaluations of the objective function and…
We systematically validate the static local mesh refinement capabilities of a recently proposed IMEX-DG scheme implemented in the framework of the deal.II library. Non-conforming meshes are employed in atmospheric flow simulations to…
Traditional computational fluid dynamics calculates the physical information of the flow field by solving partial differential equations, which takes a long time to calculate and consumes a lot of computational resources. We build a fluid…
The spread of machine learning techniques coupled with the availability of high-quality experimental and numerical data has significantly advanced numerous applications in fluid mechanics. Notable among these are the development of data…
The behavior of complex systems is determined not only by the topological organization of their interconnections but also by the dynamical processes taking place among their constituents. A faithful modeling of the dynamics is essential…
Data arrives at our senses as a continuous stream, smoothly transforming from one instant to the next. These smooth transformations can be viewed as continuous symmetries of the environment that we inhabit, defining equivalence relations…
Recent visual odometry (VO) methods incorporating geometric algorithm into deep-learning architecture have shown outstanding performance on the challenging monocular VO task. Despite encouraging results are shown, previous methods ignore…
Optical flow estimation is a fundamental problem in computer vision, yet the reliance on expensive ground-truth annotations limits the scalability of supervised approaches. Although unsupervised and semi-supervised methods alleviate this…
Constructing transferable descriptors for conformation representation of molecular and biological systems finds numerous applications in drug discovery, learning-based molecular dynamics, and protein mechanism analysis. Geometric graph…
We introduce randomized algorithms to Clifford's Geometric Algebra, generalizing randomized linear algebra to hypercomplex vector spaces. This novel approach has many implications in machine learning, including training neural networks to…
This paper presents a characteristic-based flux partitioning for the semi-implicit time integration of atmospheric flows. Nonhydrostatic models require the solution of the compressible Euler equations. The acoustic time-scale is…
We present rectified flow, a surprisingly simple approach to learning (neural) ordinary differential equation (ODE) models to transport between two empirically observed distributions \pi_0 and \pi_1, hence providing a unified solution to…
We propose a data-driven mean-curvature solver for the level-set method. This work is the natural extension to $\mathbb{R}^3$ of our two-dimensional strategy in [DOI: 10.1007/s10915-022-01952-2][1] and the hybrid inference system of [DOI:…
We formulate a hierarchical rectified flow to model data distributions. It hierarchically couples multiple ordinary differential equations (ODEs) and defines a time-differentiable stochastic process that generates a data distribution from a…
Representing graph data in a low-dimensional space for subsequent tasks is the purpose of attributed graph embedding. Most existing neural network approaches learn latent representations by minimizing reconstruction errors. Rare work…
Despite the enormous success of graph neural networks (GNNs), most existing GNNs can only be applicable to undirected graphs where relationships among connected nodes are two-way symmetric (i.e., information can be passed back and forth).…