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We present a novel hybrid strategy based on machine learning to improve curvature estimation in the level-set method. The proposed inference system couples enhanced neural networks with standard numerical schemes to compute curvature more…
We propose a deep learning strategy to estimate the mean curvature of two-dimensional implicit interfaces in the level-set method. Our approach is based on fitting feed-forward neural networks to synthetic data sets constructed from…
We present an error-neural-modeling-based strategy for approximating two-dimensional curvature in the level-set method. Our main contribution is a redesigned hybrid solver [Larios-C\'ardenas and Gibou, J. Comput. Phys. (May 2022),…
We present a machine learning framework that blends image super-resolution technologies with passive, scalar transport in the level-set method. Here, we investigate whether we can compute on-the-fly, data-driven corrections to minimize…
We propose meta-curvature (MC), a framework to learn curvature information for better generalization and fast model adaptation. MC expands on the model-agnostic meta-learner (MAML) by learning to transform the gradients in the inner…
Accurate boundary detection in high-dimensional data remains a central challenge in unsupervised learning, particularly in the presence of non-linear structures and heterogeneous densities. In this work, we introduce Mean Curvature Boundary…
Recently, machine learning has been used to substitute parts of conventional computational fluid dynamics, e.g. the cell-face reconstruction in finite-volume solvers or the curvature computation in the Volume-of-Fluid (VOF) method. The…
Here a semi-implicit formulation of the gradient augmented level set method is presented. By tracking both the level set and it's gradient accurate subgrid information is provided,leading to highly accurate descriptions of a moving…
The level-set method is a popular method for interface capturing. One of the advantages of the level-set method is that the curvature and the normal vector of the interface can be readily calculated from the level-set function. However, in…
The analysis of curves has been routinely dealt with using tools from functional data analysis. However its extension to multi-dimensional curves poses a new challenge due to its inherent geometric features that are difficult to capture…
Hyperbolic neural networks (HNNs) have demonstrated notable efficacy in representing real-world data with hierarchical structures via exploiting the geometric properties of hyperbolic spaces characterized by negative curvatures. Curvature…
Estimation of interface curvature in surface-tension dominated flows is a remaining challenge in Volume of Fluid (VOF) methods. Data-driven methods are recently emerging as a promising alternative in this domain. They outperform…
Semi-supervised learning algorithms typically construct a weighted graph of data points to represent a manifold. However, an explicit graph representation is problematic for neural networks operating in the online setting. Here, we propose…
Bilevel optimization is a powerful tool for many machine learning problems, such as hyperparameter optimization and meta-learning. Estimating hypergradients (also known as implicit gradients) is crucial for developing gradient-based methods…
We establish convergence results for a spatial semidiscretization of Mean Curvature Flow (MCF) for surfaces with fixed boundaries. Our analysis is based on Huisken's evolution equations for the mean curvature and the normal vector, enabling…
A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such…
Deep learning systems extensively use convolution operations to process input data. Though convolution is clearly defined for structured data such as 2D images or 3D volumes, this is not true for other data types such as sparse point…
In this paper, we propose a point cloud classification method based on graph neural network and manifold learning. Different from the conventional point cloud analysis methods, this paper uses manifold learning algorithms to embed point…
Point cloud analysis (such as 3D segmentation and detection) is a challenging task, because of not only the irregular geometries of many millions of unordered points, but also the great variations caused by depth, viewpoint, occlusion, etc.…
Gradient-based iterative optimization methods are the workhorse of modern machine learning. They crucially rely on careful tuning of parameters like learning rate and momentum. However, one typically sets them using heuristic approaches…