Related papers: Learning Neural Operators on Riemannian Manifolds

Understanding and mapping a new environment are core abilities of any autonomously navigating agent. While classical robotics usually estimates maps in a stand-alone manner with SLAM variants, which maintain a topological or metric…

For partial differential equations on domains of arbitrary shapes, existing works of neural operators attempt to learn a mapping from geometries to solutions. It often requires a large dataset of geometry-solution pairs in order to obtain a…

Computationally efficient surrogates for parametrized physical models play a crucial role in science and engineering. Operator learning provides data-driven surrogates that map between function spaces. However, instead of full-field…

We aim to solve the problem of manipulating deformable objects, particularly elastic bands, in real-world scenarios. However, deformable object manipulation (DOM) requires a policy that works on a large state space due to the unlimited…

Deep learning is emerging as a new paradigm for solving inverse imaging problems. However, the deep learning methods often lack the assurance of traditional physics-based methods due to the lack of physical information considerations in…

The indicator matrix plays an important role in machine learning, but optimizing it is an NP-hard problem. We propose a new relaxation of the indicator matrix and prove that this relaxation forms a manifold, which we call the Relaxed…

Multi-Task Learning (MTL) seeks to boost statistical power and learning efficiency by discovering structure shared across related tasks. State-of-the-art MTL representation methods, however, usually treat the latent representation matrix as…

Neural Operators offer a powerful, data-driven tool for solving parametric PDEs as they can represent maps between infinite-dimensional function spaces. In this work, we employ physics-informed Neural Operators in the context of…

One of the primary reasons behind the success of neural networks has been the emergence of an array of new, highly-successful optimizers, perhaps most importantly the Adam optimizer. It is widely used for training neural networks, yet…

Deep learning on non-Euclidean domains is important for analyzing complex geometric data that lacks common coordinate systems and familiar Euclidean properties. A central challenge in this field is to define convolution on domains, which…

Operator learning is a data-driven approximation of mappings between infinite-dimensional function spaces, such as the solution operators of partial differential equations. Kernel-based operator learning can offer accurate, theoretically…

Despite significant advances in the field of deep learning in ap-plications to various areas, an explanation of the learning pro-cess of neural network models remains an important open ques-tion. The purpose of this paper is a comprehensive…

Computations on a manifold often involve constructing an operator on the tangent space and computing its inverse, which can be time-consuming in many applications. In order to reduce the computational costs and preserve the benign…

We consider two Riemannian geometries for the manifold $\mathcal{M}(p,m\times n)$ of all $m\times n$ matrices of rank $p$. The geometries are induced on $\mathcal{M}(p,m\times n)$ by viewing it as the base manifold of the submersion…

Neural operators have emerged as a powerful, data-driven paradigm for learning solution operators of partial differential equations (PDEs). State-of-the-art architectures, such as the Fourier Neural Operator (FNO), have achieved remarkable…

Recent advancements in the discipline of quantum algorithms have displayed the importance of the geometry of quantum operators. Given this thrust, this paper develops a rigorous geometric framework to analyze how the Riemannian structure of…

Deep learning is the mainstream technique for many machine learning tasks, including image recognition, machine translation, speech recognition, and so on. It has outperformed conventional methods in various fields and achieved great…

The scientific computation methods development in conjunction with artificial intelligence technologies remains a hot research topic. Finding a balance between lightweight and accurate computations is a solid foundation for this direction.…

Deep neural networks can approximate functions on different types of data, from images to graphs, with varied underlying structure. This underlying structure can be viewed as the geometry of the data manifold. By extending recent advances…

We propose Lie group embedded dynamical neural networks (LieEDNN) and the corresponding learning algorithms based on gradient descent and metric projection on smooth manifold, where we treat Lie group as an intrinsic representation for…