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The remarkable success of Deep Neural Networks(DNN) is driven by gradient-based optimization, yet this process is often undermined by its tendency to produce disordered weight structures, which harms feature clarity and degrades learning…
In this work, we explore the ability of NN (Neural Networks) to serve as a tool for finding eigen-pairs of ordinary differential equations. The question we aime to address is whether, given a self-adjoint operator, we can learn what are the…
Ensemble learning has proven effective in improving predictive performance and estimating uncertainty in neural networks. However, conventional ensemble methods often suffer from redundant parameter usage and computational inefficiencies…
Operator eigenvalue problems play a critical role in various scientific fields and engineering applications, yet numerical methods are hindered by the curse of dimensionality. Recent deep learning methods provide an efficient approach to…
We propose a flexible machine-learning framework for solving eigenvalue problems of diffusion operators in moderately large dimension. We improve on existing Neural Networks (NNs) eigensolvers by demonstrating our approach ability to…
Density estimation is a fundamental problem in statistical learning. This problem is especially challenging for complex high-dimensional data due to the curse of dimensionality. A promising solution to this problem is given here in an…
We present a novel architecture for learning geometry-aware preconditioners for linear partial differential equations (PDEs). We show that a deep operator network (Deeponet) can be trained on a simple geometry and remain a robust…
While many problems in machine learning focus on learning mappings between finite-dimensional spaces, scientific applications require approximating mappings between function spaces, i.e., operators. We study the problem of learning…
t-distributed stochastic neighbor embedding (t-SNE) is a well-established visualization method for complex high-dimensional data. However, the original t-SNE method is nonparametric, stochastic, and often cannot well prevserve the global…
Deep neural networks (DNNs) have achieved remarkable success in numerous domains, and their application to PDE-related problems has been rapidly advancing. This paper provides an estimate for the generalization error of learning Lipschitz…
Neural operators have recently become popular tools for designing solution maps between function spaces in the form of neural networks. Differently from classical scientific machine learning approaches that learn parameters of a known…
In this paper, we construct approximated solutions of Differential Equations (DEs) using the Deep Neural Network (DNN). Furthermore, we present an architecture that includes the process of finding model parameters through experimental data,…
In this work, we introduce and study a class of Deep Neural Networks (DNNs) in continuous-time. The proposed architecture stems from the combination of Neural Ordinary Differential Equations (Neural ODEs) with the model structure of…
Embedding high-dimensional data onto a low-dimensional manifold is of both theoretical and practical value. In this paper, we propose to combine deep neural networks (DNN) with mathematics-guided embedding rules for high-dimensional data…
Neural operators aim to learn mappings between infinite-dimensional function spaces, but their performance often degrades on complex or irregular geometries due to the lack of geometry-aware representations. We propose the Finite Element…
Operator learning problems arise in many key areas of scientific computing where Partial Differential Equations (PDEs) are used to model physical systems. In such scenarios, the operators map between Banach or Hilbert spaces. In this work,…
In this article, we introduce a general theoretical framework to analyze non-consistent approximations of the discrete eigenmodes of a self-adjoint operator. We focus in particular on the discrete eigenvalues laying in spectral gaps. We…
Learning operators between infinitely dimensional spaces is an important learning task arising in wide applications in machine learning, imaging science, mathematical modeling and simulations, etc. This paper studies the nonparametric…
Operator regression provides a powerful means of constructing discretization-invariant emulators for partial-differential equations (PDEs) describing physical systems. Neural operators specifically employ deep neural networks to approximate…
The Lane-Emden equation, a nonlinear second-order ordinary differential equation, plays a fundamental role in theoretical physics and astrophysics, particularly in modeling the structure of stellar interiors. Also referred to as the…