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Found 20 papers, 11 papers with code
Operator learning without the adjoint
There is a mystery at the heart of operator learning: how can one recover a non-self-adjoint operator from data without probing the adjoint?
Operator learning for hyperbolic partial differential equations
We construct the first rigorously justified probabilistic algorithm for recovering the solution operator of a hyperbolic partial differential equation (PDE) in two variables from input-output training pairs.
A Mathematical Guide to Operator Learning
We explain the types of problems and PDEs amenable to operator learning, discuss various neural network architectures, and explain how to employ numerical PDE solvers effectively.
Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems
Finally, we present a suite of convergence results for the spectral information of stochastic Koopman operators.
Elliptic PDE learning is provably data-efficient
PDE learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data.
Tuning Frequency Bias in Neural Network Training with Nonuniform Data
Small generalization errors of over-parameterized neural networks (NNs) can be partially explained by the frequency biasing phenomenon, where gradient-based algorithms minimize the low-frequency misfit before reducing the high-frequency residuals.
Learning Green's functions associated with time-dependent partial differential equations
Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data.
Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems
This allows us to compute the spectral measure associated with the dynamics of a protein molecule with a 20, 046-dimensional state space and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number $>10^5$ that has a 295, 122-dimensional state space.
Arbitrary-Depth Universal Approximation Theorems for Operator Neural Networks
Here, we prove that operator NNs of bounded width and arbitrary depth are universal approximators for continuous nonlinear operators.
A generalization of the randomized singular value decomposition
The randomized singular value decomposition (SVD) is a popular and effective algorithm for computing a near-best rank $k$ approximation of a matrix $A$ using matrix-vector products with standard Gaussian vectors.
Data-driven discovery of Green's functions with human-understandable deep learning
There is an opportunity for deep learning to revolutionize science and technology by revealing its findings in a human interpretable manner.
Learning elliptic partial differential equations with randomized linear algebra
Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning the associated Green's function $G$.
Over-parametrized neural networks as under-determined linear systems
We draw connections between simple neural networks and under-determined linear systems to comprehensively explore several interesting theoretical questions in the study of neural networks.
The ultraspherical spectral element method
We introduce a novel spectral element method based on the ultraspherical spectral method and the hierarchical Poincar\'{e}-Steklov scheme for solving second-order linear partial differential equations on polygonal domains with unstructured quadrilateral or triangular meshes.
Numerical Analysis Numerical Analysis 65N35, 65N55, 65M60
A sparse spectral method on triangles
In this paper, we demonstrate that many of the computational tools for univariate orthogonal polynomials have analogues for a family of bivariate orthogonal polynomials on the triangle, including Clenshaw's algorithm and sparse differentiation operators.
Numerical Analysis 65N35
Fast Poisson solvers for spectral methods
Poisson's equation is the canonical elliptic partial differential equation.
Numerical Analysis
Why are Big Data Matrices Approximately Low Rank?
Here, we explain the effectiveness of low rank models in data science by considering a simple generative model for these matrices: we suppose that each row or column is associated to a (possibly high dimensional) bounded latent variable, and entries of the matrix are generated by applying a piecewise analytic function to these latent variables.
A nonuniform fast Fourier transform based on low rank approximation
By viewing the nonuniform discrete Fourier transform (NUDFT) as a perturbed version of a uniform discrete Fourier transform, we propose a fast, stable, and simple algorithm for computing the NUDFT that costs $\mathcal{O}(N\log N\log(1/\epsilon)/\log\!\log(1/\epsilon))$ operations based on the fast Fourier transform, where $N$ is the size of the transform and $0<\epsilon <1$ is a working precision.
Numerical Analysis
Fast computation of Gauss quadrature nodes and weights on the whole real line
The algorithm is based on Newton's method with carefully selected initial guesses for the nodes and a fast evaluation scheme for the associated orthogonal polynomial.
Numerical Analysis
