1 code implementation • 31 Jan 2024 • Nicolas Boullé, Diana Halikias, Samuel E. Otto, Alex Townsend

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?

no code implementations • 29 Dec 2023 • Christopher Wang, Alex Townsend

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.

no code implementations • 22 Dec 2023 • Nicolas Boullé, Alex Townsend

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.

1 code implementation • 21 Aug 2023 • Matthew J. Colbrook, Qin Li, Ryan V. Raut, Alex Townsend

Finally, we present a suite of convergence results for the spectral information of stochastic Koopman operators.

1 code implementation • 24 Feb 2023 • Nicolas Boullé, Diana Halikias, Alex Townsend

PDE learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data.

no code implementations • 28 May 2022 • Annan Yu, Yunan Yang, Alex Townsend

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.

no code implementations • 27 Apr 2022 • Nicolas Boullé, Seick Kim, Tianyi Shi, Alex Townsend

Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data.

1 code implementation • 29 Nov 2021 • Matthew J. Colbrook, Alex Townsend

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.

no code implementations • 23 Sep 2021 • Annan Yu, Chloé Becquey, Diana Halikias, Matthew Esmaili Mallory, Alex Townsend

Here, we prove that operator NNs of bounded width and arbitrary depth are universal approximators for continuous nonlinear operators.

no code implementations • ICLR 2022 • Nicolas Boullé, Alex Townsend

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.

2 code implementations • 1 May 2021 • Nicolas Boullé, Christopher J. Earls, Alex Townsend

There is an opportunity for deep learning to revolutionize science and technology by revealing its findings in a human interpretable manner.

no code implementations • 31 Jan 2021 • Nicolas Boullé, Alex Townsend

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 $G$.

no code implementations • 29 Oct 2020 • Austin R. Benson, Anil Damle, Alex Townsend

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.

1 code implementation • 15 Jun 2020 • Daniel Fortunato, Nicholas Hale, Alex Townsend

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

3 code implementations • NeurIPS 2020 • Nicolas Boullé, Yuji Nakatsukasa, Alex Townsend

We consider neural networks with rational activation functions.

1 code implementation • 13 Feb 2019 • Sheehan Olver, Alex Townsend, Geoff Vasil

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

1 code implementation • 30 Oct 2017 • Daniel Fortunato, Alex Townsend

Poisson's equation is the canonical elliptic partial differential equation.

Numerical Analysis

no code implementations • 21 May 2017 • Madeleine Udell, Alex Townsend

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.

4 code implementations • 17 Jan 2017 • Diego Ruiz-Antolin, Alex Townsend

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

1 code implementation • 20 Oct 2014 • Alex Townsend, Thomas Trogdon, Sheehan Olver

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

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