The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale's 18th problem
Deep learning (DL) has had unprecedented success and is now entering scientific
computing with full force. However, current DL methods typically suffer from instability, even …
computing with full force. However, current DL methods typically suffer from instability, even …
Residual dynamic mode decomposition: robust and verified Koopmanism
Dynamic mode decomposition (DMD) describes complex dynamic processes through a
hierarchy of simpler coherent features. DMD is regularly used to understand the …
hierarchy of simpler coherent features. DMD is regularly used to understand the …
The mpEDMD algorithm for data-driven computations of measure-preserving dynamical systems
MJ Colbrook - SIAM Journal on Numerical Analysis, 2023 - SIAM
Koopman operators globally linearize nonlinear dynamical systems and their spectral
information is a powerful tool for the analysis and decomposition of nonlinear dynamical …
information is a powerful tool for the analysis and decomposition of nonlinear dynamical …
Rigorous data‐driven computation of spectral properties of Koopman operators for dynamical systems
Koopman operators are infinite‐dimensional operators that globally linearize nonlinear
dynamical systems, making their spectral information valuable for understanding dynamics …
dynamical systems, making their spectral information valuable for understanding dynamics …
Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems
Koopman operators linearize nonlinear dynamical systems, making their spectral
information of crucial interest. Numerous algorithms have been developed to approximate …
information of crucial interest. Numerous algorithms have been developed to approximate …
Convergence rates for learning linear operators from noisy data
This paper studies the learning of linear operators between infinite-dimensional Hilbert
spaces. The training data comprises pairs of random input vectors in a Hilbert space and …
spaces. The training data comprises pairs of random input vectors in a Hilbert space and …
The multiverse of dynamic mode decomposition algorithms
MJ Colbrook - arxiv preprint arxiv:2312.00137, 2023 - arxiv.org
Dynamic Mode Decomposition (DMD) is a popular data-driven analysis technique used to
decompose complex, nonlinear systems into a set of modes, revealing underlying patterns …
decompose complex, nonlinear systems into a set of modes, revealing underlying patterns …
On the computation of geometric features of spectra of linear operators on Hilbert spaces
MJ Colbrook - Foundations of Computational Mathematics, 2024 - Springer
Computing spectra is a central problem in computational mathematics with an abundance of
applications throughout the sciences. However, in many applications gaining an …
applications throughout the sciences. However, in many applications gaining an …
Implicit regularization in AI meets generalized hardness of approximation in optimization--Sharp results for diagonal linear networks
JS Wind, V Antun, AC Hansen - arxiv preprint arxiv:2307.07410, 2023 - arxiv.org
Understanding the implicit regularization imposed by neural network architectures and
gradient based optimization methods is a key challenge in deep learning and AI. In this work …
gradient based optimization methods is a key challenge in deep learning and AI. In this work …
[HTML][HTML] A contour method for time-fractional PDEs and an application to fractional viscoelastic beam equations
We develop a rapid and accurate contour method for the solution of time-fractional PDEs.
The method inverts the Laplace transform via an optimised stable quadrature rule, suitable …
The method inverts the Laplace transform via an optimised stable quadrature rule, suitable …