10 papers
Sharp Computational Bounds for Spectral Types of Schrödinger Operators
Matthew J. Colbrook, George Coote
We prove sharp bounds for determining the spectral-type decomposition of Schrödinger operators in the spirit of Smale's program on the foundations of computation. For explicit one-…
The Spectral Skeleton of Chaos: Koopman Wave Packets on Poincaré Sections
Jason J. Bramburger, Shiloh-Sheray Gagnon, Rachel Morris +1
A Poincaré section replaces a flow by a return map, but for a chaotic system this map is usually known only from sampled crossings. We show that coarse transport can be read direct…
Residual-Guided Dictionary Learning for Spectrally Accurate Koopman Approximation
George Coote, Matthew J. Colbrook
Koopman theory promises linear structure in nonlinear dynamics, but numerical Koopman spectra are easy to compute and hard to trust. A finite EDMD matrix always has eigenvalues; th…
PRONE: Petrov-Galerkin Operator Learning Unifies DMD, SINDy & Koopmanism
Matthew J. Colbrook, April Herwig, J. Nathan Kutz
Data-driven dynamics often asks how to linearize a nonlinear system. We ask instead: which observables should be advanced, and where should their futures live? This leads to Petrov…
Residual Pseudospectra Reveal a Physics-Informed Koopman Backbone for Tropical Pacific Variability and ENSO Prediction
Paula Lorenzo-Sanchez, Matthew J. Colbrook, Antonio Navarra
Tropical Pacific sea-surface-temperature (SST) variability spans interacting timescales, with the ENSO as its dominant interannual expression. Yet the dynamical structure organizin…
Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning
Kelan Gray, Finlay Brown, Nicolas Boullé +1
Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be exp…
Finding Koopman Invariant Subspaces via Personalized PageRank
Hyukpyo Hong, Qin Li, Matthew J. Colbrook +1
Selecting a finite dictionary of observables whose span is Koopman-invariant is a central challenge in data-driven Koopman operator approximation. We address this problem by exploi…
Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds
Gustav Conradie, Nicolas Boullé, Jean-Christophe Loiseau +2
Koopman operator theory provides a global linear representation of nonlinear dynamics and underpins many data-driven methods. In practice, however, finite-dimensional feature space…
Weighted Birkhoff Averages Accelerate Data-Driven Methods
Maria Bou-Sakr-El-Tayar, Jason J. Bramburger, Matthew J. Colbrook
Many data-driven algorithms in dynamical systems rely on ergodic averages that converge painfully slowly. One simple idea changes this: taper the ends. Weighted Birkhoff averages c…
Noisy PDE Training Requires Bigger PINNs
Sebastien Andre-Sloan, Anirbit Mukherjee, Matthew Colbrook
Physics-Informed Neural Networks (PINNs) are increasingly used to approximate solutions of partial differential equations (PDEs), particularly in high dimensions. In real-world set…