Limits of Learning Dynamical Systems
arXiv:2409.13493 · doi:10.1137/24M1696974
Abstract
A dynamical system is a transformation of a phase space, and the transformation law is the primary means of defining as well as identifying the dynamical system. It is the object of focus of many learning techniques. Yet there are many secondary aspects of dynamical systems - invariant sets, the Koopman operator, and Markov approximations, which provide alternative objectives for learning techniques. Crucially, while many learning methods are focused on the transformation law, we find that forecast performance can depend on how well these other aspects of the dynamics are approximated. These different facets of a dynamical system correspond to objects in completely different spaces - namely interpolation spaces, compact Hausdorff sets, unitary operators and Markov operators respectively. Thus learning techniques targeting any of these four facets perform different kinds of approximations. We examine whether an approximation of any one of these aspects of the dynamics could lead to an approximation of another facet. Many connections and obstructions are brought to light in this analysis. Special focus is put on methods of learning of the primary feature - the dynamics law itself. The main question considered is the connection of learning this law with reconstructing the Koopman operator and the invariant set. The answers are tied to the ergodic and topological properties of the dynamics, and reveal how these properties determine the limits of forecasting techniques.
arXiv admin note: text overlap with arXiv:2208.05349
References in corpus (25)
- A roadmap for the computation of persistent homology
- Embedding and Approximation Theorems for Echo State Networks
- Time-series learning of latent-space dynamics for reduced-order model closure
- Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
- Nonparametric forecasting of low-dimensional dynamical systems
- Coherent structures and isolated spectrum for Perron-Frobenius cocycles
- Delay-coordinate maps and the spectra of Koopman operators
- Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques
- Machine Learning for Prediction with Missing Dynamics
- Koopman spectra in reproducing kernel Hilbert spaces
- Reproducing kernel Hilbert space compactification of unitary evolution groups
- Echo State Networks trained by Tikhonov least squares are L2(μ) approximators of ergodic dynamical systems
- Super convergence of ergodic averages for quasiperiodic orbits
- Chaos on compact manifolds: Differentiable synchronizations beyond the Takens theorem
- Quantitative Quasiperiodicity
- Trace formulas for stochastic evolution operators: Weak noise perturbation theory
- Learning strange attractors with reservoir systems
- Measuring quasiperiodicity
- Approximation of stationary processes by Hidden Markov Models
- Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator
- Indo-Pacific variability on seasonal to multidecadal timescales. Part I: Intrinsic SST modes in models and observations
- Extraction and Prediction of Coherent Patterns in Incompressible Flows through Space-Time Koopman Analysis
- Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem
- Consistent spectral approximation of Koopman operators using resolvent compactification
- Conditional expectation using compactification operators