42 research outputs found
Convergence of weak-SINDy Surrogate Models
In this paper, we give an in-depth error analysis for surrogate models
generated by a variant of the Sparse Identification of Nonlinear Dynamics
(SINDy) method. We start with an overview of a variety of non-linear system
identification techniques, namely, SINDy, weak-SINDy, and the occupation kernel
method. Under the assumption that the dynamics are a finite linear combination
of a set of basis functions, these methods establish a matrix equation to
recover coefficients. We illuminate the structural similarities between these
techniques and establish a projection property for the weak-SINDy technique.
Following the overview, we analyze the error of surrogate models generated by a
simplified version of weak-SINDy. In particular, under the assumption of
boundedness of a composition operator given by the solution, we show that (i)
the surrogate dynamics converges towards the true dynamics and (ii) the
solution of the surrogate model is reasonably close to the true solution.
Finally, as an application, we discuss the use of a combination of weak-SINDy
surrogate modeling and proper orthogonal decomposition (POD) to build a
surrogate model for partial differential equations (PDEs)
Learning High-Dimensional Nonparametric Differential Equations via Multivariate Occupation Kernel Functions
Learning a nonparametric system of ordinary differential equations (ODEs)
from trajectory snapshots in a -dimensional state space requires
learning functions of variables. Explicit formulations scale
quadratically in unless additional knowledge about system properties, such
as sparsity and symmetries, is available. In this work, we propose a linear
approach to learning using the implicit formulation provided by vector-valued
Reproducing Kernel Hilbert Spaces. By rewriting the ODEs in a weaker integral
form, which we subsequently minimize, we derive our learning algorithm. The
minimization problem's solution for the vector field relies on multivariate
occupation kernel functions associated with the solution trajectories. We
validate our approach through experiments on highly nonlinear simulated and
real data, where may exceed 100. We further demonstrate the versatility of
the proposed method by learning a nonparametric first order quasilinear partial
differential equation.Comment: 22 pages, 3 figures, submitted to Neurips 202
Dirichlet Form Theory and its Applications
Theory of Dirichlet forms is one of the main achievements in modern probability theory. It provides a powerful connection between probabilistic and analytic potential theory. It is also an effective machinery for studying various stochastic models, especially those with non-smooth data, on fractal-like spaces or spaces of infinite dimensions. The Dirichlet form theory has numerous interactions with other areas of mathematics and sciences.
This workshop brought together top experts in Dirichlet form theory and related fields as well as promising young researchers, with the common theme of developing new foundational methods and their applications to specific areas of probability. It provided a unique opportunity for the interaction between the established scholars and young researchers
Uniform global stability of switched nonlinear systems in the Koopman operator framework
In this paper, we provide a novel solution to an open problem on the global
uniform stability of switched nonlinear systems. Our results are based on the
Koopman operator approach and, to our knowledge, this is the first theoretical
contribution to an open problem within that framework. By focusing on the
adjoint of the Koopman generator in the Hardy space on the polydisk (or on the
real hypercube), we define equivalent linear (but infinite-dimensional)
switched systems and we construct a common Lyapunov functional for those
systems, under a solvability condition of the Lie algebra generated by the
linearized vector fields. A common Lyapunov function for the original switched
nonlinear systems is derived from the Lyapunov functional by exploiting the
reproducing kernel property of the Hardy space. The Lyapunov function is shown
to converge in a bounded region of the state space, which proves global uniform
stability of specific switched nonlinear systems on bounded invariant sets.Comment: 29 pages, 3 figure
Consistent spectral approximation of Koopman operators using resolvent compactification
Koopman operators and transfer operators represent dynamical systems through
their induced linear action on vector spaces of observables, enabling the use
of operator-theoretic techniques to analyze nonlinear dynamics in state space.
The extraction of approximate Koopman or transfer operator eigenfunctions (and
the associated eigenvalues) from an unknown system is nontrivial, particularly
if the system has mixed or continuous spectrum. In this paper, we describe a
spectrally accurate approach to approximate the Koopman operator on for
measure-preserving, continuous-time systems via a ``compactification'' of the
resolvent of the generator. This approach employs kernel integral operators to
approximate the skew-adjoint Koopman generator by a family of skew-adjoint
operators with compact resolvent, whose spectral measures converge in a
suitable asymptotic limit, and whose eigenfunctions are approximately periodic.
Moreover, we develop a data-driven formulation of our approach, utilizing data
sampled on dynamical trajectories and associated dictionaries of kernel
eigenfunctions for operator approximation. The data-driven scheme is shown to
converge in the limit of large training data under natural assumptions on the
dynamical system and observation modality. We explore applications of this
technique to dynamical systems on tori with pure point spectra and the Lorenz
63 system as an example with mixing dynamics.Comment: 60 pages, 7 figure