arXiv subjects

Joanna Slawinska

Publications and source records attributed to Joanna Slawinska.

18 recordsLinked to original sources

arXiv2026

Tensor network approximation of Koopman operators

We propose a tensor network framework for approximating the evolution of observables of measure-preserving ergodic systems. Our approach is based on a spectrally-convergent approximation of the skew-adjoint Koopman generator by a diagonalizable, skew-adjoint operator $W_τ$ that acts on a reproducing kernel Hilbert space $\mathcal H_τ$ with coalgebra structure and Banach algebra structure under the pointwise product of functions. Leveraging this structure, we lift the unitary evolution operators $e^{t W_τ}$ (which can be thought of as regularized Koopman operators) to a unitary evolution group on the Fock space $F(\mathcal H_τ)$ generated by $\mathcal H_τ$ that acts multiplicatively with respect to the tensor product. Our scheme also employs a representation of classical observables ($L^\infty$ functions of the state) by quantum observables (self-adjoint operators) acting on the Fock space, and a representation of probability densities in $L^1$ by quantum states. Combining these constructions leads to an approximation of the Koopman evolution of observables that is representable as evaluation of a tree tensor network built on a tensor product subspace $\mathcal H_τ^{\otimes n} \subset F(\mathcal H_τ)$ of arbitrarily high grading $n \in \mathbb N$. A key feature of this quantum-inspired approximation is that it captures information from a tensor product space of dimension $(2d+1)^n$, generated from a collection of $2d + 1$ eigenfunctions of $W_τ$. Furthermore, the approximation is positivity preserving. The paper contains a theoretical convergence analysis of the method and numerical applications to two dynamical systems on the 2-torus: an ergodic torus rotation as an example with pure point Koopman spectrum and a Stepanoff flow as an example with topological weak mixing.

math.DS↗

arXiv2026

Quantum mechanical closure of partial differential equations with symmetries

We develop a statistical framework for the dynamical closure of spatiotemporal dynamics governed by partial differential equations. Employing the mathematical framework of quantum mechanics to embed the original classical dynamics into a quantum mechanical representation, we use the space of quantum density operators to model the unresolved degrees of freedom of the original dynamics in a statistical sense, and the framework of quantum measurement to predict their contributions to the resolved dynamics. The embedded dynamics is discretized by a positivity preserving process, leading to a compressed representation that is invariant under the dynamical symmetries of the resolved dynamics. We present a data based formulation of the closure scheme and apply it to a closure problem for the shallow water equations. The numerical results demonstrate that our closure model can accurately predict the main features of the true dynamics, including for out of sample initial conditions.

math.DS↗

arXiv2025

Learning Dynamical Systems with the Spectral Exterior Calculus

We present a data-driven framework for learning dynamical systems on compact Riemannian manifolds based on the spectral exterior calculus (SEC). This approach represents vector fields as linear combinations of frame elements constructed using the eigenfunctions of the Laplacian on smooth functions, along with their gradients. Such reconstructed vector fields generate dynamical flows that consistently approximate the true system, while being compatible with the nonlinear geometry of the manifold. The data-driven implementation of this framework utilizes embedded data points and tangent vectors as training data, along with a graph-theoretic approximation of the Laplacian. In this paper, we prove the convergence of the SEC-based reconstruction in the limit of large data. Moreover, we illustrate the approach numerically with applications to dynamical systems on the unit circle and the 2-torus. In these examples, the reconstructed vector fields compare well with the true vector fields, in terms of both pointwise estimates and generation of orbits.

math.DS↗

arXiv2025

Second quantization for classical nonlinear dynamics

Using techniques from many-body quantum theory, we propose a framework for representing the evolution of observables of measure-preserving ergodic flows through infinite-dimensional rotation systems on tori. This approach is based on a class of weighted Fock spaces $F_w(\mathcal H_τ)$ generated by a 1-parameter family of reproducing kernel Hilbert spaces $\mathcal H_τ$, and endowed with commutative Banach algebra structure under the symmetric tensor product using a subconvolutive weight $w$. We describe the construction of the spaces $F_w(\mathcal H_τ)$ and show that their Banach algebra spectra, $σ(F_w(\mathcal H_τ))$, decompose into a family of tori of potentially infinite dimension. Spectrally consistent unitary approximations $U^t_τ$ of the Koopman operator acting on $\mathcal H_τ$ are then lifted to rotation systems on these tori akin to the topological models of ergodic systems with pure point spectra in the Halmos--von Neumann theorem. Our scheme also employs a procedure for representing observables of the original system by polynomial functions on finite-dimensional tori in $σ(F_w(\mathcal H_τ))$ of arbitrarily large degree, with coefficients determined from pointwise products of eigenfunctions of $U^t_τ$. This leads to models for the Koopman evolution of observables on $L^2$ built from tensor products of finite collections of approximate Koopman eigenfunctions. Numerically, the scheme is amenable to consistent data-driven implementation using kernel methods. We illustrate it with applications to Stepanoff flows on the 2-torus and the Lorenz 63 system. Connections with quantum computing are also discussed.

math.DS↗

arXiv2023

Revealing trends and persistent cycles of non-autonomous systems with operator-theoretic techniques: Applications to past and present climate dynamics

An important problem in modern applied science is characterizing the behavior of systems with complex internal dynamics subjected to external forcings from their environment. While a great variety of techniques has been developed to analyze such non-autonomous systems, many approaches rely on the availability of ensembles of experiments or simulations in order to generate sufficient information to encapsulate the external forcings. This makes them unsuitable to study important classes of natural systems such as climate dynamics where only a single realization is observed. Here, we show that operator-theoretic techniques previously developed to identify slowly decaying observables of autonomous dynamical systems provide a powerful means for identifying trends and persistent cycles of non-autonomous systems using data from a \emph{single} trajectory of the system. Using systematic mathematical analysis and prototype examples, we demonstrate that eigenfunctions of Koopman and transfer operators provide coordinates that simultaneously capture nonlinear trends and coherent modes of internal variability. In addition, we apply our framework to two real-world examples from present and past climate dynamics: Variability of sea surface temperature (SST) over the industrial era and the mid-Pleistocene transition (MPT) of Quaternary glaciation cycles. Our results provide a nonparametric representation of SST and surface air temperature (SAT) trends over the industrial era, while also capturing the response of the seasonal precipitation cycle to these trends. In addition, our paleo-climate analysis reveals the dominant glaciation cycles over the past 3 million years and the MPT with a high level of granularity.

math.DS↗

arXiv2023

Koopman spectral analysis of skew-product dynamics on Hilbert $C^*$-modules

We introduce a linear operator on a Hilbert $C^*$-module for analyzing skew-product dynamical systems. The operator is defined by composition and multiplication. We show that it admits a decomposition in the Hilbert $C^*$-module, called eigenoperator decomposition, that generalizes the concept of the eigenvalue decomposition. This decomposition reconstructs the Koopman operator of the system in a manner that represents the continuous spectrum through eigenoperators. In addition, it is related to the notions of cocycle and Oseledets subspaces and it is useful for characterizing coherent structures under skew-product dynamics. We present numerical applications to simple systems on two-dimensional domains.

math.DS↗

arXiv2022

Quantum Mechanics for Closure of Dynamical Systems

We propose a scheme for data-driven parameterization of unresolved dimensions of dynamical systems based on the mathematical framework of quantum mechanics and Koopman operator theory. Given a system in which some components of the state are unknown, this method involves defining a surrogate system in a time-dependent quantum state which determines the fluxes from the unresolved degrees of freedom at each timestep. The quantum state is a density operator on a finite-dimensional Hilbert space of classical observables and evolves over time under an action induced by the Koopman operator. The quantum state also updates with new values of the resolved variables according to a quantum Bayes' law, implemented via an operator-valued feature map. Kernel methods are utilized to learn data-driven basis functions and represent quantum states, observables, and evolution operators as matrices. The resulting computational schemes are automatically positivity-preserving, aiding in the physical consistency of the parameterized system. We analyze the results of two different modalities of this methodology applied to the Lorenz 63 and Lorenz 96 multiscale systems, and show how this approach preserves important statistical and qualitative properties of the underlying chaotic systems.

math.DS↗

arXiv2022

Data Assimilation in Operator Algebras

We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a non-abelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to new computational data assimilation schemes that are (i) automatically positivity-preserving; and (ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to data assimilation of the Lorenz 96 multiscale system and the El Nino Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification.

math.ST↗

arXiv2021

Embedding classical dynamics in a quantum computer

We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides a new operator-theoretic representation of classical dynamics by combining ergodic theory with quantum information science. The resulting quantum embedding of classical dynamics (QECD) enables efficient simulation of spaces of classical observables with exponentially large dimension using a quadratic number of quantum gates. The QECD framework is based on a quantum feature map for representing classical states by density operators on a reproducing kernel Hilbert space, $\mathcal H $, and an embedding of classical observables into self-adjoint operators on $\mathcal H$. In this scheme, quantum states and observables evolve unitarily under the lifted action of Koopman evolution operators of the classical system. Moreover, by virtue of the reproducing property of $\mathcal H$, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve an exponential quantum computational advantage, we project the state of the quantum system to a density matrix on a $2^n$-dimensional tensor product Hilbert space associated with $n$ qubits. By employing discrete Fourier-Walsh transforms, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through a quantum circuit of size $O(n)$. Furthermore, the circuit features a state preparation stage, also of size $O(n)$, and a quantum Fourier transform stage of size $O(n^2)$, which makes predictions of observables possible by measurement in the standard computational basis. We prove theoretical convergence results for these predictions as $n\to\infty$. We present simulated quantum circuit experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum System One.

quant-ph↗

arXiv2021

Spectral analysis of climate dynamics with operator-theoretic approaches

The Earth's climate system is a classical example of a multiscale, multiphysics dynamical system with an extremely large number of active degrees of freedom, exhibiting variability on scales ranging from micrometers and seconds in cloud microphysics, to thousands of kilometers and centuries in ocean dynamics. Yet, despite this dynamical complexity, climate dynamics is known to exhibit coherent modes of variability. A primary example is the El Niño Southern Oscillation (ENSO), the dominant mode of interannual (3-5 yr) variability in the climate system. The objective and robust characterization of this and other important phenomena presents a long-standing challenge in Earth system science, the resolution of which would lead to improved scientific understanding and prediction of climate dynamics, as well as assessment of their impacts on human and natural systems. Here, we show that the spectral theory of dynamical systems, combined with techniques from data science, provides an effective means for extracting coherent modes of climate variability from high-dimensional model and observational data, requiring no frequency prefiltering, but recovering multiple timescales and their interactions. Lifecycle composites of ENSO are shown to improve upon results from conventional indices in terms of dynamical consistency and physical interpretability. In addition, the role of combination modes between ENSO and the annual cycle in ENSO diversity is elucidated.

physics.ao-ph↗

arXiv2021

Reproducing kernel Hilbert space compactification of unitary evolution groups

A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. It is based on an approximation of the generator of the system by a compact operator $W_τ$ on a reproducing kernel Hilbert space (RKHS). A key tool is that $W_τ$ is skew-adjoint , and thus can be characterized by a unique projection-valued measure, discrete by compactness, and an associated orthonormal basis of eigenfunctions. These eigenfunctions can be ordered in terms of a Dirichlet energy on the RKHS, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, the regularized generator has a well-defined Borel functional calculus allowing the construction of a unitary evolution group $\{ e^{t W_τ} \}_{t\in\mathbb{R}}$ on the RKHS, which approximates the unitary Koopman evolution group of the original system. We establish convergence results for the spectrum and Borel functional calculus of the regularized generator to those of the original system in the limit $τ\to 0^+$. Convergence results are also established for a data-driven formulation, where these operators are approximated using finite-rank operators obtained from observed time series. An advantage of working with an RKHS structure is that one can perform pointwise evaluation and interpolation through bounded linear operators, which is not possible in $L^p$ spaces. This enables the out of sample evaluation of data-approximated eigenfunctions, as well as data-driven forecasts initialized with pointwise initial data (as opposed to probability densities in $L^p$). The pattern extraction and prediction framework is numerically applied to three instances of ergodic dynamical systems with atomic and continuous spectra.

math.DS↗

arXiv2019

Extended-range statistical ENSO prediction through operator-theoretic techniques for nonlinear dynamics

Forecasting the El Nino-Southern Oscillation (ENSO) has been a subject of vigorous research due to the important role of the phenomenon in climate dynamics and its worldwide socioeconomic impacts. Over the past decades, numerous models for ENSO prediction have been developed, among which statistical models approximating ENSO evolution by linear dynamics have received significant attention owing to their simplicity and comparable forecast skill to first-principles models at short lead times. Yet, due to highly nonlinear and chaotic dynamics (particularly during ENSO initiation), such models have limited skill for longer-term forecasts beyond half a year. To resolve this limitation, here we employ a new nonparametric statistical approach based on analog forecasting, called kernel analog forecasting (KAF), which avoids assumptions on the underlying dynamics through the use of for nonlinear kernel methods for machine learning and dimension reduction of high-dimensional datasets. Through a rigorous connection with Koopman operator theory for dynamical systems, KAF yields statistically optimal predictions of future ENSO states as conditional expectations, given noisy and potentially incomplete data at forecast initialization. Here, using industrial-era Indo-Pacific sea surface temperature (SST) as training data, the method is shown to successfully predict the Nino 3.4 index in a 1998-2017 verification period out to a 10-month lead, which corresponds to an increase of 3-7 months over a benchmark linear inverse model (LIM), while significantly improving upon the ENSO predictability "spring barrier". An analysis of a 1300-yr control integration of a comprehensive climate model (CCSM4) further demonstrates that the enhanced predictability afforded by KAF holds over potentially much longer leads, extending to 24 months versus 18 months in the benchmark LIM.

physics.ao-ph↗

arXiv2018

Spatiotemporal pattern extraction by spectral analysis of vector-valued observables

We present a data-driven framework for extracting complex spatiotemporal patterns generated by ergodic dynamical systems. Our approach, called Vector-valued Spectral Analysis (VSA), is based on an eigendecomposition of a kernel integral operator acting on a Hilbert space of vector-valued observables of the system, taking values in a space of functions (scalar fields) on a spatial domain. This operator is constructed by combining aspects of the theory of operator-valued kernels for machine learning with delay-coordinate maps of dynamical systems. In contrast to conventional eigendecomposition techniques, which decompose the input data into pairs of temporal and spatial modes with a separable, tensor product structure, the patterns recovered by VSA can be manifestly non-separable, requiring only a modest number of modes to represent signals with intermittency in both space and time. Moreover, the kernel construction naturally quotients out dynamical symmetries in the data, and exhibits an asymptotic commutativity property with the Koopman evolution operator of the system, enabling decomposition of multiscale signals into dynamically intrinsic patterns. Application of VSA to the Kuramoto-Sivashinsky model demonstrates significant performance gains in efficient and meaningful decomposition over eigendecomposition techniques utilizing scalar-valued kernels.

math.DS↗

arXiv2018

The Antarctic circumpolar wave and its seasonality: Intrinsic traveling modes and ENSO teleconnections

Interannual variability in the Southern Ocean is investigated via nonlinear Laplacian spectral analysis (NLSA), an objective eigendecomposition technique for nonlinear dynamical systems that can simultaneously recover multiple timescales from data with high skill. Applied to modeled and observed sea surface temperature and sea ice concentration data, NLSA recovers the wavenumber-2 eastward propagating signal corresponding to the Antarctic circumpolar wave (ACW). During certain phases of its lifecycle, the spatial patterns of this mode display a structure that can explain the statistical origin of the Antarctic dipole pattern. Another group of modes have combination frequencies consistent with the modulation of the annual cycle by the ACW. Further examination of these newly identified modes reveals that they can have either eastward or westward propagation, combined with meridional pulsation reminiscent of sea ice reemergence patterns in the Arctic. Moreover, they exhibit smaller-scale spatial structures, and explain more Indian Ocean variance than the primary ACW modes. We attribute these modes to teleconnections between ACW and the tropical Indo-Pacific Ocean; in particular, fundamental ENSO modes and their associated combination modes with the annual cycle recovered by NLSA. Another mode extracted from the Antarctic variables displays an eastward propagating wavenumber-3 structure over the Southern Ocean, but exhibits no strong correlation to interannual Indo-Pacific variability.

physics.ao-ph↗

arXiv2018

Vector-Valued Spectral Analysis of Space-Time Data

Identifying coherent spatiotemporal patterns generated by complex dynamical systems is a central problem in many science and engineering disciplines. Here, we combine ideas from the theory of operator-valued kernels with delay-embedding techniques in dynamical systems to develop of a method for objective identification of spatiotemporal coherent patterns, without prior knowledge of the state space and/or the dynamical laws of the system generating the data. A key aspect of this method is that it operates on a space of vector-valued observables using a kernel measure of similarity that takes into account both temporal and spatial degrees of freedom (in contrast, classical kernel techniques such as PCA utilize aggregate measures of similarity between 'snapshots'). As a result, spectral decomposition of data via our approach yields a significantly more efficient and physically meaningful representation of complex spatiotemporal signals than conventional methods based on scalar-valued kernels. We demonstrate this behavior with applications to an oscillator model and sea surface temperature data from a climate model.

physics.data-an↗

arXiv2017

Data-Driven Koopman Analysis of Tropical Climate Space-Time Variability

We study nonlinear dynamics of the Earth's tropical climate system. For that, we apply a recently developed technique for feature extraction and mode decomposition of spatiotemporal data generated by ergodic dynamical systems. The method relies on constructing low-dimensional representations (temporal patterns) of signals using eigenfunctions of Koopman operators governing the evolution of observables in ergodic dynamical systems. We apply this technique to a variety of tropical climate datasets and extract a multiscale hierarchy of spatiotemporal patterns on diurnal to interannual timescales. In particular, we detect without prefiltering the input data modes operating on intraseasonal and shorter timescales that correspond to propagation of organized convection. We discuss the salient properties of these propagating features and in particular we focus on how the activity of certain types of these traveling patterns is related to lower-frequency dynamics. As an extension of this work, we discuss their potential predictability based on a range of nonparametric techniques and potential advances related to understanding the deterministic and stochastic aspects of the variability of these modes.

physics.ao-ph↗

arXiv2016

Indo-Pacific variability on seasonal to multidecadal timescales. Part II: Multiscale atmosphere-ocean linkages

The coupled atmosphere-ocean variability of the Indo-Pacific on interannual to multidecadal timescales is investigated in a millennial control run of CCSM4 and in observations using a family of modes recovered in Part~I of this work from unprocessed SST data through nonlinear Laplacian spectral analysis (NLSA). It is found that ENSO and combination modes of ENSO with the annual cycle exhibit a seasonally synchronized southward shift of equatorial surface zonal winds and thermocline adjustment consistent with terminating El Nino and La Nina events. The surface wind patterns associated with these modes also generate teleconnections between the Pacific and Indian Oceans, leading to a pattern of SST anomalies characteristic of the Indian Ocean dipole. Fundamental and combination modes representing the tropospheric biennial oscillation (TBO) in CCSM4 are also found to be consistent with mechanisms for seasonally synchronized biennial variability of the Asian-Australian monsoon and Walker circulation. On longer timescales, the leading multidecadal pattern recovered by NLSA from Indo-Pacific SST data in CCSM4, referred to as west Pacific multidecadal mode (WPMM), is found to significantly modulate ENSO and TBO activity with periods of negative SST anomalies in the western tropical Pacific favoring stronger ENSO and TBO variability. Physically, this behavior is attributed to the fact that cold WPMM phases feature anomalous decadal westerlies in the tropical central Pacific and easterlies in the tropical Indian Ocean, as well as an anomalously deep thermocline in the eastern Pacific cold tongue. Moreover, despite the relatively low SST variance explained by this mode, the WPMM is found to correlate significantly with decadal precipitation over Australia in CCSM4.

physics.ao-ph↗

arXiv2016

Indo-Pacific variability on seasonal to multidecadal timescales. Part I: Intrinsic SST modes in models and observations

The variability of Indo-Pacific SST on seasonal to multidecadal timescales is investigated using a recently introduced technique called nonlinear Laplacian spectral analysis (NLSA). Through this technique, drawbacks associated with ad hoc pre-filtering of the input data are avoided, enabling recovery of low-frequency and intermittent modes not previously accessible via classical approaches. Here, a multiscale hierarchy of spatiotemporal modes is identified for Indo-Pacific SST in millennial control runs of CCSM4 and CM3 and in HadISST data. On interannual timescales, a mode with spatiotemporal patterns corresponding to the fundamental component of ENSO emerges, along with ENSO-modulated annual modes consistent with combination mode theory. The ENSO combination modes also feature prominent activity in the Indian Ocean, explaining significant fraction of the SST variance in regions associated with the Indian Ocean dipole. A pattern resembling the tropospheric biennial oscillation emerges in addition to ENSO and the associated combination modes. On multidecadal timescales, the dominant NLSA mode in the model data is predominantly active in the western tropical Pacific. The interdecadal Pacific oscillation also emerges as a distinct NLSA mode, though with smaller explained variance than the western Pacific multidecadal mode. Analogous modes on interannual and decadal timescales are also identified in HadISST data for the industrial era, as well as in model data of comparable timespan, though decadal modes are either absent or of degraded quality in these datasets.

physics.geo-ph↗