arXiv subjects

Derek DeSantis

Publications and source records attributed to Derek DeSantis.

14 recordsLinked to original sources

arXiv2026

Koopman early warning signals for bifurcation and rate-induced tipping

Abrupt transitions in complex systems are often preceded by early warning signals. However, most indicators rely on the notion of critical slowing down and do not generally extend to rate-induced tipping where transitions can occur without local loss of stability. This is problematic in stochastic, nonautonomous systems where internal variability and time-varying variables interact to shape tipping onset. We use Koopman operator theory to develop a unified early warning framework for both bifurcation and rate-induced tipping in stochastic systems. Our approach builds on residual Koopman mode decomposition that measures discrepancies between dynamics and their finite-dimensional approximation, and extends it to the control setting by augmenting the observable space with time-varying control variables. In idealized examples, the resulting indicators recover expected signatures near bifurcation points and improve detection in rate-induced regimes where classical indicators fail. We further show that learned embeddings through deep learning outperform prescribed dictionaries, especially in a high-dimensional setting. Applied to simulations of the Atlantic Meridional Overturning Circulation, our Koopman-based indicators distinguish tipping from non-tipping trajectories and reveal interpretable spectral signatures prior to critical transition.

nlin.CD↗

arXiv2026

Viability of Tensor Train Methods for Geophysical Fluid Dynamics

Tensor train (TT) methods have recently gained popularity for accelerating the solving of systems of PDEs. Here, we evaluate the performance of TT methods in the context of geophysical fluid dynamics (GFD) using the shallow water equations and a discretization scheme employed by the ocean component of the Energy Exascale Earth System Model (E3SM). Through a suite of four test cases of increasing complexity, we evaluate TT methods in terms of how much TT is able to compress the model state, the error incurred by the TT approximation, and the speedup obtained by TT versus an optimal standard non-TT implementation in a representative subproblem. We show that though TT is able to effectively compress and speed up simple flows, it struggles to efficiently represent more complex states that are common in realistic GFD applications.

physics.flu-dyn↗

arXiv2025

Breaking the Curse of Dimensionality: Solving Configurational Integrals for Crystalline Solids by Tensor Networks

Accurately evaluating configurational integrals for dense solids remains a central and difficult challenge in the statistical mechanics of condensed systems. Here, we present a novel tensor network approach that reformulates the high-dimensional configurational integral for identical-particle crystals into a sequence of computationally efficient summations. We represent the integrand as a high-dimensional tensor and apply tensor-train (TT) decomposition together with a custom TT-cross interpolation scheme. This approach avoids the need to explicitly construct the full tensor, which would otherwise be computationally intractable. We introduce tailored rank-1 and rank-2 schemes optimized for sharply peaked Boltzmann probability densities, typical in crystalline solids. When applied to the calculation of internal energy and pressure-temperature curves for crystalline copper (Cu) and argon (Ar), as well as the alpha-to-beta phase transition in tin (Sn), our method accurately reproduces molecular dynamics simulation results using tight-binding, machine learning (HIP-NN), and MEAM potentials, all within seconds of computation time.

cond-mat.stat-mech↗

arXiv2025

Deep Koopman operator framework for causal discovery in nonlinear dynamical systems

We use a deep Koopman operator-theoretic formalism to develop a novel causal discovery algorithm, Kausal. Causal discovery aims to identify cause-effect mechanisms for better scientific understanding, explainable decision-making, and more accurate modeling. Standard statistical frameworks, such as Granger causality, lack the ability to quantify causal relationships in nonlinear dynamics due to the presence of complex feedback mechanisms, timescale mixing, and nonstationarity. This presents a challenge in studying many real-world systems, such as the Earth's climate. Meanwhile, Koopman operator methods have emerged as a promising tool for approximating nonlinear dynamics in a linear space of observables. In Kausal, we propose to leverage this powerful idea for causal analysis where optimal observables are inferred using deep learning. Causal estimates are then evaluated in a reproducing kernel Hilbert space, and defined as the distance between the marginal dynamics of the effect and the joint dynamics of the cause-effect observables. Our numerical experiments demonstrate Kausal's superior ability in discovering and characterizing causal signals compared to existing approaches of prescribed observables. Lastly, we extend our analysis to observations of El Ni\~no-Southern Oscillation highlighting our algorithm's applicability to real-world phenomena. Our code is available at https://github.com/juannat7/kausal.

cs.LG↗

arXiv2024

Space-Time Spectral Element Tensor Network Approach for Time Dependent Convection Diffusion Reaction Equation with Variable Coefficients

In this paper, we present a new space-time Petrov-Galerkin-like method. This method utilizes a mixed formulation of Tensor Train (TT) and Quantized Tensor Train (QTT), designed for the spectral element discretization (Q1-SEM) of the time-dependent convection-diffusion-reaction (CDR) equation. We reformulate the assembly process of the spectral element discretized CDR to enhance its compatibility with tensor operations and introduce a low-rank tensor structure for the spectral element operators. Recognizing the banded structure inherent in the spectral element framework's discrete operators, we further exploit the QTT format of the CDR to achieve greater speed and compression. Additionally, we present a comprehensive approach for integrating variable coefficients of CDR into the global discrete operators within the TT/QTT framework. The effectiveness of the proposed method, in terms of memory efficiency and computational complexity, is demonstrated through a series of numerical experiments, including a semi-linear example.

math.NA↗

arXiv2024

Nonlocal, Pattern-aware Response and Feedback Framework for Regional Climate Change

We devise a pattern-aware feedback framework for representing the forced climate response using a suite of Green's function experiments with solar radiation perturbations. By considering the column energy balance, a comprehensive linear response function (CLRF) forimportant climate variables and feedback quantities such as moist static energy, sea surface temperature, albedo, cloud optical depth, and lapse rate is learned from Green's function data. The learned CLRF delineates the effects of the energy diffusion in both the ocean and atmosphere and the pattern-aware feedbacks from the aforementioned radiatively active processes. The CLRF can then be decomposed into forcing-response mode pairs which are in turn used to construct a reduced-order model (ROM) describing the dominant dynamics of climate responses. These mode pairs capture nonlocal effects and teleconnections in the climate and thus, make the ROM apt for capturing regional features of climate change response. A key observation is that the CLRF captures the polar amplified response as the most excitable mode of the climate system and this mode is explainable in the data-learned pattern-aware feedback framework. The ROM can be used for predicting the response for a given forcing and for reconstructing the forcing from a given response; we demonstrate these capabilities for independent forcing pattern.

physics.ao-ph↗

arXiv2024

Causal Discovery in Nonlinear Dynamical Systems using Koopman Operators

We present a theory of causality in dynamical systems using Koopman operators. Our theory is grounded on a rigorous definition of causal mechanism in dynamical systems given in terms of flow maps. In the Koopman framework, we prove that causal mechanisms manifest as particular flows of observables between function subspaces. While the flow map definition is a clear generalization of the standard definition of causal mechanism given in the structural causal model framework, the flow maps are complicated objects that are not tractable to work with in practice. By contrast, the equivalent Koopman definition lends itself to a straightforward data-driven algorithm that can quantify multivariate causal relations in high-dimensional nonlinear dynamical systems. The coupled Rossler system provides examples and demonstrations throughout our exposition. We also demonstrate the utility of our data-driven Koopman causality measure by identifying causal flow in the Lorenz 96 system. We show that the causal flow identified by our data-driven algorithm agrees with the information flow identified through a perturbation propagation experiment. Our work provides new theoretical insights into causality for nonlinear dynamical systems, as well as a new toolkit for data-driven causal analysis.

math.DS↗

arXiv2024

High-order Tensor-Train Finite Volume Method for Shallow Water Equations

In this paper, we introduce a high-order tensor-train (TT) finite volume method for the Shallow Water Equations (SWEs). We present the implementation of the $3^{rd}$ order Upwind and the $5^{th}$ order Upwind and WENO reconstruction schemes in the TT format. It is shown in detail that the linear upwind schemes can be implemented by directly manipulating the TT cores while the WENO scheme requires the use of TT cross interpolation for the nonlinear reconstruction. In the development of numerical fluxes, we directly compute the flux for the linear SWEs without using TT rounding or cross interpolation. For the nonlinear SWEs where the TT reciprocal of the shallow water layer thickness is needed for fluxes, we develop an approximation algorithm using Taylor series to compute the TT reciprocal. The performance of the TT finite volume solver with linear and nonlinear reconstruction options is investigated under a physically relevant set of validation problems. In all test cases, the TT finite volume method maintains the formal high-order accuracy of the corresponding traditional finite volume method. In terms of speed, the TT solver achieves up to 124x acceleration of the traditional full-tensor scheme.

math.NA↗

arXiv2024

How optimal control of polar sea-ice depends on its tipping points

Several Earth system components are at a high risk of undergoing rapid and irreversible qualitative changes or `tipping', due to increasing climate warming. Potential tipping elements include Arctic sea-ice, Atlantic meridional overturning circulation, and tropical coral reefs. Amidst such immediate concerns, it has become necessary to investigate the feasibility of arresting or even reversing the crossing of tipping thresholds using feedback control. In this paper, we study the control of an idealized diffusive energy balance model (EBM) for the Earth's climate; this model has two tipping points due to strong co-albedo feedback. One of these tipping points is a `small icecap' instability responsible for a rapid transition to an ice-free climate state under increasing greenhouse gas (GHG) forcing. We develop an optimal control strategy for the EBM under different climate forcing scenarios with the goal of reversing sea ice loss while minimizing costs. We find that effective control is achievable for such a system, but the cost of reversing sea-ice loss nearly quadruples for an initial state that has just tipped as compared to a state before reaching the tipping point. We also show that thermal inertia may delay tipping leading to an overshoot of the critical GHG forcing threshold. This may offer a short intervention window (overshoot window) during which the control required to reverse sea-ice loss only scales linearly with intervention time. While systems with larger system inertia may have longer overshoot windows, this increased elbow room comes with a steeper rise in the requisite control once the intervention is delayed past this window. Additionally, we find that the requisite control to restore sea-ice is localized in the polar region.

physics.ao-ph↗

arXiv2023

Dynamic Basis Function Interpolation for Adaptive In Situ Data Integration in Ocean Modeling

We propose a new method for combining in situ buoy measurements with Earth system models (ESMs) to improve the accuracy of temperature predictions in the ocean. The technique utilizes the dynamics \textit{and} modes identified in ESMs alongside buoy measurements to improve accuracy while preserving features such as seasonality. We use this technique, which we call Dynamic Basis Function Interpolation, to correct errors in localized temperature predictions made by the Model for Prediction Across Scales Ocean component (MPAS-O) with the Global Drifter Program's in situ ocean buoy dataset.

physics.ao-ph↗

arXiv2020

Factorization of Binary Matrices: Rank Relations, Uniqueness and Model Selection of Boolean Decomposition

The application of binary matrices are numerous. Representing a matrix as a mixture of a small collection of latent vectors via low-rank decomposition is often seen as an advantageous method to interpret and analyze data. In this work, we examine the factorizations of binary matrices using standard arithmetic (real and nonnegative) and logical operations (Boolean and $\mathbb{Z}_2$). We examine the relationships between the different ranks, and discuss when factorization is unique. In particular, we characterize when a Boolean factorization $X = W \land H$ has a unique $W$, a unique $H$ (for a fixed $W$), and when both $W$ and $H$ are unique, given a rank constraint. We introduce a method for robust Boolean model selection, called BMF$k$, and show on numerical examples that BMF$k$ not only accurately determines the correct number of Boolean latent features but reconstruct the pre-determined factors accurately.

math.NA↗

arXiv2020

Coarse-Grain Cluster Analysis of Tensors with Application to Climate Biome Identification

A tensor provides a concise way to codify the interdependence of complex data. Treating a tensor as a d-way array, each entry records the interaction between the different indices. Clustering provides a way to parse the complexity of the data into more readily understandable information. Clustering methods are heavily dependent on the algorithm of choice, as well as the chosen hyperparameters of the algorithm. However, their sensitivity to data scales is largely unknown. In this work, we apply the discrete wavelet transform to analyze the effects of coarse-graining on clustering tensor data. We are particularly interested in understanding how scale effects clustering of the Earth's climate system. The discrete wavelet transform allows classification of the Earth's climate across a multitude of spatial-temporal scales. The discrete wavelet transform is used to produce an ensemble of classification estimates, as opposed to a single classification. Information theoretic approaches are used to identify important scale lenghts in clustering The L15 Climate Datset. We also discover a sub-collection of the ensemble that spans the majority of the variance observed, allowing for efficient consensus clustering techniques that can be used to identify climate biomes.

cs.LG↗

arXiv2019

Nonnegative Canonical Tensor Decomposition with Linear Constraints: nnCANDELINC

There is an emerging interest in tensor factorization applications in big-data analytics and machine learning. To speed up the factorization of extra-large datasets, organized in multidimensional arrays (aka tensors), easy to compute compression-based tensor representations, such as Tucker and Tensor Train formats, are used to approximate the initial large-tensor. Further, tensor factorization is used to extract latent features that can facilitate discoveries of new mechanisms and signatures hidden in the data, where the explainability of the latent features is of principal importance. Nonnegative tensor factorization extracts latent features that are naturally sparse and parts of the data, which makes them easily interpretable. However, to take into account available domain knowledge and subject matter expertise, additional constraints often need to be imposed, which lead us to Canonical decomposition with linear constraints (CANDELINC), a Canonical Polyadic Decomposition with rank deficient factors. In CANDELINC, Tucker compression is used as a pre-processing step, which leads to a larger residual error but to more explainable latent features. Here, we propose a nonnegative CANDELINC (nnCANDELINC) accomplished via a specific nonnegative Tucker decomposition; we refer to as minimal or canonical nonnegative Tucker. We derive several results required to understand the specificity of nnCANDELINC, focusing on the difficulties of preserving the nonnegative rank to its Tucker core and comparing the real-valued to the nonnegative case. Finally, we demonstrate nnCANDELINC performance on synthetic and real-world examples.

math.NA↗

arXiv2018

Operator algebras generated by left invertibles

Operator algebras generated by partial isometries and their adjoints form the basis for some of the most well studied classes of C*-algebras. The primary object of this paper is the norm-closed operator algebra generated by a left invertible $T$ together with its Moore-Penrose inverse $T^\dagger$. We denote this algebra by $\mathfrak{A}_T$. In the isometric case, $T^\dagger = T^*$ and $\mathfrak{A}_T$ is a representation of the Toeplitz algebra. Of particular interest is the case when $T$ satisfies a non-degeneracy condition called analytic. We show that $T$ is analytic if and only if $T^*$ is Cowen-Douglas. When $T$ is analytic with Fredholm index $-1$, the algebra $\mathfrak{A}_T$ contains the compact operators, and any two such algebras are boundedly isomorphic if and only if they are similar.

math.OA↗