Low-dimensional Model Research Articles

Stochastic disturbance with finite-time chaos stabilization and synchronization for a fractional-order nonautonomous hybrid nonlinear complex system via a sliding mode control

This paper provides a novel approach to studying the chaos control and synchronization of fractional-order hybrid Darcy–Brinkman systems (FOHDBs). We use the truncated Galerkin method to transform the system of partial differential equations (PDEs) into ordinary differential equations (ODEs) based on Fourier modes in a nonlinear low-dimensional Brinkman model. In the nonlinear complex system, hybrid nanoparticles and fluid flow physical characteristics are considered to be external disturbances. As a result, the implications of model uncertainty and external disruptions are fully absorbed into the problem, further complicating it and resulting in highly complex and unpredictable behaviors. In addition, To assure the occurrence of the sliding motion in a finite amount of time, an appropriate robust fractional sliding mode technique is proposed. Sliding motion is shown to occur in finite time using the fractional Lyapunov stability. Following that, we discussed control techniques for achieving infinitesimally close to an equilibrium of chaotic states and tracking errors. It is important to note that FOHDBs and external stochastic disturbances can be controlled using the proposed fractional nonsingular terminal sliding mode control technique. Finally, the effectiveness and applicability of the suggested finite-time control technique are graphically illustrated by the analytical and numerical simulations.

Efficient interpolation of molecular properties across chemical compound space with low-dimensional descriptors

We demonstrate accurate data-starved models of molecular properties for interpolation in chemical compound spaces with low-dimensional descriptors. Our starting point is based on three-dimensional, universal, physical descriptors derived from the properties of the distributions of the eigenvalues of Coulomb matrices. To account for the shape and composition of molecules, we combine these descriptors with six-dimensional features informed by the Gershgorin circle theorem. We use the nine-dimensional descriptors thus obtained for Gaussian process regression based on kernels with variable functional form, leading to extremely efficient, low-dimensional interpolation models. The resulting models trained with 100 molecules are able to predict the product of entropy and temperature (S × T) and zero point vibrational energy (ZPVE) with the absolute error under 1 kcal mol−1 for >78 % and under 1.3 kcal mol−1 for >92 % of molecules in the test data. The test data comprises 20 000 molecules with complexity varying from three atoms to 29 atoms and the ranges of S × T and ZPVE covering 36 kcal mol−1 and 161 kcal mol−1, respectively. We also illustrate that the descriptors based on the Gershgorin circle theorem yield more accurate models of molecular entropy than those based on graph neural networks that explicitly account for the atomic connectivity of molecules.

Gaussian-process-regression-based method for the localization of exceptional points in complex resonance spectra

Resonances in open quantum systems depending on at least two controllable parameters can show the phenomenon of exceptional points (EPs), where not only the eigenvalues but also the eigenvectors of two or more resonances coalesce. Their exact localization in the parameter space is challenging, in particular in systems, where the computation of the quantum spectra and resonances is numerically very expensive. We introduce an efficient machine learning algorithm to find EPs based on Gaussian process regression (GPR). The GPR-model is trained with an initial set of eigenvalue pairs belonging to an EP and used for a first estimation of the EP position via a numerically cheap root search. The estimate is then improved iteratively by adding selected exact eigenvalue pairs as training points to the GPR-model. The GPR-based method is developed and tested on a simple low-dimensional matrix model and then applied to a challenging real physical system, viz., the localization of EPs in the resonance spectra of excitons in cuprous oxide in external electric and magnetic fields. The precise computation of EPs, by taking into account the complete valence band structure and central-cell corrections of the crystal, can be the basis for the experimental observation of EPs in this system.

DeepION: A Deep Learning-Based Low-Dimensional Representation Model of Ion Images for Mass Spectrometry Imaging.

Mass spectrometry imaging (MSI) is a high-throughput imaging technique capable of the qualitative and quantitative in situ detection of thousands of ions in biological samples. Ion image representation is a technique that produces a low-dimensional vector embedded with significant spectral and spatial information on an ion image, which further facilitates the distance-based similarity measurement for the identification of colocalized ions. However, given the low signal-to-noise ratios inherent in MSI data coupled with the scarcity of annotated data sets, achieving an effective ion image representation for each ion image remains a challenge. In this study, we propose DeepION, a novel deep learning-based method designed specifically for ion image representation, which is applied to the identification of colocalized ions and isotope ions. In DeepION, contrastive learning is introduced to ensure that the model can generate the ion image representation in a self-supervised manner without manual annotation. Since data augmentation is a crucial step in contrastive learning, a unique data augmentation strategy is designed by considering the characteristics of MSI data, such as the Poisson distribution of ion abundance and a random pattern of missing values, to generate plentiful ion image pairs for DeepION model training. Experimental results of rat brain tissue MSI show that DeepION outperforms other methods for both colocalized ion and isotope ion identification, demonstrating the effectiveness of ion image representation. The proposed model could serve as a crucial tool in the biomarker discovery and drug development of the MSI technique.

A Systematic Algorithm to Compute Hazardous Area Extent for Two-Phase Releases Based on an Improved Low-Dimensional (Simple) Dispersion Model

A Systematic Algorithm to Compute Hazardous Area Extent for Two-Phase Releases Based on an Improved Low-Dimensional (Simple) Dispersion Model

Covering convection with a thermal blanket: numerical simulation and stochastic modelling

Adding moving boundaries to convective fluids is known to result in non-trivial and surprising dynamics, leading to spectacular geoformations ranging from kilometre-scale karst terrains to planetary-scale plate tectonics. On the one hand, the moving solid alters the surrounding flow field, but on the other hand, the flow modifies the motion and shape of the solid. This leads to a two-way coupling that is significant in the study of fluid–structure interactions and in the understanding of geomorphologies. In this work, we investigate the coupling between a floating plate and the convective fluid below it. Through numerical experiments, we show that the motion of this plate is driven by the flow beneath. However, the flow structure is also modified by the presence of the plate, leading to the ‘thermal blanket’ effect where the trapped heat beneath the plate results in buoyant and upwelling flows that in turn push the plate away. By analysing this two-way coupling between moving boundary and fluid, we are able to capture the dynamical behaviours of this plate through a low-dimensional stochastic model. Geophysically, the thermal blanket effect is believed to drive the continental drift, therefore understanding this mechanism has significance beyond fluid dynamics.

Reaction coordinate flows for model reduction of molecular kinetics.

In this work, we introduce a flow based machine learning approach called reaction coordinate (RC) flow for the discovery of low-dimensional kinetic models of molecular systems. The RC flow utilizes a normalizing flow to design the coordinate transformation and a Brownian dynamics model to approximate the kinetics of RC, where all model parameters can be estimated in a data-driven manner. In contrast to existing model reduction methods for molecular kinetics, RC flow offers a trainable and tractable model of reduced kinetics in continuous time and space due to the invertibility of the normalizing flow. Furthermore, the Brownian dynamics-based reduced kinetic model investigated in this work yields a readily discernible representation of metastable states within the phase space of the molecular system. Numerical experiments demonstrate how effectively the proposed method discovers interpretable and accurate low-dimensional representations of given full-state kinetics from simulations.

Capturing the edge of chaos as a spectral submanifold in pipe flows

An extended turbulent state can coexist with the stable laminar state in pipe flows. We focus here on short pipes with additional discrete symmetries imposed. In this case, the boundary between the coexisting basins of attraction, often called the edge of chaos, is the stable manifold of an edge state, which is a lower-branch travelling wave solution. We show that a low-dimensional submanifold of the edge of chaos can be constructed from velocity data using the recently developed theory of spectral submanifolds (SSMs). These manifolds are the unique smoothest nonlinear continuations of non-resonant spectral subspaces of the linearized system at stationary states. Using very low-dimensional SSM-based reduced-order models, we predict transitions to turbulence or laminarization for velocity fields near the edge of chaos.

Low-dimensional modeling approach promises more efficient climate predictions

Novel dynamical systems approach may lead to better forecasting using less computational power.

Optimal dosage protocols for mathematical models of synergy of chemo- and immunotherapy.

The release of tumor antigens during traditional cancer treatments such as radio- or chemotherapy leads to a stimulation of the immune response which provides synergistic effects these treatments have when combined with immunotherapies. A low-dimensional mathematical model is formulated which, depending on the values of its parameters, encompasses the 3 E's (elimination, equilibrium, escape) of tumor immune system interactions. For the escape situation, optimal control problems are formulated which aim to revert the process to the equilibrium scenario. Some numerical results are included.

Enhancing Seismic Response in Frame Structures through Rigid Connections to Structures with Vibro-Impacting Mass

In this paper, a promising approach is studied that can efficiently mitigate seismic effects on a frame structure by coupling it with an protection system. Various devices are employed to achieve this objective, including tuned mass dampers, dynamic mass absorbers, elastoplastic dampers, and rocking rigid walls. This paper delves into the efficacy of a vibro-impacting nonlinear energy sink in reducing seismic effects on a frame structure. More precisely, a supplementary apparatus, consisting of an auxiliary structure equipped with a vibro-impacting nonlinear energy sink, is rigidly linked to the first story of the targeted frame structure. The seismic response of this coupled system is derived through a dynamically equivalent, low-dimensional model. As a result of the rigid connection between the frame structure and the protection system, the low-dimensional model includes only three degrees of freedom: two displacements that represent the motion of the frame structure, which is rigidly connected to the external structure, while the third characterizes the motion of the vibro-impacting mass. For the vibro-impacting nonlinear energy sink, an ideal model, which assumes instantaneous impacts, is used for the vibro-impacting mass. The proposed model is used for an in-depth parametric analysis, and the outcomes are presented in gain maps that illustrate the effectiveness of the coupling within a designated parameter plane. The findings demonstrate that the coupling with the external structure, which is equipped with a vibro-impacting mass, effectively mitigates displacements and drifts in the frame structure across a broad range of parameter values that define the protection system.

Multi-Dimensional Modeling of Cerebral Hemodynamics: A Systematic Review.

The Circle of Willis (CoW) describes the arterial system in the human brain enabling the neurovascular blood supply. Neurovascular diseases like intracranial aneurysms (IAs) can occur within the CoW and carry the risk of rupture, which can lead to subarachnoid hemorrhage. The assessment of hemodynamic information in these pathologies is crucial for their understanding regarding detection, diagnosis and treatment. Multi-dimensional in silico approaches exist to evaluate these hemodynamics based on patient-specific input data. The approaches comprise low-scale (zero-dimensional, one-dimensional) and high-scale (three-dimensional) models as well as multi-scale coupled models. The input data can be derived from medical imaging, numerical models, literature-based assumptions or from measurements within healthy subjects. Thus, the most realistic description of neurovascular hemodynamics is still controversial. Within this systematic review, first, the models of the three scales (0D, 1D, 3D) and second, the multi-scale models, which are coupled versions of the three scales, were discussed. Current best practices in describing neurovascular hemodynamics most realistically and their clinical applicablility were elucidated. The performance of 3D simulation entails high computational expenses, which could be reduced by analyzing solely the region of interest in detail. Medical imaging to establish patient-specific boundary conditions is usually rare, and thus, lower dimensional models provide a realistic mimicking of the surrounding hemodynamics. Multi-scale coupling, however, is computationally expensive as well, especially when taking all dimensions into account. In conclusion, the 0D-1D-3D multi-scale approach provides the most realistic outcome; nevertheless, it is least applicable. A 1D-3D multi-scale model can be considered regarding a beneficial trade-off between realistic results and applicable performance.

Credible School Value-Added with Undersubscribed School Lotteries

Abstract We introduce two empirical strategies harnessing the randomness in school assignment mechanisms to measure school value-added. The first estimator controls for the probability of school assignment, treating take-up as ignorable. We test this assumption using randomness in assignments. The second approach uses assignments as instrumental variables (IVs) for low-dimensional models of value-added and forms empirical Bayes posteriors from these IV estimates. Both strategies solve the underidentification challenge arising from school undersubscription. Models controlling for assignment risk and lagged achievement in Denver and New York City yield reliable value-added estimates. Estimates from models with lower-quality achievement controls are improved by IV.

Exact hierarchical reductions of dynamical models via linear transformations

Dynamical models described by ordinary differential equations (ODEs) are a fundamental tool in the sciences and engineering. Exact reduction aims at producing a lower-dimensional model in which each macro-variable can be directly related to the original variables, and it is thus a natural step towards the model’s formal analysis and mechanistic understanding. We present an algorithm which, given a polynomial ODE model, computes a longest possible chain of exact linear reductions of the model such that each reduction refines the previous one, thus giving a user control of the level of detail preserved by the reduction. This significantly generalizes over the existing approaches which compute only the reduction of the lowest dimension subject to an approach-specific constraint. The algorithm reduces finding exact linear reductions to a question about representations of finite-dimensional algebras. We provide an implementation of the algorithm, demonstrate its performance on a set of benchmarks, and illustrate the applicability via case studies. Our implementation is freely available at https://github.com/x3042/ExactODEReduction.jl.

Tropical forest characterisation using parametric SAR tomography at P band and low-dimensional models

Tropical forest characterisation using parametric SAR tomography at P band and low-dimensional models

Low-Dimensional Equivalent Models and Multithreading-Based Parallel EMT Simulation Method for Multi-converter Systems

Low-Dimensional Equivalent Models and Multithreading-Based Parallel EMT Simulation Method for Multi-converter Systems

A Multi-Model Framework to Explore ADHD Diagnosis from Neuroimaging Data

Attention Deficit Hyperactivity Disorder (ADHD) is a frequent neurodevelopmental disorder in children that is commonly diagnosed subjectively. The objective detection of ADHD based on neuroimaging data has been a complex problem with low ranges of accuracy, possibly due to (among others) complex diagnostic processes, the high number of features considered and imperfect measurements in data collection. Hence, reliable neuroimaging biomarkers for detecting ADHD have been elusive. To address this problem we consider a recently proposed multi-model selection method called Sparse Wrapper AlGorithm (SWAG), which is a greedy algorithm that combines screening and wrapper approaches to create a set of low-dimensional models with good predictive power. While preserving the previous levels of accuracy, SWAG provides a measure of importance of brain regions for identifying ADHD. Our approach also provides a set of equally-performing and simple models which highlight the main feature combinations to be analyzed and the interactions between them. Taking advantage of the network of models resulting from this approach, we confirm the relevance of the frontal and temporal lobes as well as highlight how the different regions interact to detect the presence of ADHD. In particular, these results are fairly consistent across different learning mechanisms employed within the SWAG (i.e. logistic regression, linear and radial-kernel support vector machines) thereby providing population-level insights, as well as delivering feature combinations that are smaller and often perform better than those that would be used if employing their original versions directly.

Uncovering acoustic signatures of pore formation in laser powder bed fusion

We present a machine learning workflow to discover signatures in acoustic measurements that can be utilized to create a low-dimensional model to accurately predict the location of keyhole pores formed during additive manufacturing processes. Acoustic measurements were sampled at 100 kHz during single-layer laser powder bed fusion (LPBF) experiments, and spatio-temporal registration of pore locations was obtained from post-build radiography. Power spectral density (PSD) estimates of the acoustic data were then decomposed using non-negative matrix factorization with custom varvec{k}-means clustering (NMFvarvec{k}) to learn the underlying spectral patterns associated with pore formation. NMFvarvec{k} returned a library of basis signals and matching coefficients to blindly construct a feature space based on the PSD estimates in an optimized fashion. Moreover, the NMFvarvec{k} decomposition led to the development of computationally inexpensive machine learning models which are capable of quickly and accurately identifying pore formation with classification accuracy of supervised and unsupervised label learning greater than 95% and 90%, respectively. The intrinsic data compression of NMFk, the relatively light computational cost of the machine learning workflow, and the high classification accuracy makes the proposed workflow an attractive candidate for edge computing toward in-situ keyhole pore prediction in LPBF.

Lower-Dimensional Model of the Flow and Transport Processes in Thin Domains by Numerical Averaging Technique

In this work, we present a lower-dimensional model for flow and transport problems in thin domains with rough walls. The full-order model is given for a fully resolved geometry, wherein we consider Stokes flow and a time-dependent diffusion–convection equation with inlet and outlet boundary conditions and zero-flux boundary conditions for both the flow and transport problems on domain walls. Generally, discretizations of a full-order model by classical numerical schemes result in very large discrete problems, which are computationally expensive given that sufficiently fine grids are needed for the approximation. To construct a computationally efficient numerical method, we propose a model-order-reduction numerical technique to reduce the full-order model to a lower-dimensional model. The construction of the lower-dimensional model for the flow and the transport problem is based on the finite volume method and the concept of numerical averaging. Numerical results are presented for three test geometries with varying roughness of walls and thickness of the two-dimensional domain to show the accuracy and applicability of the proposed scheme. In our numerical simulations, we use solutions obtained from the finite element method on a fine grid that can resolve the complex geometry at the grid level as the reference solution to the problem.

Data driven models of the Madden-Julian Oscillation: understanding its evolution and ENSO modulation

Understanding the physical mechanisms of the Madden-Julian Oscillation (MJO) and its evolution is a major concern within the climate community. Its main importance relies on its ability to act as a source of predictability within the intra-seasonal time-scale in tropical and extratropical regions, therefore filling the gap between weather and climate forecasts. However, most atmospheric general circulation models fail to correctly represent MJO’s evolution, and their prediction skills are still far from MJO’s theoretical predictability. In this work we infer low dimensional models of the MJO from data by applying a recently developed machine learning technique, the Sparse Identification of Non-linear Dynamics (SINDy). We use the daily-mean outgoing longwave radiation MJO index (OMI) as input data to infer bi-dimensional climatological models of the MJO, and analyse the inferred models during El Niño and La Niña years. This approach allows us to diagnose the MJO’s behaviour in OMI’s phase space. Our results show that MJO can be most frequently represented by a harmonic oscillator, which represents the MJO’s eastward propagation and characteristic period. Upon this basic oscillatory behaviour, we find that small non-linear corrections play a fundamental role in representing MJO’s non-uniform speed of propagation, explaining its acceleration over the Pacific Ocean region. Particularly, we find that MJO’s evolution is most frequently non-linear [linear] during El Niño [La Niña] years. Overall, our work shows that SINDy can robustly model MJO’s evolution as a linear oscillator with small non-linear corrections, contributing to understand the MJO’s dynamics and dependency on El Niño-Southern Oscillation.