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Dimensionality reduction of complex dynamical systems.

ABSTRACT: One of the outstanding problems in complexity science and engineering is the study of high-dimensional networked systems and of their susceptibility to transitions to undesired states as a result of changes in external drivers or in the structural properties. Because of the incredibly large number of parameters controlling the state of such complex systems and the heterogeneity of its components, the study of their dynamics is extremely difficult. Here we propose an analytical framework for collapsing complex N-dimensional networked systems into an S+1-dimensional manifold as a function of S effective control parameters with S << N. We test our approach on a variety of real-world complex problems showing how this new framework can approximate the system's response to changes and correctly identify the regions in the parameter space corresponding to the system's transitions. Our work offers an analytical method to evaluate optimal strategies in the design or management of networked systems.

SUBMITTER: Tu C

PROVIDER: S-EPMC7753969 | biostudies-literature | 2021 Jan

REPOSITORIES: biostudies-literature

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Dimensionality reduction of complex dynamical systems.

Tu Chengyi C D'Odorico Paolo P Suweis Samir S

iScience 20201209 1

One of the outstanding problems in complexity science and engineering is the study of high-dimensional networked systems and of their susceptibility to transitions to undesired states as a result of changes in external drivers or in the structural properties. Because of the incredibly large number of parameters controlling the state of such complex systems and the heterogeneity of its components, the study of their dynamics is extremely difficult. Here we propose an analytical framework for coll ...[more]

PMID: 33364591

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Project description:The prediction of extreme events, from avalanches and droughts to tsunamis and epidemics, depends on the formulation and analysis of relevant, complex dynamical systems. Such dynamical systems are characterized by high intrinsic dimensionality with extreme events having the form of rare transitions that are several standard deviations away from the mean. Such systems are not amenable to classical order-reduction methods through projection of the governing equations due to the large intrinsic dimensionality of the underlying attractor as well as the complexity of the transient events. Alternatively, data-driven techniques aim to quantify the dynamics of specific, critical modes by utilizing data-streams and by expanding the dimensionality of the reduced-order model using delayed coordinates. In turn, these methods have major limitations in regions of the phase space with sparse data, which is the case for extreme events. In this work, we develop a novel hybrid framework that complements an imperfect reduced order model, with data-streams that are integrated though a recurrent neural network (RNN) architecture. The reduced order model has the form of projected equations into a low-dimensional subspace that still contains important dynamical information about the system and it is expanded by a long short-term memory (LSTM) regularization. The LSTM-RNN is trained by analyzing the mismatch between the imperfect model and the data-streams, projected to the reduced-order space. The data-driven model assists the imperfect model in regions where data is available, while for locations where data is sparse the imperfect model still provides a baseline for the prediction of the system state. We assess the developed framework on two challenging prototype systems exhibiting extreme events. We show that the blended approach has improved performance compared with methods that use either data streams or the imperfect model alone. Notably the improvement is more significant in regions associated with extreme events, where data is sparse.

Project description:Oxidation of organic compounds in the atmosphere produces an immensely complex mixture of product species, posing a challenge for both their measurement in laboratory studies and their inclusion in air quality and climate models. Mass spectrometry techniques can measure thousands of these species, giving insight into these chemical processes, but the datasets themselves are highly complex. Data reduction techniques that group compounds in a chemically and kinetically meaningful way provide a route to simplify the chemistry of these systems but have not been systematically investigated. Here we evaluate three approaches to reducing the dimensionality of oxidation systems measured in an environmental chamber: positive matrix factorization (PMF), hierarchical clustering analysis (HCA), and a parameterization to describe kinetics in terms of multigenerational chemistry (gamma kinetics parameterization, GKP). The evaluation is implemented by means of two datasets: synthetic data consisting of a three-generation oxidation system with known rate constants, generation numbers, and chemical pathways; and the measured products of OH-initiated oxidation of a substituted aromatic compound in a chamber experiment. We find that PMF accounts for changes in the average composition of all products during specific periods of time but does not sort compounds into generations or by another reproducible chemical process. HCA, on the other hand, can identify major groups of ions and patterns of behavior and maintains bulk chemical properties like carbon oxidation state that can be useful for modeling. The continuum of kinetic behavior observed in a typical chamber experiment can be parameterized by fitting species' time traces to the GKP, which approximates the chemistry as a linear, first-order kinetic system. The fitted parameters for each species are the number of reaction steps with OH needed to produce the species (the generation) and an effective kinetic rate constant that describes the formation and loss rates of the species. The thousands of species detected in a typical laboratory chamber experiment can be organized into a much smaller number (10-30) of groups, each of which has a characteristic chemical composition and kinetic behavior. This quantitative relationship between chemical and kinetic characteristics, and the significant reduction in the complexity of the system, provides an approach to understanding broad patterns of behavior in oxidation systems and could be exploited for mechanism development and atmospheric chemistry modeling.

Dataset Information

Dimensionality reduction of complex dynamical systems.

Publications

Dimensionality reduction of complex dynamical systems.

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