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Gajamannage K, Jayathilake DI, Park Y, Bollt EM. Recurrent neural networks for dynamical systems: Applications to ordinary differential equations, collective motion, and hydrological modeling. Chaos 2023; 33:013109. [PMID: 36725658 PMCID: PMC9822653 DOI: 10.1063/5.0088748] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/20/2022] [Accepted: 12/07/2022] [Indexed: 06/18/2023]
Abstract
Classical methods of solving spatiotemporal dynamical systems include statistical approaches such as autoregressive integrated moving average, which assume linear and stationary relationships between systems' previous outputs. Development and implementation of linear methods are relatively simple, but they often do not capture non-linear relationships in the data. Thus, artificial neural networks (ANNs) are receiving attention from researchers in analyzing and forecasting dynamical systems. Recurrent neural networks (RNNs), derived from feed-forward ANNs, use internal memory to process variable-length sequences of inputs. This allows RNNs to be applicable for finding solutions for a vast variety of problems in spatiotemporal dynamical systems. Thus, in this paper, we utilize RNNs to treat some specific issues associated with dynamical systems. Specifically, we analyze the performance of RNNs applied to three tasks: reconstruction of correct Lorenz solutions for a system with a formulation error, reconstruction of corrupted collective motion trajectories, and forecasting of streamflow time series possessing spikes, representing three fields, namely, ordinary differential equations, collective motion, and hydrological modeling, respectively. We train and test RNNs uniquely in each task to demonstrate the broad applicability of RNNs in the reconstruction and forecasting the dynamics of dynamical systems.
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Affiliation(s)
- K. Gajamannage
- Department of Mathematics and Statistics, Texas A&M University—Corpus Christi, Corpus Christi, Texas 78412, USA
| | - D. I. Jayathilake
- Department of Physical and Environmental Sciences, Texas A&M University—Corpus Christi, Corpus Christi, Texas 78412, USA
| | - Y. Park
- Department of Mathematics and Statistics, Texas A&M University—Corpus Christi, Corpus Christi, Texas 78412, USA
| | - E. M. Bollt
- Department of Electrical and Computer Engineering and The Clarkson Center for Complex Systems, Clarkson University, Potsdam, New York 13699, USA
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Pilkiewicz KR, Lemasson BH, Rowland MA, Hein A, Sun J, Berdahl A, Mayo ML, Moehlis J, Porfiri M, Fernández-Juricic E, Garnier S, Bollt EM, Carlson JM, Tarampi MR, Macuga KL, Rossi L, Shen CC. Decoding collective communications using information theory tools. J R Soc Interface 2020; 17:20190563. [PMID: 32183638 PMCID: PMC7115225 DOI: 10.1098/rsif.2019.0563] [Citation(s) in RCA: 19] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2019] [Accepted: 02/28/2020] [Indexed: 02/03/2023] Open
Abstract
Organisms have evolved sensory mechanisms to extract pertinent information from their environment, enabling them to assess their situation and act accordingly. For social organisms travelling in groups, like the fish in a school or the birds in a flock, sharing information can further improve their situational awareness and reaction times. Data on the benefits and costs of social coordination, however, have largely allowed our understanding of why collective behaviours have evolved to outpace our mechanistic knowledge of how they arise. Recent studies have begun to correct this imbalance through fine-scale analyses of group movement data. One approach that has received renewed attention is the use of information theoretic (IT) tools like mutual information, transfer entropy and causation entropy, which can help identify causal interactions in the type of complex, dynamical patterns often on display when organisms act collectively. Yet, there is a communications gap between studies focused on the ecological constraints and solutions of collective action with those demonstrating the promise of IT tools in this arena. We attempt to bridge this divide through a series of ecologically motivated examples designed to illustrate the benefits and challenges of using IT tools to extract deeper insights into the interaction patterns governing group-level dynamics. We summarize some of the approaches taken thus far to circumvent existing challenges in this area and we conclude with an optimistic, yet cautionary perspective.
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Affiliation(s)
- K. R. Pilkiewicz
- Environmental Laboratory, U.S. Army Engineer Research and Development Center (EL-ERDC), Vicksburg, MS, USA
| | | | - M. A. Rowland
- Environmental Laboratory, U.S. Army Engineer Research and Development Center (EL-ERDC), Vicksburg, MS, USA
| | - A. Hein
- National Oceanic and Atmospheric Administration, Santa Cruz, CA, USA
- University of California, Santa Cruz, CA, USA
| | - J. Sun
- Department of Mathematics, Clarkson University, Potsdam, NY, USA
| | - A. Berdahl
- School of Aquatic and Fishery Sciences, University of Washington, Seattle, WA, USA
| | - M. L. Mayo
- Environmental Laboratory, U.S. Army Engineer Research and Development Center (EL-ERDC), Vicksburg, MS, USA
| | - J. Moehlis
- Department of Mechanical Engineering, University of California, Santa Barbara, CA, USA
| | - M. Porfiri
- Department of Mechanical and Aerospace Engineering and Department of Biomedical Engineering, New York University Tandon School of Engineering, Brooklyn, NY, USA
| | | | - S. Garnier
- Department of Biological Sciences, New Jersey Institute of Technology, Newark, NJ, USA
| | - E. M. Bollt
- Department of Mathematics, Clarkson University, Potsdam, NY, USA
| | - J. M. Carlson
- Department of Physics, University of California, Santa Barbara, CA, USA
| | - M. R. Tarampi
- Department of Psychology, University of Hartford, West Hartford, CT, USA
| | - K. L. Macuga
- School of Psychological Science, Oregon State University, Corvallis, OR, USA
| | - L. Rossi
- Department of Mathematical Sciences, University of Delaware, Newark, DE, USA
| | - C.-C. Shen
- Department of Computer and Information Sciences, University of Delaware, Newark, DE, USA
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Santitissadeekorn N, Bollt EM. The infinitesimal operator for the semigroup of the Frobenius-Perron operator from image sequence data: vector fields and transport barriers from movies. Chaos 2007; 17:023126. [PMID: 17614680 DOI: 10.1063/1.2742932] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
In this paper, we present an approach to approximate the Frobenius-Perron transfer operator from a sequence of time-ordered images, that is, a movie dataset. Unlike time-series data, successive images do not provide a direct access to a trajectory of a point in a phase space; more precisely, a pixel in an image plane. Therefore, we reconstruct the velocity field from image sequences based on the infinitesimal generator of the Frobenius-Perron operator. Moreover, we relate this problem to the well-known optical flow problem from the computer vision community and we validate the continuity equation derived from the infinitesimal operator as a constraint equation for the optical flow problem. Once the vector field and then a discrete transfer operator are found, then, in addition, we present a graph modularity method as a tool to discover basin structure in the phase space. Together with a tool to reconstruct a velocity field, this graph-based partition method provides us with a way to study transport behavior and other ergodic properties of measurable dynamical systems captured only through image sequences.
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Affiliation(s)
- N Santitissadeekorn
- Department of Mathematics and Computer Science, Clarkson University, Potsdam, New York 13699-5815, USA
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Bollt EM, Stanford T, Lai YC, Zyczkowski K. Validity of threshold-crossing analysis of symbolic dynamics from chaotic time series. Phys Rev Lett 2000; 85:3524-3527. [PMID: 11030937 DOI: 10.1103/physrevlett.85.3524] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2000] [Revised: 08/07/2000] [Indexed: 05/23/2023]
Abstract
A practical and popular technique to extract the symbolic dynamics from experimentally measured chaotic time series is the threshold-crossing method, by which an arbitrary partition is utilized for determining the symbols. We address to what extent the symbolic dynamics so obtained can faithfully represent the phase-space dynamics. Our principal result is that such a practice can lead to a severe misrepresentation of the dynamical system. The measured topological entropy is a Devil's staircase-like, but surprisingly nonmonotone, function of a parameter characterizing the amount of misplacement of the partition.
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Affiliation(s)
- EM Bollt
- Mathematics Department, 572 Holloway Road, U.S. Naval Academy, Annapolis, Maryland 21402-5002, USA
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Davidchack RL, Lai YC, Bollt EM, Dhamala M. Estimating generating partitions of chaotic systems by unstable periodic orbits. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 2000; 61:1353-1356. [PMID: 11046413 DOI: 10.1103/physreve.61.1353] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/1999] [Indexed: 05/23/2023]
Abstract
An outstanding problem in chaotic dynamics is to specify generating partitions for symbolic dynamics in dimensions larger than 1. It has been known that the infinite number of unstable periodic orbits embedded in the chaotic invariant set provides sufficient information for estimating the generating partition. Here we present a general, dimension-independent, and efficient approach for this task based on optimizing a set of proximity functions defined with respect to periodic orbits. Our algorithm allows us to obtain the approximate location of the generating partition for the Ikeda-Hammel-Jones-Moloney map.
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Affiliation(s)
- RL Davidchack
- Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045, USA
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