Dealing with Dynamics through a Data-informed Dynamical Systems Theory

Address the presence of significant non-linearities, oscillations, and other complex dynamics and feedback loops: We will employ dynamical systems, real algebraic geometry, and topology to develop data-informed algorithms that satisfy mathematical constraints during operation of intelligent machines and provide performance guarantees in terms of safety, efficiency, and robustness.

Research Highlights

1. A Novel Combinatorial Framework Based on Topological Data Analysis

Due to its robustness and flexibility Topological Data Analysis (TDA) is ideal to extract reliable information about the dynamics of the underlying system obtained from limited and sparse data, such as robot trajectories or GPS-based systems. In general, TDA methods solve problems without the need to tune parameters or relying on predetermined models, enabling accurate predictions even with inaccurate models. This feature is explored in a study where sparse data was used to identify dynamic behaviors like fixed points, periodic orbits, connecting orbits, bistability, and chaotic dynamics. A novel combinatorial framework based on TDA is used to estimate Regions of Attraction (RoAs) in data-driven controllers, where verification is crucial for understanding the conditions under which a controller can be safely applied to solve a task. Compared to traditional methods based on numerical approximations of Lyapunov functions, this method offers greater data efficiency and improved accuracy. Moreover, its versatility makes it applicable not only for analyzing dynamics but also for synthesizing controllers to enhance hybrid solutions and identify the physical limitations of robotic systems.

2. Data Informed Dynamical Systems

Modern dynamical systems are often understood through data rather than first principals. By modelling data through a special class of Gaussian processes–Brownian motion–we were able to provide a precise probability that a given topological characterization of the dynamics was accurate. While this setting is limited, it gives us a rigorous theoretical understanding of the interplay between observed data and long-term dynamics, setting the groundwork for future, and more general, studies.

3. Topological Tools for Analyzing the Global Dynamics of Robot Controllers

Results of the combinatorial analysis for a pendulum operating under a learned controller so as to reach the (0,0) state in the phase space.
(left) The method outputs the Morse graph. Nodes identify recurrent dynamics and edges indicate the directions of nonrecurrent dynamics.
(right) The colored regions indicate regions of attraction. The color coding matches the regions of phase space with the recurrent information of the Morse graph.
For safe deployment of robots or synthesis of effective hybrid controllers it is important to be able to understand the global dynamics. Of particular interest is the identification of attractors and their regions of attraction. We are developing a topological framework to provide an effective and explainable analysis the global dynamics of robot controllers (including data driven controllers). Our approach probes the dynamics locally by forward propagating short trajectories over a state-space discretization, and from this information builds a combinatorial representation of the underlying system’s state space and non-linear dynamics.
This representation is summarized via a directed acyclic graph, called a Morse graph (see Figure 1), that provides insights about attractors and their regions of attraction (see Figure 2) and more generally the recurrent and nonrecurrent dynamics.
We use the Morse graph to identify physical limitations of the robotic system and/or to identify how to synthesize controllers to form improved hybrid controllers.

Papers

  • Factor model for high-dimensional tensor time series (with discussion), by R. Chen, D. Yang, and C.H. Zhang in Journal of American Statistical Association, 117, 94-132.
  • Generalized autoregressive moving average models with GARCH errors, by T. Zheng, H. Xiao, and R. Chen, in Journal of Time Series Analysis, 43, 125-146.
  • Rank determination in tensor factor model, by Y. Han, R. Chen, and C.-H. Zhang, in Electronic Journal of Statistics, 16, 1726-1803.
  • KoPA: Automated Kronecker product approximation, by C. Cai, R. Chen, and H. Xiao, in Journal of Machine Learning Research, 2023, 23, 1-44.
  • Modeling dynamic transport network with matrix factor models: an application to international trade flow, by Y. Chen, and R. Chen, in J. Data Science, 2022, 1-18. in press also at [pdf]
  • Forecasting the U.S. unemployment rate: another look, by H. Xiao, R. Chen, and J. Guerard, in Wilmott, 2022, 122.
  • Individualized group learning, by C. Cai, M. Xie, and R. Chen, in Journal of American Statistical Association, 2023, 118, 622-638.
  • Composite index construction with expert opinion, by R. Chen, Y. Ji, G. Jiang, R. Xie, and P. Zhu, in Journal of Business & Economic Statistics, 2023, 41, 67-79.
  • Hybrid Kronecker product decomposition and approximation, by C. Cai, R. Chen, and H. Xiao, in J. Computational and Graphical Statistics, 2023, 32, 838-852.
  • Reduced rank autoregressive models for matrix time series, by H. Xiao, Y. Han, and R. Chen, in Journal of Business & Economic Statistics, 2023+, in press.
  • Simultaneous decorrelation of matrix time series, by Y. Han, R. Chen, C.-H. Zhang, and Q. Yao, in Journal of American Statistical Association, 2023+, in press. also at [pdf]
  • Resampling strategy in Sequential Monte Carlo for constrained sampling problems, by C. Cai, M. Lin, and R. Chen, in Statistica Sinica, 2023+, in press.
    also at [pdf]
  • State space emulation and annealed Sequential Monte Carlo for High Dimensional Optimization, by C. Cai, and R. Chen, in Statistica Sinica, 2023+, in press.
    also at [pdf]
  • Computing the Conley Index: a Cautionary Tale, by K. Mischaikow and C. Weibel, in SIAM Applied Algebra and Geometry, 2023.
  • Identifying Nonlinear Dynamics with High Confidence from Sparse Data, by Bogdan Batko, Marcio Gameiro, Ying Hung, William Kalies, Konstantin Mischaikow, and Ewerton Vieira, in SIAM Journal on Applied Dynamical Systems, 2023.
  • Conditioned Weiner Processes as Nonlinearities: A Rigorous Probabilistic Analysis of Dynamics by Konstantin Mischaikow and Cameron Thieme in Journal of Computational Dynamics, 2023, 10(3): 371-386. doi: 10.3934/jcd.2023004.
  • Data-Efficient Characterization of the Global Dynamics of Robot Controllers with Confidence Guarantees by E. Vieira, A. Sivaramakrishnan, Yao Song, E. Granados, M. Gameiro, K. Mischaikow, Ying Hung and K. E. Bekris in International Conference on Robotics and Automation , (ICRA) 2023.
  • Identifying Nonlinear Dynamics with High Confidence from Sparse Data by B. Botko, M. Gameiro, Y. Hung, W. Kalies, K. Mischaikow and E. Vieira in SIAM Journal on Applied Dynamical Systems to appear.
  • Effective non-prehensile manipulation via persistent homology & Monte-Carlo tree search by Ewerton Vieira, Kai Gao, Daniel Nakhimovich, Kostas E. Bekris, and Jingjin Yu in 18th International Symposium of Experimental Robotics (ISER 2023), 2023.
  • Morse Graphs: Topological Tools for Analyzing the Global Dynamics of Robot Controllers , by E. Vieira, E. Granados, A. Sivaramakrishnan, M. Gameiro, K. Mischaikow and K. E. Bekris submitted to The International Journal of Robotics Research (IJRR).
  • Identifying Nonlinear Dynamics with High Confidence from Sparse Data, by Batko, B., Gameiro, M., Hung, Y., Kalies, W, Mischaikow, K, and Vieira, Ewerton,
    arXiv, (2022}, https://doi.org/10.48550/arxiv.2206.13779.
  • Data-Efficient Characterization of the Global Dynamics of Controllers with Confidence Guarantees, by Ewerton R. Vieira, Edgar Granados, Aravind Sivaramakrishnan, Yao Song, Marcio Gameiro, Ying Hung, Konstantin Mischaikow, Kostas E. Bekris, preprint.
  • Morse Graphs: Topological Tools for Analyzing the Global Dynamics of Robot Controllers by Ewerton Vieira, Edgar Granados, Aravind Sivaramakrishnan, Marcio Gameiro, Konstantin Mischaikow, and Kostas E. Bekris, WAFR 2022.
  • Persistent homology for effective non-prehensile manipulation by Ewerton Vieira, Daniel Nakhimovich, Kai Gao, Rui Wang, Jingjin Yu, and Kostas E. Bekris, IEEE International Conference on Robotics and Automation (ICRA), 2022.
  • Covering action on Conley index theory by D.V.S. Lima, M.R. Da Silveira, E.R. Vieira, ErgodicTheory and Dynamical Systems 1-33. doi:10.1017/etds.2022.13 (2022).
  • Subspace Differential Privacy, by Jie Gao, Ruobin Gong, Fang-Yi Yu, Proceedings of the 36th AAAI Conference on Artificial Intelligence (AAAI-22), February 22 – March 1st, 2022. [pdf]
  • Persistent homology with non-contractible preimages, by Konstantin Mischaikow and Charles Weibel, Homology, Homotopy and Application, accepted, 2021.

  • Equilibria and their Stability in Networks with Steep Sigmoidal Nonlinearities, by W. Duncan, T. Gedeon, H. Kokubu, K. Mischaikow, and H. Oka, SIADS 20, 2021, p. 2108-2141.

  • Lattice Structures for Attractors III, by William D. Kalies, Konstantin Mischaikow, and Robert C. A. M. Vandervorst, Journal Dynamics Differential Equations, accepted, 2021,

  • A computational framework for the connection matrix theory, by Shaun Harker, Konstantin Mischaikow, and Kelly Spendlove, Journal of Applied and Computational Topology 5, 2021, p. 459–529.
  • Rational design of complex phenotype via network models, by Marcio Gameiro, Tomas Gedeon, Shane Kepley, and Konstantin Mischaikow, PLoS Computational Biology 17(7), 2021, e1009189.
  • Computing Linear Extensions for Polynomial Posets Subject to Algebraic Constraints, by Shane Kepley, Konstantin Mischaikow, and Lun Zhang, SIAM J. Appl. Algebra Geom. 5, 2021, 388–416.
  • Quantitative measure of memory loss in complex spatiotemporal systems, by Miroslav Kramar, Lenka Kovalvˇinova ́, Konstantin Mischaikow and Lou Kondic, Chaos, 31, 033126, DOI: 10.1063/5.0033419, March 2021.
  • Mapping parameter spaces of biological switches, by Rocky Diegmiller, Lun Zhang, Marcio Gameiro, Justin Barr, Jasmin Imran Alsous, Paul Schedl, Stanislav Y. Shvartsman, and Konstantin Mischaikow, PLoS Computational Biology, 17 e1008711, DOI: 10.1371/journal.pcbi.1008711, Feb. 2021.
  • Contractibility of a persistence map preimage, by Jacek Cyranka, Konstantin Mischaikow, and Charles Weibel, Journal of Applied and Computational Topology, 4 (2020) 509-523.
  • Interaction network analysis in shear thickening suspensions, by Marcio Gameiro, Abhinendra Singh, Lou Kondic, Konstantin Mischaikow, and Jeffrey F. Morris, Physical Review Fluids, 5:034307 2020.
  • Combinatorial models of global dynamics: learning cycling motion from data, by Ulrich Bauer, David Hien, Oliver Junge, Konstantin Mischaikow, and Max Snijders, ENOC2020: 10th European Nonlinear Dynamics Conference, 2020, Accepted.
  • Conley index approach to sampled dynamics, by Bogdan Batko, Konstantin Mischaikow, Marian Mrozek and Mateusz Przybylski, SIADS, 19, DOI10.1137/19M1254404 (2020) 665-704.
  • Safe and Effective Picking Paths in Clutter given Discrete Distributions of Object Poses, by Rui Wang, Chaitanya Mitash, Shiyang Lu, Daniel Boehm, and Kostas E. Bekris, in IEEE/RSJ International Confernece on Intelligent Robots and Systems (IROS), 2020, [pdf] [bib]

  • Refined Analysis of Asymptotically-Optimal Kinodynamic Planning in the State-Cost Space, by Michal Kleinbort, Edgar Granados, Kiril Solovey, Riccardo Bonalli, Kostas E. Bekris, and Dan Halperin, in IEEE International Conference on Robotics and Automation (ICRA), 2020, [pdf] [bib]

  • That and There: Judging the Intent of Pointing Actions with Robotic Arms, by Malihe Alikhani, Baber Khalid, Rahul Shome, Chaitanya Mitash, Kostas Bekris and Matthew Stone, in Proceedings of AAAI, 2020, [pdf] [git] [bib]

  • Synchronized Multi-Arm Rearrangement Guided by Mode Graphs with Capacity Constraints, by Rahul Shome and Kostas E. Bekris, in The 14th International Workshop on the Algorithmic Foundations of Robotics (Wafr), 2020, [pdf] [bib]

  • Pushing the Boundaries of Asymptotic Optimality in Integrated Task and Motion Planning, by Rahul Shome, Daniel Nakhimovich and Kostas E. Bekris, in The 14th International Workshop on the Algorithmic Foundations of Robotics (Wafr), 2020, [pdf] [bib]

  • Factor Models for High-Dimensional Tensor Time Series, by Rong Chen, Dan Yang, and Cun-Hui Zhang, submitted to Journal of American Statistical Association, 2020, [pdf] [bib]

  • Factor Models for High-Dimensional Tensor Time Series, by Rong Chen, Dan Yang, and Cun-hui Zhang, arXiv preprint, 2020, [pdf] [bib]

  • Tensor Factor Model Estimation by Iterative Projection, by Yuefeng Han, Rong Chen, Dan Yang, and Cun-Hui Zhang, arXiv preprint, 2020, [pdf] [bib]

  • Autoregressive models for matrix-valued time series, by Rong Chen, Han Xiao, and Dan Yang, in Journal of Econometrics, Elsevier, 2020, [pdf] [bib]

  • KoPA: Automated Kronecker Product Approximation, by Chencheng Cai, Rong Chen, and Han Xiao, arXiv preprint, 2020, [pdf] [bib]

  • Hybrid Kronecker Product Decomposition and Approximation, by Chencheng Cai, Rong Chen, and Han Xiao, arXiv preprint, 2020, [pdf] [bib]

  • Stable Matrix Completion using Properly Configured Kronecker Product Decomposition, by Chencheng Cai, Rong Chen, and Han Xiao, arXiv preprint, 2020, [pdf] [bib]

  • Individualized Group Learning, by Chencheng Cai, Rong Chen, and Min-ge Xie, arXiv preprint, 2020, [pdf] [bib]

  • Threshold factor models for high-dimensional time series, by Xialu Liu and Rong Chen, in Journal of Econometrics, Elsevier, 2020, [pdf] [bib]

Preprints:

  • CP factor model for dynamic tensors, by Y. Han, C.-H. Zhang, and R. Chen, in Journal of American Statistical Association, under revision.
  • Multivariate spatial-temporal modeling with latent low-dimensional dynamics, by Y. Chen, X. Yun, R. Chen, and Q. Yao, in Journal of American Statistical Association, under revision. also at pdf
  • Tensor factor model estimation by iterative projection, by Y. Han, R. Chen, D. Yang, and C.-H. Zhang, in Annal of Statistics, under revision. also at pdf
  • Matrix completion using Kronecker product approximation, by C. Cai, R. Chen, and H. Xiao, in Journal of the Royal Statistical Society, Series B, under revision. also at pdf
  • MORALS: Analysis of High-Dimensional Robot Controllers via Topological Tools in a Latent Space by E. Vieira, A. Sivaramakrishnan, E. Granados, S. Tangirala, K. Mischaikow, and K. E. Bekris, Submitted.
  • Global Dynamics of Ordinary Differential Equations: Rook Fields, Connection Matrices, and Ramp System by W. Duncan, M. Gameiro, T. Gedeon, K. Kokubo, K. Mischaikow, H. Oka, B. Rivas and E. Vieira, in preparation.

Presentations

  • Kronecker Product Approximation for Matrix Approximation, Denoising and Completion – by Rong Chen, Hong Kong University, 2023.
  • New Approaches to Statistical Learning of Modern Time Series – by Rong Chen, Hong Kong University, 2023.
  • Dynamic Factor Model for Matrix/Tensor Time Series – by Rong Chen, Hong Kong University of Science and Technology, 2023.
  • Dynamic Factor Model for Matrix/Tensor Time Series – by Rong Chen, Joint Statistical Meeting, Toronto, 2023.
  • Time Varying Dynamic Transport Networks and Matrix Factor Models – by Rong Chen, The 2023 Workshop on Statistical Network Analysis and Beyond, Anchorage, 2023.
  • Simultaneous Decorrelation of Matrix Time Series – by Rong Chen, New York University, 2023.
  • Simultaneous Decorrelation of Matrix Time Series – by Rong Chen, University of Rochester, 2023.
  • Functional Quantile Autoregression – by Rong Chen, Workshop on Emerging New Topics in Functional Data Analysis, Singapore, 2023.
  • From Regulatory Networks to Dynamics of ODEs – by K. Mischaikow, Lorentz Center, Leiden, Netherlands, October 2023.
  • Identifying Nonlinear Dynamics from Sparse Data – by K. Mischaikow, KiPAS Dynamics Days, Tokyo, Japan, August 2023.
  • A combinatorial/homological framework for continuous nonlinear dynamics – by K. Mischaikow, International Congress Industrial and Applied Mathematics, Tokyo, Japan, August 2023.
  • Identifying Nonlinear Dynamics from Sparse Data – by K. Mischaikow, TDA week 2023, Kyoto, Japan, August 2023.
  • Algorithmic approach to the global dynamics of multi-parameter systems of ODEs – by K. Mischaikow, FoCM, Paris, June 2023.
  • Identifying Nonlinear Dynamics with High Confidence from Sparse Data – by K. Mischaikow, VU Amsterdam, June 2023.
  • Identifying Nonlinear Dynamics with High Confidence from Sparse Data – by K. Mischaikow, Flatiron Institute, April 2023.
  • Identifying Nonlinear Dynamics with High Confidence from Sparse Data – by K. Mischaikow, Boston University, April 2023.
  • Identifying Nonlinear Dynamics with High Confidence from Sparse Data, Randomness in Topology and its Applications – by K. Mischaikow, Institute for Mathematical and Statistical Innovation, Chicago, March 2023.
  • Computing the Global Dynamics of Parameterized Systems of ODEs – by K. Mischaikow, Minicourse at the 13th Americas Conference on Diff. Equations and Nonlinear Analysis and ICMC Summer Meeting on Differential Equations – 2023 Chapter, Sao Carlos, Brazil, February 2023.
  • Towards an algorithm for identifying the global dynamics of multi-parameter systems of ordinary differential equations – by K. Mischaikow, Discretization in Geometry and Dynamics, Magdeburg, Germany, October 2022.
  • Identifying Nonlinear Dynamics with High Confidence from Sparse Data – by K. Mischaikow, International Workshop on Dynamics, Optimization, and Computation, Paderborn, Germany, September 2022.
  • Conditioned Wiener processes as nonlinearities: A rigorous probabilistic analysis of dynamics – by Konstantin Mischaikow, Cameron Thieme, at the Third Symposium for Machine Learning and Dynamical Systems at the Fields Institute, September 26-30, 2022.
  • DATA INSPIRE: Morse Graphs for Robot Controllers – by E. R. Vieira HDR2 From Harnessing the Data Revolution (DR) to Harvesting DR, Alexandria, VA, USA.
  • Data-Efficient Characterization of the Global Dynamics of Robot Controllers with Confidence Guarantees, – by E. R. Vieira International Conference on Robotics and Automation (ICRA), 2023, London, UK.
  • Combinatorial description of global dynamics via piecewise linear models – by E. R. Vieira, January 22-24, 2023 at the University of São Paulo, São Carlos SP Brazil.
  • Moving Beyond ODE Analysis: Combinatorics and Algebraic Topology for Understanding Global Dynamics in Multi-Scale Systems – by E. R. Vieira, ICIAM (International Conference on Industrial and Applied Mathematics) 2023 TOKYO, August 20-25, 2023, Waseda University, Tokyo, Japan.
  • Conley Complexes for Parameterized ODE Systems. – by E. R. Vieira, Computational Mathematics Seminar, Krakow, Poland.
  • Identifying Nonlinear Dynamics with High Confidence from Sparse Time Series Data, Applied Topology in Frontier Sciences – talk by Konstantin Mischaikow, online, July 2022
  • Solving Systems of Ordinary Differential Equations via Combinatorial Homological Algebra – talk by Konstantin Mischaikow, DyToComp, Bedlewo, Poland, June 2022
  • Identifying Nonlinear Dynamics with High Confidence from Sparse Time Series Data – talk by Konstantin Mischaikow, AMSS-YMSC-BIMSA Joint Seminar on Progress of Topology and Its Applications, May 2022
  • We have Data and Computers, why do we need Math? – talk by Konstantin Mischaikow, Oklahoma University, April, 2022
  • We have Data and Computers, why do we need Math? – talk by Konstantin Mischaikow, RUMA, Rutgers, March, 2022
  • Solving Systems of Ordinary Differential Equations via Combinatorial Homological Algebra – talk by Konstantin Mischaikow, ICMC Summer Meeting on Differential Equations, U. Sao Paulo, Sao Carlos, Brazil, February 2022
  • Global Dynamics of Ordinary Differential Equations: Ramp Systems, Rook Fields, and Connection Matrices – talk by Konstantin Mischaikow, Dynamics Seminar, VU Amsterdam, November 2021
  • Global Dynamics of Ordinary Differential Equations: Ramp Systems, Rook Fields, and Connection Matrices – talk by Konstantin Mischaikow, Conf\’erence \`a la m\’emoire de Genevi\`eve Raugel, Paris, November 2021
  • Solving Systems of Ordinary Differential Equations via Combinatorial Homological Algebra, Beyond Topological Data Analysis – talk by Konstantin Mischaikow, online, August 2021
  • Global Dynamics of Ordinary Differential Equations: Ramp Systems, Rook Fields, and Connection Matrices, Dynamics Seminar – seminar talk by Konstantin Mischaikow at VU Amsterdam, November 2021
  • Global Dynamics of Ordinary Differential Equations: Ramp Systems, Rook Fields, and Connection Matrices – seminar talk by Konstantin Mischaikow at Conference a la memoire de Genevieve Raugel, Paris, November 2021
  • Solving Systems of Ordinary Differential Equations via Combinatorial Homological Algebra, Beyond Topological Data Analysis – seminar talk by Konstantin Mischaikow, online, August 2021
  • Data Driven Dynamics, Dynamics, Topology, and Robotic Control – seminar talk by Konstantin Mischaikow at Rutgers, May 2021
  • Identifying dynamics of networks, Hot Topics: Topological Insights in Neuroscience – – seminar talk by Konstantin Mischaikow at MSRI, Berkeley, May 2021
  • Identifying dynamics from finite data – seminar talk by Konstantin Mischaikow at SIP Seminars, Rutgers, March 2021
  • Understanding nonlinear dynamics with finite data – Konstantin Mischaikow, Mini-courses (Three lectures) Brummer & Partners, MathDataLab, KTH, Stockholm, March 2021
  • A Computationally Efficient Combinatorial Algebraic Topological Approach to Analyzing the Dynamics of Networks – talk by Konstantin Mischaikow, Short Course: Topological Data Analysis, DSOFT and GSNP APS March Meeting, 2021
  • Wherefore computer assisted proofs in dynamics – seminar talk by Konstantin Mischaikow at CRM-CAMP Colloquium, February 2021
  • Nonlinear Dynamics in an Age of Heuristic Science – seminar talk by Konstantin Mischaikow at Applied Topology Network, January 2021
  • DSGRN: An efficient tool for understanding regulatory networks – seminar talk by Konstantin Mischaikow at CQB, Rutgers, October 2020
  • Dynamic Clades: A coarse approach to nonlinear dynamics – seminar talk by Konstantin Mischaikow at DynamIC, Imperial College, London, October 2020
  • Nonlinear Dynamics for Data Driven Science – seminar talk by Konstantin Mischaikow at Second Symposium on MLDS, Fields Institute, September 2020
  • An Approach to Solving x’=? – colloquium by Konstantin Mischaikow at Cornell University, January 2020
  • An Approach to Solving x’=? – seminar talk by Konstantin Mischaikow at Rutgers, March 2020
  • An Approach to Solving x’=? – seminar talk by Konstantin Mischaikow at Yeshiva, April 2020
  • Data, Nonlinear Dynamics, and Algebraic Topology – workshop talk by Konstantin Mischaikow at CODATA-IIASA, February 2020
  • DSGRN: An efficient tool for understanding regulatory networks – seminar talk by Konstantin Mischaikow at ASHBi Distinguished Seminar, October 2019
  • The DSGRN Database: Identifying Network Function with Network Topology – seminar talk by Konstantin Mischaikow at Artificial Intelligence for Synthetic Biology, Arlington VA, November 2019
  • The DSGRN Database for Dynamics of Gene Regulatory Networks – seminar talk by Konstantin Mischaikow at Southeast Center for Mathematics and Biology, Georgia Tech, February 2020
  • Generating Motion for Adaptive Robotics – seminar talk by Kostas Bekris at Computational Robotics, AI and Biomedicine Lab, Rice University, June 25, 2020