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

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

  • 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, accepted to Workshop on the Algorithmic Foundations of Robotics (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]

Presentations

  • 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