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Dynamical Systems In Machine Learning

Public syllabus for 2025-2026

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Academic overview

Programme
BD
Period
Year 1, Semester 1
Credits
5
Weeks
14

Curriculum placement

Appears in study plans

BD (2025 - 2027) Year 1, Semester 1 Elective subject, group 2

Teaching team

Course coordinator
Eva Kaslik
Seminar coordinators
Eva Kaslik

Learning time distribution

Total
Curriculum Lecture Practice Total Weekly Lecture Practice
42 28 14 3 2 1
Exam hours
8
Individual Study Bibliography study Field study Homework Tutoring Others
83 20 10 40 5 0
Overall
125

Learning outcomes

Knowledge

  • Foundations of continuous and discrete dynamical systems theory.
  • Modeling machine learning algorithms as dynamical systems.
  • Nonlinear dynamics and chaos in machine learning models.
  • Dimensionality reduction techniques using dynamical systems.
  • Continuous and discrete-time systems applications in AI.
  • Computational tools for simulating dynamical systems.
  • Scientific research methodology at the intersection of dynamical systems and AI.

Skills

  • Modeling and analyzing machine learning algorithms through dynamical systems theory.
  • Implementing recurrent neural networks as dynamical systems.
  • Conducting stability and convergence analysis.
  • Applying dynamical systems for dimensionality reduction in high-dimensional datasets.
  • Using advanced simulation tools (e.g. Python) for dynamical system analysis.
  • Conducting research in machine learning with dynamical systems applications.

Responsibility

  • Adhere to ethical and professional standards in research.
  • Respect confidentiality and intellectual property in collaborations.
  • Commit to lifelong learning in dynamical systems and machine learning.
  • Maintain academic integrity and accurate representation of competencies.
  • Preserve autonomy and independence in professional opinions and research.

Online platform

(none)

Course content

Content Methods Obs
L1. Introduction. Definition and examples of dynamical systems. Continuous and discrete-time systems. Phase space, state space, and evolution law. Examples (gradient descent, etc.) Discourse, conversation, illustration by examples 2 hours
L2. Linear Continuous-Time Dynamical Systems. Autonomous ODEs and linear vector fields. Jordan normal form, spectral decomposition, and stability of linear systems. Invariant sets and subspaces. Examples (robotics, etc.) Discourse, conversation, illustration by examples 2 hours
L3. Nonlinear Continuous-Time Dynamical Systems. Equilibrium states and Lyapunov stability. Linearization of nonlinear systems. Limit cycles and deterministic chaos. Examples (CTRNNs, etc.) Discourse, conversation, illustration by examples 2 hours
L4. Discrete-Time Dynamical Systems. Orbits, fixed points, periodic points, and cycles. Cobweb diagrams. Bifurcations. Examples (Logistic maps, population models, and iterative algorithms in optimization, etc.) Discourse, conversation, illustration by examples 2 hours
L5. Chaos and Fractals. Chaos theory and sensitive dependence. Sharkovskii’s theorem. Fractal dimension. Fractals. Examples (Mandelbrot and Julia sets, Lorenz attractor, applications in meteorology and financial systems) Discourse, conversation, illustration by examples 2 hours
L6. Dimensionality Reduction. Singular Value Decomposition (SVD) and Principal Component Analysis (PCA). Dimensionality reduction in machine learning. Examples (Feature extraction for image and text data, face recognition, and data compression) Discourse, conversation, illustration by examples 2 hours
L7. Clustering and Classification. Supervised vs. unsupervised learning. Examples: k-means clustering, hierarchical clustering, Gaussian mixture models, and decision trees. Discourse, conversation, illustration by examples 2 hours
L8. Neural Networks as Dynamical Systems. Discrete-Time Recurrent Neural Networks (DTRNNs) and Continuous-Time Recurrent Neural Networks (CTRNNs). Reservoir Computing. Discourse, conversation, illustration by examples 2 hours
L9. Machine Learning Algorithms as Dynamical Systems. Stability and convergence analysis of optimization algorithms. Reinforcement learning. Examples: Convergence in supervised learning, actor-critic methods in reinforcement learning, etc.. Discourse, conversation, illustration by examples 2 hours
L10. Stochastic Dynamical Systems in Machine Learning. Stochastic processes in dynamical systems. Related algorithms. Examples: Random walks, Markov processes, and applications in online learning and probabilistic models. Discourse, conversation, illustration by examples 2 hours
L11. Advanced Topics: Koopman Operator Theory. Dynamic Mode Decomposition (DMD). Koopman operator theory for linearization of nonlinear systems, Dynamic Mode Decomposition for capturing system dynamics from data. Discourse, conversation, illustration by examples 2 hours
L12. Advanced Topics: Data-Driven Dynamical Systems. Approaches to modeling dynamical systems directly from data, machine learning methods for inferring system dynamics. Discourse, conversation, illustration by examples 2 hours
L13-L14. Review. Project presentations. Additional topics. Discourse, conversation, illustration by examples 2 hours

Course bibliography

[1] ELAYDI, S. An introduction to difference equations. Springer Science & Business Media, 2005. [2] CULL, P.l; FLAHIVE, M.; ROBSON, R. Difference equations: from rabbits to chaos. Springer, 2005. [3] MURPHY, K. P., Machine learning: A probabilistic perspective. MIT Press, 2012 [4] DEVANEY, R. L. An Introduction to Chaotic Dynamical Systems, Westview Press, 2003. [5] GROS, C. Complex and Adaptive Dynamical systems. Springer, 2008. [6] BRUNTON, S., KUTZ, J.L. Data-Driven Science and Engineering Machine Learning, Dynamical Systems, and Control (2nd ed.), Cambridge University Press, 2021.

Seminar content

Content Methods Obs
Labs 1-7 follow closely the topics discussed at the lectures. Dialogue with students, cooperative learning, modeling, case studies 14 hours
Bibliography: The same as for the lectures.

Seminar bibliography

The content is in accordance with the structure of similar courses offered by other universities and it covers the main aspects of using metaheuristic algorithms in solving real world problems.

Corroboration

(none)

AI tools guidance

(none)

Evaluation and delivery

Activity Criteria Methods Percentage
C
  • Knowledge of Theoretical Concepts. Understanding and application of key dynamical systems and machine learning theories.
  • Identification of Methods and Algorithms. Ability to select appropriate methods and algorithms relevant to the problem at hand.
  • Practical Problem-Solving.
  • Effective implementation of suitable techniques to solve a practical problem, supported by analysis and results.
  • Project presentation (report, software application, oral presentation)
  • 70.0%
S
  • Usage of Software Tools.
  • Competence in using software tools (e.g., Python) to implement and solve lab assignments.
  • Mathematical Analysis and Understanding. Ability to conduct mathematical analysis of the dynamical systems presented in the assignments.
  • Presentation of Results.
  • Clarity and accuracy in presenting results (e.g., visualizations, graphs, and explanations).
  • Applications at the lab and homeworks
  • 30.0%

Performance standards

Achieves a minimum of 50% overall in the combined course evaluations (project and lab assignments). Project: Must score at least 50% of the available points on the project. Lab Assignments: Must successfully complete at least 3 out of 5 lab assignments, demonstrating minimal proficiency in both the software tools and mathematical analysis. The final grade is calculated as a weighted average of the marks from the components listed in sections 9.4 and 9.5. A passing grade requires an overall average of at least 5, without the need for each individual component to be at least 5. During each exam session (including retakes), the grade is computed using the same method. Students may only retake exams for components where their current mark is below 5, unless they specifically request to be re-examined for other components.

Additional info

(none)