Бакалавриат
2026/2027
Классическое и нейросетевое моделирование
ID 1144347
Статус:
Курс обязательный (Прикладной анализ данных)
Где читается:
Факультет компьютерных наук
Когда читается:
4-й курс, 1-3 модуль
Охват аудитории:
для своего кампуса
Язык:
английский
Кредиты:
9
Контактные часы:
96
Course Syllabus
Abstract
Modern research data analysis increasingly requires going beyond purely statistical or purely deterministic approaches. Real-world problems ranging from processing the results of physical experiments to modeling epidemiological, biological, or engineering processes pose fundamental questions for analysts: how to work with systems where data is sparse but theory is abundant; how to extract hidden parameters from observations; how to combine the accuracy of classical models with the flexibility of neural networks. This course offers a systematic view to modeling by integration three fundamental approaches: complex systems theory, classical numerical methods, and modern physics-informed neural network architectures. Designed for fourth-year students with advanced programming and data analysis skills, the course focuses on developing a research engineering culture, the ability to select a descriptive language based on the nature of the object and the nature of the data.
Learning Objectives
- to form students' systematic understanding of dynamical systems, methods of their qualitative and numerical analysis, including bifurcation analysis
- to master methods for reconstructing ordinary differential equations from time series, forecasting time series of dynamical systems, and calculating attractor characteristics from time series
- to introduce the basic principles of synergetics, fundamental synergetic models, and blow-up regimes
- to develop practical skills in mathematical modeling, numerical analysis, and working with modern software tools
- to introduce students to classical and neural methods for solving ordinary and partial differential equations
- to develop an understanding of physics-informed neural networks and their relationship to classical numerical methods
- to provide practical experience in training PINNs for forward and high-dimensional problems
- to teach students to formulate and solve inverse problems using sparse and noisy data
- to introduce methods of sensitivity analysis and uncertainty quantification
- to develop skills in validating, comparing, and critically evaluating classical and neural modelling approaches
Expected Learning Outcomes
- know basic concepts and methods of qualitative analysis of dynamical systems
- be able to build and investigate mathematical models of dynamical systems
- be able to perform analytical and numerical bifurcation analysis
- be able to reconstruct dynamical equations from time series and make forecasts
- be able to compute time series characteristics: dimension, Lyapunov exponents, entropy
- be able to analyze synergetic models and determine conditions for the emergence of blow-up regimes
- have skills in working with modern software packages for numerical modeling and analysis
- have skills in methods of qualitative theory of differential equations
- have skills in processing and analyzing time series of dynamical systems
- have skills in modeling self-organization processes and blow-up regimes
- be able to formulate initial and boundary value problems for ODEs and PDEs
- be able to Reduce PDEs to ODE systems through spatial discretization
- be able to construct and train PINNs for forward and inverse problems
- be able to impose initial, boundary and physical constraints
- be able to validate neural solutions against analytical or classical numerical solutions
- be able to identify unknown parameters from sparse noisy observations
- be able to perform local and global sensitivity analysis
- be able to Propagate uncertainty and obtain posterior predictive estimates
- have skills in implementing classical and neural differential-equation solvers using modern software
- have skills in applying automatic differentiation to differential operators and sensitivities
- have skills in designing physics-informed losses and collocation strategies
- have skills in comparing classical methods and PINNs
- have skills in constructing parametric PINNs
- have skills in applying Monte Carlo, ensemble and Bayesian methods
- have skills in interpreting sensitivity indices, posterior distributions and uncertainty intervals
Course Contents
- Dynamical Systems
- Bifurcation Analysis of Dynamical Systems
- Numerical Bifurcation Analysis of Cauchy Problems
- Reconstruction of ODEs from Time Series
- Forecasting Time Series of Dynamical Systems
- Analysis of Time Series of Dynamical Systems: Calculating Series and Attractor Dimensions from Data
- Mathematical Formulation of ODE and PDE Problems
- Classical Numerical Methods for Partial Differential Equations
- Physics-Informed Neural Networks for ODEs and PDEs
- Neural Solution of High-Dimensional PDEs and the Schrödinger Equation
- Inverse Problems and Parameter Identification
- Parametric PINNs and Sensitivity Analysis
- Uncertainty Quantification and Bayesian Inverse Problems
- Basic Principles of Synergetics
- Basic Synergetic Models
- Blow-up Regimes
Interim Assessment
- 2026/2027 2nd moduleFinal = 0.5* Module 1 + 0.5* Module 2, where Module 1 = 0.4 * Laboratory work 1 + 0.6 * Colloquium 1; Module 2 = 0.4 * homework 1 + 0.6 * homework 2
- 2026/2027 3rd module0.6 * Colloquium 2 + 0.4 * Laboratory work 2
