Кто читает:: Департамент больших данных и информационного поиска

Статус:: Курс обязательный

Когда читается:: 4-й курс, 1-3 модуль

Преподаватели

Громов Василий Александрович

Тараканов Александр Александрович

Томащук Корней Кириллович

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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

Assessment Elements

Laboratory work 1
Colloquium 1
Homework 1
Homework 2
Laboratory work 2
Colloquium 2

Interim Assessment

2026/2027 2nd module
Final = 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 module
0.6 * Colloquium 2 + 0.4 * Laboratory work 2

Bibliography

Recommended Core Bibliography

The complete guide to capital markets for quantitative professionals, Kuznetsov, A., 2007

Recommended Additional Bibliography

Foundations of complex-system theories : in economics, evolutionary biology, and statistical physics, Auyang, S.Y., 1999

Authors

Tarakanov Aleksandr Aleksandrovich
TOMASHCHUK KORNEY KIRILLOVICH
Derkach Denis Aleksandrovich
Gromov Vasilii Aleksandrovich

Бакалаврская программа «Прикладной анализ данных»

Контакты

Classical and Neural Modeling