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Abstract: Dynamical Systems and Neural Networks
In this course students shall learn the fundamental principles governing complex systems via the theory of dynamical systems. We shall mainly rely on PDEs and ODEs as prototypical models to capture the dynamics of various systems from vehicle motion to the spreading of diseases. In the first quarter of the course, we shall make a brief recapitulation of the basic topics such as wellposedness, phase space diagrams, stability theory and bifurcation theory, classes of PDEs and also look at standard numerical methods for solving ODEs and PDEs. With the emergence of neural networks as universal function approximators they offer compelling advantages for dealing with complex and high-dimensional PDEs and ODEs. The course shall focus on understanding how neural networks and the concepts of machine learning can be used to obtain solutions to different types of dynamical systems especially the ones specified using PDEs and ODEs. Going further, we shall also focus on the problem of estimating unobserved states and the problem of controlling the system states based on partial observations with the help of neural networks. Finally, we shall see how the three core blocks of prediction, estimation and control can be integrated seamlessly in a network to obtain a generic unified algorithm for solving dynamical problems. By the course's end, students will have gained a solid foundational understanding of dynamical systems, ODEs, PDEs and neural networks, equipped with analytical and computational tools to address interdisciplinary problems effectively.
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Contents
• Recap on the basics of ODEs and PDEs from the viewpoint of functional theory
• Recap on numerical methods for ODEs (deterministic methods: Euler, RK, finite difference, finite elements)
• Neural networks (NNs) and approximation theorems
• NNs as a method for solving differential functional equations
• NNs as a method for solving functional state estimation problem
• NNs as a method for solving functional optimization problem
• Different types of NN design for PDEs and ODEs
• Physics informed neural networks, neural differential equations and diffusion models
• Applications in process automation, automated robot and vehicle control
Competencies / intended learning achievements
From the lecture students are able to
• refresh the basic foundational concepts and techniques to specify, analyze and solve ODEs and PDEs
• refresh the basic concepts of probability and statistics
• become familiar with stochastic and statistical methods for solving differential equations.
• utilize neural networks and approximation theorems to solve differential equations effectively.
• utilize neural networks for estimating unobserved states of the dynamical system.
• utilize neural networks for synthesizing actuation signals and to design automatic control algorithms for dynamical system.
From the (integrated) tutorials students are able to
• become proficient in using relevant python libraries for solving ODEs/PDEs
• apply monte-carlo method for solving ODEs
• apply physics-informed neural networks and related models to real-world problems involving differential equations
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