12-758 Special Topics: Elements of Continuum Mechanics, Controls, and Data Science
12 units, Carnegie Mellon University
This course will explore connections between the three subjects mentioned in the title. It will comprise three parts. The first part will focus on tensor algebra and matrix representations, various types of linear transformations important for both Continuum Mechanics and Data Science, and methods to analyze them like Rectangular and General Cartesian bases, Orthogonal tensors, invertible tensors, symmetric tensors, positive definite tensors, the eigenvalue problem, the Cayley-Hamilton, the Polar Decomposition, and the Singular Value Decomposition (SVD/PCA) theorems. The second part of the course will introduce the types of questions one wants to apply continuum mechanics to, through a toy problem of the inelasticity of a 1-d bar. Notions of elasticity (order), dissipation (disorder/entropy), and internal variables interpreted as controls will be introduced. Basic operational ideas from the Calculus of Variations will be studied to set up the problem as a nonlinear optimization within the framework of the Pontryagin Maximum Principle (PMP). The problem will be discretized, and two different optimization formulations - Least Squares, and one based on Convex Duality - will be studied to solve the discrete optimization problem using the Ordinary and Stochastic Gradient Descent algorithms of Data Science. The third part of the course will introduce the general principles of continuum mechanics: the balance laws of mass, linear momentum, angular momentum, energy, and entropy, in the bulk and at surfaces of discontinuity; and the principles of material frame indifference and material symmetry for formulating constitutive equations.
Prerequisites: none listed
Offered: Fall 2026
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