Course Content:  This is a broad introduction to the theory of ordinary differential equations and dynamical systems. The main objective is to develop methods for approximating and analyzing differential equations. This includes the fundamental theory of the existence and uniqueness of solutions and the qualitative analysis of nonlinear equations. Topics include: analytical methods for solving ODEs; approximate solutions of ODEs; theory and conditions for existence and uniqueness of solutions of IVPs; BVP and self-adjoint operators; Strum-Loiuville theory;  systems of linear ODEs; nonlinear systems; conservative systems; reversible systems; Lyapunov Stability; qualitative theory of dynamical systems; bifurcation theory.

Tenative Course Content:  The plan is to cover theoretical and computational approaches for learning dynamical systems from data. A potential list of topics include:

• Proper Orthogonal Decomposition (POD)

• Dynamic Mode Decomposition (DMD)

• Discrete Empirical Interpolation Method (DEIM)

• Operator Inference

• Sparse Learning (convex and nonconvex), Bounded Orthogonal Systems (BOS)

• Sparse Approximations to Differential Operators

• Random Feature Methods

• Neural Network models for approximating solutions to differential equations

• Operator Learning and Multi-Operator Learning