This study explains a stochastic and variational approach to the finite difference method applied to first order Hamilton-Jacobi equations generated by Hamiltonians of the Tonelli class. In Section 1, the connection between Hamilton-Jacobi equations and Hamiltonian dynamics is stated, which implies the necessity to introduce weak solutions. In Section 2, the representation formula of a local classical solution obtained via the method of characteristics is extended to a globally defined function employing Tonelli’s calculus of variations. This function is called a value function in optimal control theory, and it is a weak solution of the Hamilton-Jacobi equation in the sense of viscosity solutions. In Section 3, reasoning similar to the previous section is demonstrated for the discretized Hamilton-Jacobi equations with the most elementary finite difference method, where a stochastic feature appears due to numerical viscosity. The convergence of approximation is proven in terms of stochastic and variational techniques, yielding a viscosity solution, its derivative and characteristic curves all at once. This approach is substantially applied to weak Kolmogorov-Arnold-Moser (KAM) theory.

The recent applications of topological data analysis in machine learning are reviewed in this paper. Simplicial and persistent homology were briefly explained and then two such applications were described. The first application is a topological study of the activation of neural networks, and the second application is a convergence result for the stochastic subgradient method for topological loss functions.