Differential Equation
A differential equation (DE) relates one or more unknown functions and their derivatives. Basically, while we often do not know the explicit function of real-world systems because their form is too complex or unknown, we can model how the function changes (derivatives) based on its or another functions current state.
Ordinary Differential Equation (ODE)¶
An ODE is a DE that depends only on a single independent variable (e.g. time). In contrast, Partial Differential Equations (PDEs) depend on more than one independent variable (e.g. x-y coordinates). In the context of machine learning, ODEs often stand in contrast to the #Stochastic Differential Equation (SDE), where the progression has a random element.
Stochastic Differential Equation (SDE)¶
The SDE is a DE that models a stochastic process, like stock prices, random growth or particle behavior. Its typically notated as:
where \(W\) is a Wiener Process ("random" Brownian motion). The \(f(X_{t},t)\) term is often called the drift-term, denoting the general direction of the functions evolution, whereas the \(g(X_{t},t)\) term denotes the stochastic uncertainty of the process. The \(dW_{t}\) is an [[Itô's Integral]] from Stochastic Calculus rather than a standard integral from Newtonian Calculus.