Although I don't know in detail, I am aware of the following facts regarding the usefulness of gradients in some domains of artificial intelligence, especially in optimization.
First order gradient: It quantifies the rate of change of a function with respect to its inputs. It is useful in artificial intelligence, especially in gradient-based algorithms, to know about the direction in which the parameters need to be updated.
Second-order gradient: It somehow quantifies the curvature of the function. It is used in artificial intelligence, especially in optimization, to know whether the function has convex or concave portions.
With this context, I want to know whether there is any significance for higher-order gradients in artificial intelligence? Note that higher order refers to the order $\ge 3$.