I posted the same question to Math Stackexchange earlier without much luck, so I am posting here.


I have a space $\mathcal{F}$ of vector-valued functions $f(\in\mathcal{F}):\mathbb{R}^d \rightarrow \mathbb{R}^n$. Consider a function $\Pi: \mathcal{F} \rightarrow \mathcal{F}$:

$$ \Pi (f) (x) = \sum_{i=1}^{P} K \left(x,x_i\right) f(x_i) $$

where $x\in\mathbb{R}^d$, and $K:\mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}^{n\times n}$. To clarify, $\Pi (f) (x)$ means "evaluate $\Pi (f) \in \mathcal{F}$ at $x$".

Our dynamical equation for a function $f$ is:

$$ \frac{\partial f_t}{\partial t}=\Pi(f_t) $$

$t$ is a time in $\mathbb{R}$. Apparently, assuming $f_0$ as the initial condition, the solution is

$$ f_t = \exp(t\Pi) f_0$$

but I am not sure how to prove this. The solution for a matrix differential equation has the same form and I can prove it for that, but I am not sure how to do it for this functional case.

To provide the context, this is from the neural tangent kernel paper by Jacot et al.