Jacobi and Ramanujan to the rescue!

One of the Jacobi theta functions gives us the identity

$\vartheta (z ; \tau) = 1 + \displaystyle 2 \sum_{n=1}^{\infty} {\left( e^{\pi i \tau} \right)}^{n^2} \cos (2 \pi n z)$

Setting $n=r$ , $z = 0$ and $\tau = i$ , we get

$\vartheta (0 ; i) = 1 + \displaystyle 2 \sum_{r=1}^{\infty} {\left( e^{-\pi} \right)}^{r^2} \implies \sum_{r=1}^{\infty} \dfrac{1}{e^{\pi r^2}} = \dfrac 12 \left[ \vartheta (0 ; i) - 1 \right]$

The explicit value of $\vartheta (0 ; i)$ is given by

$\varphi \left( e^{-\pi} \right) = \vartheta (0 ; i) = \dfrac{\sqrt[4]{\pi}}{\Gamma \left(\frac 34 \right)}$

Hence

$\displaystyle \sum_{r=1}^{\infty} \dfrac{1}{e^{\pi r^2}} = \dfrac 12 \left[ \dfrac{\pi^{\frac 14}}{\Gamma \left(\frac 34 \right)} - 1 \right]$

So, $a+b = 4+3 = \boxed{7}$ .