# Fit Range Determination with Machine Learning

One of the most tedious and error-prone things in my work in Lattice QCD is the manual choice of fit ranges. While reading up on Keras, deep neural networks and machine learning and how experimental the whole field is, I thought about just trying the fit range selection with deep learning.

We have correlation functions $C(t)$ which behave as $\sum_n A_n \exp(-E_n t)$ plus noise. The $E_n$ are the energies of the state $n$, the $A_n$ are the respective amplitudes. We are interested in extracting the smallest of the $E_n$, the ground state energy. We use that for sufficiently large times $t$ the term with the smallest energy dominates the expression. Without loss of generality we say $E_0 < E_1 < \ldots$ and formally write $$ \lim_{t \to \infty} C(t) = A_0 \exp(-E_0 t) \,. $$

By taking the *effective mass* as defined by
$$ m_\text{eff}(t) = - \log\left(\frac{C(t)}{C(t+1)}\right) $$
we get $m_\text{eff}(t) \sim E_0$ in the region of large $t$. There are more subtleties involed (back-propagation, thermal states), which we will ignore here. The effective mass is expected to be constant in some region of the data where $t$ is sufficiently large such that the higher states have decayed; yet the exponentially decaying signal-to-noise-ratio is still sufficiently good. An example for such an effective mass is the following.