PhyLSTM loss scaling

NOTE: This has been proven to be unneccessary as the $\dot{B}$ is calculated from $B$ at all times, and can even be derived from $B$ when calculating the loss.

As seen in the PhyLSTM flow note, in the original PhyLSTM implementation by Francesco, the $B$ is processed as

$B_s=\frac{B-{\mu }_B}{{\sigma }_B}$ and then

$$\dot{B_s}=\frac{d{B}_s}{dt}=\frac{1}{{\sigma }_B}\frac{dB}{dt}=\frac{1}{{\sigma }_B}\dot{B}$$

Hence for the loss function

$$\begin{array}{r}\mathcal{L}=\sum _i^5{w}_i{\mathcal{L}}_i\end{array}$$

with

$$\begin{array}{rl}{\mathcal{L}}_1& =||B-\hat{B}|{|}^2\\ & =||({\sigma }_B{B}_s+{\mu }_B)-({\sigma }_B{\hat{B}}_s+{\mu }_B)|{|}^2\\ & ={\sigma }_B^2||B_s-{\hat{B}}_s|{|}^2\end{array}$$

and similarly

\begin{align} \mathcal{L}_2 & = || \dot{B} - \hat{\dot{B}} ||^2 \\ & = \sigma_\dot{B}^2||\dot{B}_s - \hat{\dot{B}}_s ||^2 \end{align}

But the scale by the $\sigma$ are baked into the loss weights $w_1$ and $w_2$ . However, for the third loss term

\begin{align} \mathcal{L}_3 & = || \hat{\dot{B}} - \dot{\hat{B}} ||^2 \\ & = || (\sigma_\dot{B}\hat{\dot{B}}_s +\mu_\dot{B} ) - \frac{d}{dt}(\sigma_B \hat{B}_s + \mu_B ) ||^2 \\ & = || (\sigma_\dot{B}\hat{\dot{B}}_s +\mu_\dot{B} ) - \sigma_B\dot{\hat{B}}_s ||^2 \end{align}

But in the case ${\dot{B}}_s=\frac{1}{{\sigma }_B}\dot{B}$ the third loss term becomes

$${\mathcal{L}}_3=||{\sigma }_B{\hat{\dot{B}}}_s-\frac{d}{dt}({\sigma }_B{\hat{B}}_s+{\mu }_B)|{|}^2={\sigma }_B^2||{\hat{\dot{B}}}_s-{\dot{\hat{B}}}_s|{|}^2$$

In other words, when using $\dot{B}$ that is acquired independently from $B$, the ${\mathcal{L}}_3$ must be calculated explicitly with the unscaled $\hat{\dot{B}}$.