PhyLSTM flow

Not sure all information here is up to date.

Old flow

Model input: $I_s,B_{s,nl},\dot{B_{s,nl}}$ Model output: ${\hat{B}}_s,{\hat{\dot{B}}}_s$

Pre-processing

Removing the linear component of the field

$$\begin{array}{rl}{B}_{nl}& =B-(kI+m)\\ {\dot{B}}_{nl}& =\frac{\text{d}{B}_{nl}}{\text{d}t}=\dot{B}-k\dot{I}\end{array}$$

Computing the scaled variables

$$\begin{array}{rl}{I}_s& =\frac{I_s-{\mu }_I}{{\sigma }_I}\\ B_{s,nl}& =\frac{B_{nl}-{\mu }_{B_{nl}}}{{\sigma }_{B_{nl}}}\\ {\dot{B}}_{s,nl}& =\frac{\text{d}{B}_{s,nl}}{\text{d}t}=\frac{1}{{\sigma }_{B_{nl}}}{\dot{B}}_{nl}\end{array}$$

PhyLSTM

Applying PhyLSTM:

$$({\hat{B}}_{s,nl},{\hat{\dot{B}}}_{s,nl})=\text{PhyLSTM}(I_s):=f(I_s)$$

Loss

Computing the loss, the $\text{MSE}$ is proportional to the square of the L2 norm and MSE:

$$\begin{array}{r}\mathcal{L}(B_{s,nl},{\dot{B}}_{s,nl},{\hat{B}}_{s,nl},{\hat{\dot{B}}}_{s,nl})=\text{MSE}(B_{s,nl},{\hat{B}}_{s,nl})+MSE({\dot{B}}_{s,nl},{\hat{\dot{B}}}_{s,nl})\end{array}$$

and with the Bouc-Wen loss components.

Postprocessing

Unscaling the variables. The $\dot{B}$ is not unscaled normally as we are only interested $B$, but is provided here for completeness. Note that we do not have ${\sigma }_{{\dot{B}}_{\text{nl}}}$ and ${\mu }_{{\dot{B}}_{\text{nl}}}$

$$\begin{array}{rl}{\hat{B}}_{nl}& ={\sigma }_{B_{nl}}{\hat{B}}_{s,nl}+{\mu }_{B_{nl}}\\ ({\hat{\dot{B}}}_{nl}& ={\sigma }_{{\dot{B}}_{nl}}{\hat{\dot{B}}}_{s,nl}+{\mu }_{{\dot{B}}_{nl}})\end{array}$$

Adding back the linear component

$$\begin{array}{rl}\hat{B}& ={\hat{B}}_{nl}+(kI+m)\\ \hat{\dot{B}}& ={\hat{\dot{B}}}_{nl}+k\dot{I}\end{array}$$

New flow

This is a new data flow proposed to refactor the PhyLSTM code, baking in a running normalization of the variables $I$ and $B$, however this is deemed impractical as the running normalization does not work with

Model input: $I,B,\dot{B}$ Model output: $\hat{B},\hat{\dot{B}}$

Pre-processing (in PhyLSTMModule)

Compute the scaling parameters (after transformation of the parameter to avoid lookahead bias):

$$\begin{array}{rl}{\mu }_I& =E[I]:=\frac{1}{N}\sum _i^N{I}_i\\ {\sigma }_I& =\sqrt{\sum _i^N(I_i-{\mu }_i{)}^2}=\sqrt{E[I^2]-{\mu }_I^2}\\ {\mu }_B,{\sigma }_B& =\dots \end{array}$$

Removing the linear component of the field

$$\begin{array}{rl}{B}_{nl}& =B-(kI+m)\\ {\dot{B}}_{nl}& =\frac{\text{d}{B}_{nl}}{\text{d}t}=\dot{B}-k\dot{I}\end{array}$$

Scale $I$

$$\begin{array}{rl}{I}_s& =\frac{I_s-{\mu }_I}{{\sigma }_I}\end{array}$$

PhyLSTM

Applying PhyLSTM:

$$({\hat{B}}_{s,nl},{\hat{\dot{B}}}_{s,nl})=\text{PhyLSTM}(I_s):=f(I_s)$$

And compute the linear part using the fixed linear layer:

$$\begin{array}{rl}{B}_{ln}& =kI+m\\ {\dot{B}}_{ln}& =k\dot{I}\end{array}$$

Loss

Computing the loss, the $\text{MSE}$ is proportional to the square of the L2 norm and MSE:

$$\begin{array}{r}\mathcal{L}(B_{s,nl},{\dot{B}}_{s,nl},{\hat{B}}_{s,nl},{\hat{\dot{B}}}_{s,nl})=\text{MSE}(B_{s,nl},{\hat{B}}_{s,nl})+MSE({\dot{B}}_{s,nl},{\hat{\dot{B}}}_{s,nl})\end{array}$$ $$\begin{array}{rl}||\hat{B}-B|{|}_2^2& =||({\hat{B}}_{nl}+B_{ln})-B|{|}_2^2\\ & =||({\sigma }_{B_{nl}}\underset{PhyLSTM(I_s)}{\underbrace{{\hat{B}}_{s,nl}}}+{\mu }_{B_{nl}}+kI+m)-B|{|}_2^2\\ & =||{\sigma }_{B_{nl}}{\hat{B}}_{s,nl}+{\mu }_{B_{nl}}-(B-(kI+m))|{|}_2^2\\ & =||{\sigma }_{B_{nl}}{\hat{B}}_{s,nl}+{\mu }_{B_{nl}}-B_{nl}|{|}_2^2\\ & =||{\sigma }_{B_{nl}}{\hat{B}}_{s,nl}+{\mu }_{B_{nl}}-({\sigma }_{B_{nl}}{B}_{s,nl}+{\mu }_{B_{nl}})|{|}_2^2\\ & =||{\hat{B}}_{nl}-B_{nl}|{|}_2^2\\ & ={\sigma }_{B_{nl}}^2||{\hat{B}}_{s,nl}-B_{s,nl}||\end{array}$$ || \hat{\dot{B}} - \dot{B} ||^2_2 & = || (\hat{\dot{B}}_{nl} + \dot{B}_{ln}) - \dot{B} ||^2_2 \\ & = || (\sigma_{\dot{B}_{nl}} \underbrace{\hat{\dot{B}}_{s, nl}}_{PhyLSTM(I_s)} + \mu_{\dot{B}_{nl}} + k\dot{I}) - \dot{B} ||^2_2 \\ & = || \sigma_{\dot{B}_{nl}} \hat{\dot{B}}_{s, nl} + \mu_{\dot{B}_{nl}} - \dot{B}_{nl} ||^2_2 \\ & = || \sigma_{\dot{B}_{nl}} \hat{\dot{B}}_{s, nl} + \mu_{\dot{B}_{nl}} - (\sigma_{\dot{B}_{nl}} \dot{B}_{s, nl} + \mu_{\dot{B}_{nl}}) ||^2_2 \\ & = || \hat{\dot{B}}_{nl} - \dot{B}_{nl} ||^2_2 \\ & = \sigma^2_{\dot{B}_{nl}} || \hat{\dot{B}}_{s, nl} - \dot{B}_{s, nl} ||^2_2 \end{align}$$ ### Postprocessing (in PhyLSTMModule) Unscaling the variables

\begin{align} \hat{B}{nl} & = \sigma{B_{nl}} \hat{B}{s, nl} + \mu{B_{nl}} \ (~ \hat{\dot{B}}{nl} &= \sigma{\dot{B}{nl}}\hat{\dot{B}}{s, nl} + \mu_{\dot{B}_{nl}} ~) \end{align}

$$Addingbackthelinearcomponent$$

\begin{align} \hat{B} & = \hat{B}{nl} + B{ln} = \sigma_{B_{nl}} \hat{B}{s, nl} + \mu{B_{nl}} + kI + m \ \hat{\dot{B}} & = \hat{\dot{B}}{nl} + \dot{B}{ln} = \sigma_{\dot{B}{nl}}\hat{\dot{B}}{s, nl} + \mu_{\dot{B}_{nl}} + k\dot{I} \end{align}

### Computing the mean of the nonlinear part For $B_{nl}$ $$\begin{cases} \mu_{B_{nl}} & = \mu_B - (k\mu_I + m) \\ \sigma_{B_{nl}} & = \sqrt{\sigma_B^2 + k^2\sigma_I^2} \end{cases}$$ and $\dot{B}_{nl}$ $$\begin{cases} \mu_{\dot{B}_{nl}} & = \mu_\dot{B} - k\mu_\dot{I} \\ \sigma_{\dot{B}_nl} & = \sqrt{\sigma_\dot{B}^2 + k^2\sigma_\dot{I}^2} \end{cases}$$ #### Proof

\mu_{B_{nl}} = E[B_{nl}] = E[B - (kI+m)] = E[B] - (kE[I] + m) = \mu_B - k\mu_I - m

$$\begin{align} \sigma_{B_{nl}} &= \sqrt{E[B_{nl}^2]-E[B_{nl}]^2} \\ & = \sqrt{E[B_{nl}^2] - \mu_{nl}^2} \\ & = \sqrt{E[(B-(kI+m))^2] - \mu_{nl}^2} \\ & = \sqrt{E[B^2 - 2 BkI - 2 B m + I^2 k^2 + 2 kIm + m^2]-\mu_{B_{nl}}^2} \\ & = \sqrt{(\sigma^2_B + \mu^2_B) - 2k\mu_I\mu_B - 2m\mu_B + k^2(\sigma_I^2 + \mu_I^2) + 2km\mu_I + m^2 - \mu_{B_{nl}}^2} \\ & = \sqrt{\sigma_B^2 + k^2\sigma_I^2} \end{align}$$ because $$\begin{align} E[B_{nl}] & = E[(B-(kI+m))^2] \\ & = E[B^2 - 2 BkI - 2 B m + I^2 k^2 + 2 kIm + m^2] \\ & = E[B^2] - 2kE[I]E[B]- 2mE[B]+ k^2E[I^2] + 2kmE[I] + m^2 \\ & = (\sigma^2_B + \mu^2_B) - 2k\mu_I\mu_B - 2m\mu_B + k^2(\sigma_I^2 + \mu_I^2) + 2km\mu_I + m^2 \end{align}$$ and

E[X^2] = \sigma^2 + E[X]^2 = \sigma^2+\mu

and $$\begin{align} \mu_{B_{nl}}^2 & = (\mu_B - k\mu_I - m)^2 \\ & = \mu_B^2 - 2k\mu_B\mu_I - 2m\mu_B + 2km\mu_I + k^2\mu_I^2 + m^2 \end{align}$$ For the derivative $\dot{B}_{nl}$ the same proof follows, but without the constant $m$. ### Eddy current decay The original PhyLSTM follows

\begin{align} o & = \textrm{LSTM}(I)\ z & = \textrm{FC}(o), \quad z = {\hat{B}, \hat{\dot{B}},\hat{r}} \end{align}

Assuming that eddy currents are captured in the LSTM output $o$, we could simply compute

y = \textrm{FC}2(o), \quad y={\hat{I}\textrm{e}, \hat{\dot{I}}_\textrm{e}}

$$sothat$$

\hat{B}_\textrm{e} = c\hat{I}_e

$$and$$

\hat{B} \rightarrow \hat{B} + \hat{B}_\textrm{e}

Simply by following equations $\eqref{dotIe},\eqref{Ie},\eqref{Be}$ adding to the objective function

\begin{align} \mathcal{L}4 & = \left|\left| \frac{d}{dt}\hat{I}\textrm{e}-\hat{\dot{I}}_\textrm{e} \right|\right|^2_2 \

\mathcal{L}5 & = \left|\left| \hat{\dot{I}}\textrm{e} - (a\dot{I}+b\hat{I}_\textrm{e}) \right|\right|^2_2 \end{align}

$$Weshouldbeabletomodeleddycurrentdecaysinthebouc-wenhysteresismodel$$