SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} [ \nabla \cdot C (\mathbf {w}) ] _ {j} = [ \nabla \cdot \nabla \nabla G (\mathbf {w}) ] _ {j} \\ = \sum_ {i} \frac {\partial}{\partial w _ {i}} \frac {\partial}{\partial w _ {i}} \frac {\partial}{\partial w _ {j}} G (\mathbf {w}) \\ = \frac {\partial}{\partial w _ {j}} \sum_ {i} \frac {\partial}{\partial w _ {i}} \frac {\partial}{\partial w _ {i}} G (\mathbf {w}) \tag {61} \\ = \frac {\partial}{\partial w _ {j}} \operatorname {T r} (C (\mathbf {w})). \\ \end{array}
$$
So far, we can infer that  $\nabla \cdot C = \nabla \operatorname{Tr}(C)$ . According to Fokker-Planck equation( [26]), we have:
$$
\begin{array}{l} \frac {\partial p _ {t} (\mathbf {w})}{\partial t} = - \sum_ {i = 1} ^ {d} \frac {\partial}{\partial \mathbf {w} _ {i}} \left[ \nabla F _ {D} (\mathbf {w}) p _ {t} (\mathbf {w}) \right] + \frac {1}{2} \eta \sum_ {i = 1} ^ {d} \frac {\partial}{\partial \mathbf {w} _ {i}} \left[ \sum_ {j} ^ {d} \frac {\partial}{\partial \mathbf {w} _ {j}} \left[ C (\mathbf {w}) p _ {t} (\mathbf {w}) \right] \right] \\ = - \sum_ {i = 1} ^ {d} \frac {\partial}{\partial \mathbf {w} _ {i}} [ \nabla F _ {D} (\mathbf {w}) p _ {t} (\mathbf {w}) ] + \frac {1}{2} \eta \sum_ {i = 1} ^ {d} \frac {\partial}{\partial \mathbf {w} _ {i}} [ p _ {t} (\mathbf {w}) \nabla \cdot C + p _ {t} (\mathbf {w}) C \nabla_ {w} \log p _ {t} (\mathbf {w}) ] \\ = - \sum_ {i = 1} ^ {d} \frac {\partial}{\partial \mathbf {w} _ {i}} \left[ \nabla F _ {D} (\mathbf {w}) p _ {t} (\mathbf {w}) - \frac {1}{2} \eta \left[ \nabla \cdot C (\mathbf {w}) + C (\mathbf {w}) \nabla_ {w} \log p _ {t} (\mathbf {w}) \right] p _ {t} (\mathbf {w}) \right] \\ = - \sum_ {i = 1} ^ {d} \frac {\partial}{\partial \mathbf {w} _ {i}} \left[ \nabla F _ {D} (\mathbf {w}) p _ {t} (\mathbf {w}) - \frac {1}{2} \eta [ \nabla \operatorname {T r} (C (\mathbf {w})) + C (\mathbf {w}) \nabla_ {w} \log p _ {t} (\mathbf {w}) ] p _ {t} (\mathbf {w}) \right]. \tag {62} \\ \end{array}
$$
Therefore, the theorem is proven.