Combine Deep Learning, Bayesian Statistics, and Physics for:
in inverse problems.
Observed $y$
$f_\theta$
Unknown $x$
$$ y $$
$$f_\theta$$
$$ \mbox{True } x $$
Let's try to understand the neural network output by looking at the loss function
$$ \mathcal{L} = \sum_{(x_i, y_i) \in \mathcal{D}} \parallel x_i - f_\theta(y_i)\parallel^2 \quad \simeq \quad \int \parallel x - f_\theta(y) \parallel^2 \ p(x,y) \ dx dy $$Probably not, this is a randomly generated person: thispersondoesntexist.com
We want to solve for the Maximum A Posterior solution:
$$\arg \max - \frac{1}{2} \parallel {\color{Orchid} y} - {\color{SkyBlue} x} \parallel_2^2 + \log p_\theta({\color{SkyBlue} x})$$ This can be done by gradient descent as long as one has access to $\frac{\color{orange} d \color{orange}\log \color{orange}p\color{orange}(\color{orange}x\color{orange})}{\color{orange} d \color{orange}x}$.But what about uncertainty quantification?
$\Longrightarrow$ We can see which parts of the image are well constrained by data, and which regions are uncertain.
Thank you!