An interesting analogy, closer to ML, would be to look at neuroscience. It's an older field than ML, and it seems that the physics perspective has been fairly productive, even though not successful at providing a grand unified theory of cognition yet. Some examples: 

• Using methods from electric circuits to explain neurons (Hodgkin-Huxley model, cable theory)
• Dynamical systems to explain phenomena like synchronization in neuronal oscillations (ex: Kuramoto model)
• Ising models to model some collective behaviour of neurons
• Information theory is commonly used in neuroscience to analyze neural data and model the brain (ex: efficient coding hypothesis)
• Attempts at general theories of cognition like predictive processing, or the free energy principle which also have a strong physics inspiration (drawing from statistical physics, and the least action principle) 

I can recommend the book Models  of the Mind, from Grace Lindsay, which gives an overview of the many way physics contributed to neuroscience. 

In principle, one might think that it would be easier to make progress using a physics perspective on AI than in neuroscience, for example because it is easier to do experiments in AI (in neuroscience we do not have access to the value of the weights, we do not always have access to all the neurons, and often it is not possible to intervene on the system).

Interesting! Perhaps one way to not be fooled by such situations could be to use a non-parametric statistical test. For example, we could apply permutation testing: by resampling the data to break its correlation structure and performing PCA on each permuted dataset, we can form a null distribution of eigenvalues. Then, by comparing the eigenvalues from the original data to this null distribution, we could assess whether the observed structure is unlikely under randomness. Specifically, we’d look at how extreme each original eigenvalue is relative to those from the permuted data to decide if we can reject the null hypothesis that the data is random. Could be computationally prohibitive though, but maybe there are some ways to go around that (reducing the number of permutations, downsampling the data....). Also note that failing to reject the null does not mean that the data is random. Somehow it does not look like these kind of tests are common practice in interpretability?

Right, I got confused because I thought your problem was about trying to define a measure of optimisation power - for ex analogous to the Yudkowsky measure - that was also referring to a utility function, while being invariant from scaling and translation but this is different from asking

"what fraction of the default expected utility comes from outcomes at least as good as this one?’"

Nice recommendations! In addition to brain enthusiasts being useful for empirical work, there also are theoretical tools from system neuroscience that could be useful for AI safety. One area in particular would be for interpretability: if we want to model a network at various levels of "emergence", recent development in information decomposition and multivariate information theory to move beyond pairwise interaction in a neural network might be very useful. Also see recent publications to model synergestic information  and dynamical independance to perhaps automate macro variables discovery which could also be well worth exploring to study higher levels of large ML models. This would actually require both empirical and theoretical work as once the various measures of information decomposition are clearer one would need to empirically estimate test them and use them in actual ML systems for interpretability if they turn out to be meaningful.

Sure, I'm happy to read/discuss your ideas about this topic.

I am not sure about what computer aided analysis mean but one possibility could be to have formal ethical theories and prove some theorem inside their formal framework. But this raises questions about the sort of formal framework that one could use to 'prove theorems' under ethics in a meaningful way.