Automatic best-method metamethod

[pavelkomarov]

As @nathankutz said ode45 is so widely used because it's adaptive and finds the most rock-solid of several methods under the hood without user input, we'd similarly like to have a metamethod in this package that can run some optimizations and tests below the surface and give you back its best guess of a smooth derivative. I'm LaTeXing a flow chart that will govern some of this thing's operation, but I'm still slightly unclear on how we might automagically decide on some of the forks in the road. More to come as I grapple with those granularly.

[pavelkomarov]

One way to accomplish this would be to just line up a bunch of calls to optimize, given a user's data and the good guess of tvgamma from the old paper. Fairly expensive to run, but it would get the job done. Not sure there is a cheap way to do this. Some alternative optimization method may take less compute, although things like Bayesian optimization are inherently more sequential and so can end up taking longer if sufficient parallelization is available.

[pavelkomarov]

I've created some code to do this. Unfortunately, in the absence of ground truth, it tends to greatly favor velocity from the TVR module, because that method directly pushes down the second term of the loss function:

$$L = \text{RMSE} \bigg( \text{trapz}(\mathbf{\hat{\dot{x}}}(\Phi)) + \mu, \mathbf{y} \bigg) + \gamma \bigg({TV}\big(\mathbf{\hat{\dot{x}}}(\Phi)\big)\bigg)$$

That makes this loss kind of less helpful. I can try excluding regularizing on velocity directly.

Update: Indeed, excluding velocity from suggest_method's list leads to more interesting, better answers, so I've simply done that in #132 and added an explanation to the docstring about why certain methods are skipped (runtime too long, too fragile, etc.).