Fit surrogate model on existing population

[schmoelder]

This PR implements a Surrogate class for fitting GPs on existing data from optimizations.

Supersedes #45

To do

Note, there is some WIP in #152 which will also affect this PR. Should we already rebase onto that branch s.t. we can adapt the corresponding interfaces but risk some friction should there be some more changes upstream?

• Fit GP on existing population
• Provide method to evaluate objective functions, constraint functions etc.
• Validate surrogate model accuracy with (simple) test cases
• Test if optimizers converge to true solution using surrogate

Open questions

Alternative surrogate modeling approches

Currently, only GPs are implemented. However, other surrogate models can be envisioned (e.g. ANNs). To improve a more modular architecture, we could subclasses a SurrogateBase.

Follow-up projects

Once this is merged, we can also start working on other features which would improve or apply the surrogate models. Eventually, these should be moved to their own issues / PRs but for now, this is just a collection of ideas.

The interface

Currently, the SurrogateModel class somewhat mimics an OptimizationProblem as it also provides methods for estimating objectives, nonlinear constraints etc. To demonstrate this, compare the OptimizationProblem

sequenceDiagram
    User->>+OptimizationProblem: evaluate_objectives(x)
    OptimizationProblem->>+User: f(x)

Loading

with the SurrogateModel

sequenceDiagram
    User->>+SurrogateModel: estimate_objectives(x)
    SurrogateModel->>+User: f*(x)

Loading

However, it is important to note that the SurrogateModel will never provide all functionality of the OptimizationProblem, such as specifying variables, constraints etc. Hence, it cannot directly be used as an OptimizationProblem e.g. to interface with an Optimizer.

To me, this means we should rethink the architecture and consider what exactly does the SurrogateModel replace? In the context of an OptimizationProblem, I would say, it actually replaces the evaluation toolchain (that which returns the values x->f/g/m/...).

Consequently, we should consider moving the evaluation toolchain from the OptimizationProblem to its own module (which in the process would also make the OptimizationProblem less of a "god class" and would even allow reusing the toolchain in other places) and introduce an EvaluationInterface.

The architecture would then look something like the following:

sequenceDiagram
    User->>+OptimizationProblem: evaluate_objectives(x)
    OptimizationProblem->>+EvaluationInterface: evaluate(x)
    EvaluationInterface->>+OptimizationProblem: f(x)
    OptimizationProblem->>+User: f(x)

Loading

Where the EvaluationInterface is then implemented by both the EvaluationPipeline (i.e. the current "toolchain") and the SurrogateModel:

classDiagram
    class OptimizationProblem {
        evaluate_objectives(np.ndarray): np.ndarray
    }
    OptimizationProblem "1" *-- "1" EvaluationInterface
    
    class EvaluationInterface {
        <<interface>>
        +evaluate(np.ndarray): np.ndarray
    }
    
    SurrogateModel <|-- EvaluationInterface
    EvaluationPipeline <|-- EvaluationInterface

Loading

Conditioned optimization problems

One of the original ideas for this projects came from optimization problems where we want to fix the value of one of the variables and then run the optimization to find the best point given this value.

For this purpose, we should implement a ConditionedOptimizationProblem which wraps the original OptimizationProblem and provides an interface where the fixed variables are removed. While this is trivial to implement for bound constrained problems, it becomes potentially more complicated for problems with linear and nonlinear constraints.

Plots

For process design, we are often not really interested in just the optimal point but in the general topology of the parameter space. E.g. we are interested in the contours of regions with a given purity. Finely sampling the parameter space would be very expensive so the idea could be to use a surrogate model for this purpose. See also partial dependence plots and #33.