Optimization of expensive model with many parameters

I have a physical model which takes $sim50$ parameters and gives $sim2000$ outputs taking tens of minutes to run. I need to optimize these parameters to give outputs as close as possible to data. The problem of course is that it is expensive to evaluate and, probably worse, that there are so many parameters.

The best suggestion I have found so far is to use some kind of surrogate model and optimize that instead. However, these surrogate models are always (as far as I can see) for functions with just one output, i.e. the cost function. This is of course still an option here as I need some way to quantify how good the model is, so I am trying to minimise the $chi^2$. Then I can for example use Bayesian optimization or a quadratic surrogate to optimize it.

My issue with this is that $chi^2$ is like the ‘distance’ between the model result and the data in the high dimensional output space. This feels like throwing away a huge amount of information, as any optimization method based on just the cost function is only using information about this distance, and not the actual behaviour of the model. Being a physical model, certain parameters affect the outputs in particular ways, and one can fit to the data to some extent by hand. This is done without any reference to the $chi^2$ explicitly, but being a human means it will not be perfect. It also feels similar to an ‘inverse problem’, where one tries to find the most likely parameters for given data.

My questions are then: does there perhaps exist a way to create some kind of surrogate for the full model rather than just for the $chi^2$ in order to replicate the insights one uses when searching by hand instead of just looking at the ‘distance’? Even putting the optimization problem aside, this would still be extremely helpful in seeing how different parameters affect the output, giving a better understanding of the physics, but I fear using something like machine learning would require too great a number of evaluations. Then, regarding just the optimization problem, even if there does exist a way of creating such a surrogate model, would it be worth it compared to simply trying to optimize the $chi^2$ directly? Lastly, would the idea of the inverse problem help at all, i.e. could there be some way of taking the many outputs and ‘projecting’ them onto the most likely parameters, or is this just another way of stating the same problem?

Extra information: The calculation is not particularly noisy. There are no constraints on the parameters but fitting by hand has already given a good idea of where I should be looking around. I have also identified what I think are the $sim 15$ most important parameters in case it will be too difficult to optimise so many.