Algorithm for exploring a parameter space I have a research problem which has a general outline as follows:

I have a 2D parameter space, $(\alpha, \beta)$ which serve as initial conditions on some set of partial differential equations. Given values for $\alpha, \beta$ I can integrate this set of PDEs numerically to obtain a result in a 2D phase space $(x,y)$.

Now, suppose I want to work backwards and find the values of $(\alpha, \beta)$, which generate a particular $(x,y)$. I can see a general, crude algorithm as,


Choose a random $(\alpha, \beta)$
Integrate PDE to obtain $(x,y)$
Inspect how close to the desired result we are
Adjust $(\alpha, \beta)$. 
Repeat


I feel like there must be a smart/fast way to do this. Can anyone point me in the direction of some algorithms/resources which may be able to accomplish this goal? Thanks
Problem statement: You have a function $f:\mathbb{R}^2 \to \mathbb{R}^2$ where you know how to compute $f$ but not $f'$.  Given $(x,y)$, you want to find $(\alpha,\beta)$ such that $f(\alpha,\beta)=(x,y)$.

One approach is to use mathematical techniques for black-box optimization.  Define $g:\mathbb{R}^2 \to \mathbb{R}$ by

$$g(\alpha,\beta) = \|f(\alpha,\beta)-(x,y)\|^2,$$

i.e., the squared L2-norm of how far off the answer is.  Then your goal is to find $(\alpha,\beta)$ that minimizes $g$. Such a minimum will be a solution to your original problem.

The minimization problem could be solved using gradient descent, Newton's method, or some other iterative method.

The iterative methods will often require you to be able to compute the gradient of this function.  The gradient can be computed in a black-box way by using

$${\partial g \over \partial \alpha}(\alpha,\beta) = {g(\alpha+\epsilon,\beta) - g(\alpha-\epsilon,\beta) \over 2\epsilon}$$

where $\epsilon$ is a sufficiently small number, and similarly for ${\partial g \over \partial \beta}$.

This method assumes that $f$ is a "nice" function: e.g., continuous and differential and doesn't change too quickly, if you make small changes to its inputs.