Try looking at the literature for Radial Basis function inerpolation

The weights are determined by minimizing an objective function, which in some way describes the difference (mean-squared error) between the network output (predicted values) and the measured values from the testing data set.This optimization can be solved by linear regression, Gradient descent,  Levenberg–Marquardt algorithm etc. 

The determination of the weights is generally called  training. You will find more information if you look specifically for training of RBFN or ANN in the literature.

to determine how the weights are determined in this function, let us discuss the Radial Basis function(RBF) network itself as follows:

An RBF network consists of one input layer, one hidden layer and one output layer as shown in Figure 1.

Each dimension of the input pattern x corresponds to the input layer neuron. Each input layer neuron is connected to all the hidden layer neurons. The hidden layer consists of n neurons and one bias neuron. The weight vector between the input layer and hidden layer neuron i corresponds to its center vector ci.  Each hidden layer neuron computes the following Gaussian kernel function as it is in the equation_1 figure.

where ci and σi in equation_1  are the center and the width of hidden layer neuron i, respectively. If the width σi is small, the RBF decreases rapidly. If it is large, the RBF decreases slowly. The output layer consists of m neurons. The hidden layer and the output layer are fully connected.

Each output  neuron produces the linear weighted summation as it is in the equation_2 figure. where wij in equation_2  is the weights between hidden layer neuron i and output layer neuron j.

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In the paper below you can find the linear system of equations you need to solve for the training of a RBFN.

https://academic.oup.com/gji/article/214/1/460/4975275

One of the easiest method that' is Least square method is use to determine the weights in Radial Basis Function Network.