  The numerical analysis of causal fermion systems is advanced by employing
differentiable programming methods. The causal action principle for weighted
counting measures is introduced for general values of the integer parameters
$f$ (the particle number), $n$ (the spin dimension) and $m$ (the number of
spacetime points). In the case $n=1$, the causal relations are clarified
geometrically in terms of causal cones. Discrete Dirac spheres are introduced
as candidates for minimizers for large $m$ in the cases $n=1, f=2$ and $n=2,
f=4$. We provide a thorough numerical analysis of the causal action principle
for weighted counting measures for large $m$ in the cases $n=1,2$ and
$f=2,3,4$. Our numerical findings corroborate that all minimizers for large $m$
are good approximations of the discrete Dirac spheres. In the example $n=1,
f=3$ it is explained how numerical minimizers can be visualized by projected
spacetime plots. Methods and prospects are discussed to numerically investigate
settings in which hitherto no analytic candidates for minimizers are known.
