  A PCA-based, machine learning version of the SPH method is proposed. In the
present scheme, the smoothing tensor is computed to have their eigenvalues
proportional to the covariance's principal components, using a modified octree
data structure, which allows the fast estimation of the anisotropic
self-regulating kNN. Each SPH particle is the center of such an optimal kNN
cluster, i.e., the one whose covariance tensor allows the find of the kNN
cluster itself according to the Mahalanobis metric. Such machine learning
constitutes a fixed point problem. The definitive (self-regulating) kNN cluster
defines the smoothing volume, or properly saying, the smoothing ellipsoid,
required to perform the anisotropic interpolation. Thus, the smoothing kernel
has an ellipsoidal profile, which changes how the kernel gradients are
computed. As an application, it was performed the simulation of collapse and
fragmentation of a non-magnetic, rotating gaseous sphere. An interesting
outcome was the formation of protostars in the disc fragmentation, shown to be
much more persistent and much more abundant in the anisotropic simulation than
in the isotropic case.
