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A PCA-based, machine learning version of the SPH method is proposed. In thepresent scheme, the smoothing tensor is computed to have their eigenvaluesproportional to the covariance's principal components, using a modified octreedata structure, which allows the fast estimation of the anisotropicself-regulating kNN. Each SPH particle is the center of such an optimal kNNcluster, i.e., the one whose covariance tensor allows the find of the kNNcluster itself according to the Mahalanobis metric. Such machine learningconstitutes a fixed point problem. The definitive (self-regulating) kNN clusterdefines the smoothing volume, or properly saying, the smoothing ellipsoid,required to perform the anisotropic interpolation. Thus, the smoothing kernelhas an ellipsoidal profile, which changes how the kernel gradients arecomputed. As an application, it was performed the simulation of collapse andfragmentation of a non-magnetic, rotating gaseous sphere. An interestingoutcome was the formation of protostars in the disc fragmentation, shown to bemuch more persistent and much more abundant in the anisotropic simulation thanin the isotropic case.

Covariance-based smoothed particle hydrodynamics. A machine-learning application to simulating disc fragmentation