Daniele Mortari

The K-vector Range Searching Technique is a range searching technique that can be applied to fast retrieve data from any static database. The k-vector technique was initially proposed to identify stars observed by star trackers on board spacecraft. Then, it has been applied to solve different kinds of problems belonging to different fields, such as: 1) nonlinear functions inversion and intersection, 2) extensive sampling data generation with assigned analytical (or numerical) distribution, 3) find approximate solutions of nonlinear Diophantine equations, and 4) iso-surface identification for 3-dimensional data distributions and level set analysis.

The Theory of functional connections (TFC) is a mathematical framework generalizing interpolation. TFC derives analytical functionals representing all possible functions subject to a set of linear constraints. These functionals restrict the whole space of functions to just the subspace that fully satisfies the constraints. Using these functionals, constrained optimization problems are transformed into unconstrained problems. Then, already available and optimized solution methods can be used. The TFC theory has been developed for multivariate rectangular domains subject to absolute, integral, relative, and linear combinations of constraints.^[11][12][13] Numerically efficient applications of TFC have already been implemented in optimization problems, especially in solving differential equations.^[14][15] In this area, TFC has unified initial, boundary, and multi-value problems by providing fast solutions at machine-error accuracy. This approach has already been applied to solve, in real-time, direct optimal control problems, such as autonomous landing on a large planetary body.^[16] Additional applications of TFC are found in nonlinear programming and calculus of variations,^[17] in Radiative Transfer^[18]Compartmental models in epidemiology,^[19] and in Machine learning,^[20] where orders of magnitude improvements in speed and accuracy are obtained thanks to the search-space restriction enabled by TFC.

Implementations of TFC in neural networks were first proposed by the Deep-TFC framework, then by the X-TFC using an Extreme learning machine, and by the Physics-informed neural networks (PINN). In particular, TFC allowed PINN to overcome the unbalanced gradients problem that often causes PINNs to struggle to accurately learn the underlying differential equation solution.