Optimized neural forms for solving ordinary differential equations

A critical issue in approximating solutions of ordinary differential equations using neural networks is the exact satisfaction of the boundary or initial conditions. For this purpose, neural forms have been introduced, i.e., functional expressions that depend on neural networks which, by design, satisfy the prescribed conditions exactly. Expanding upon prior progress, the present work contributes in three distinct aspects. First, it presents a novel formalism for crafting optimized neural forms. Second, it outlines a method for establishing an upper bound on the absolute deviation from the exact solution. Third, it introduces a technique for converting problems with Neumann or Robin conditions into equivalent problems with parametric Dirichlet conditions. The proposed optimized neural forms were numerically tested on a set of diverse problems, encompassing first-order and second-order ordinary differential equations, as well as first-order systems. Stiff and delay differential equations were also considered. The obtained solutions were compared against solutions obtained via Runge-Kutta methods and exact solutions wherever available. The reported results and analysis verify that in addition to the exact satisfaction of the boundary or initial conditions, optimized neural forms provide closed-form solutions of superior interpolation capability and controllable overall accuracy.

Eliminating Multicollinearity Issues in Neural Network Ensembles: Incremental, Negatively Correlated, Optimal Convex Blending

Given a {features, target} dataset, we introduce an incremental algorithm that constructs an aggregate regressor, using an ensemble of neural networks. It is well known that ensemble methods suffer from the multicollinearity issue, which is the manifestation of redundancy arising mainly due to the common training-dataset. In the present incremental approach, at each stage we optimally blend the aggregate regressor with a newly trained neural network under a convexity constraint which, if necessary, induces negative correlations. Under this framework, collinearity issues do not arise at all, rendering so the method both accurate and robust.