Discovery of Differential Equations from Numerical Data

This paper proposes a method of discovering some kinds of differential equations with interval coefficients, which characterize or explain numerical data obtained by scientific observations and experiments. Such numerical data inevitably involve some ranges of errors, and hence they are represented by closed intervals in this paper. Based on these intervals, we design some interval inclusions which approximate integral equations equivalent to the differential equations. Interval coefficients of the differential equations are determined by solving the interval inclusions. Many combinations of interval coefficients can be obtained from numerical data. We find out a differential equation whose coefficients consist of intersections of the computed interval coefficients. The refutability of the differential equation is also discussed. Our discovering method is verified by some simulations