Data-driven modeling for the motion of a sphere falling through a non-Newtonian fluid

Learning physics by data for the motion of a sphere falling in a non-Newtonian fluid

“…where ω 2 = β 6 /2β 9 , c = β 3 /β 6 and 1 − d = 2β 1 β 9 /β 3 β 6 . What has been learned by Lasso in [9] is to find the coefficients to minimize Eζ 2 , and takes the result as ζ = 0. However, since the vectors {v ′′2 , v ′′ v, v ′′ , v} of data are linear independent, ζ is not a pure mathematical zero but is still a random distribution.…”

“…holds in order to reproduce the variance of the distribution of velocity v which means γ(0) = var(η). Now we just need to get an empirical distribution of v to simulate the behavior of a falling ball by our main algorithm (9). In our problem, the data of velocity are collected from the physical experiment in [7] which are showed in Figure 1.…”

“…Based on the histogram of v shown in Figure 3, the superposition of two uniform, two Wigner semicircle distributions and two triangular distributions on the interval [1,4] ∪ [7,9] are used first as a rule of thumb to describe the distribution of the velocity v in Example 1. Then considering the expectation and variance of the velocity samples, we give more precise distributions of the velocity v in Example 2.…”

“…From Figure 1, we observe that the samples of velocity mainly distribute in two parts, which would be regarded as the intervals [1,4] and [7,9] as a rule of thumb when getting into the equilibrium state. So the empirical distribution of v could be chosen as the superposition of two uniform, two Wigner semicircle or two triangular distributions (F u2 , F W 2 , F t2 ) supported on the intervals [1,4] and [7,9] with the weights determined by the ratio of the sudden acceleration in equilibrium state.…”

“…Wu and Zhang [9][10] developed a data-driven model by using the Lasso (Least absolute shrinkage and selection operator) to the second order quadric algebraic differential equation (ADE). Then the model has been compressed and only four terms are remained in the space of {v, v…”