Table 3.

Inference of a physical pendulum

	Box is flat	Box is rotated −1.57rad	Box is rotated −2.6rad
General form	dθ/dt = αω + β
	dω/dt = γsin(δθ + ϕ) + f(θ, ω)
Fraction of models matching
Control experiment	4 of 30 (13%)	26 of 30 (87%)	6 of 30 (20%)
Full algorithm	21 of 30 (70%)	29 of 30 (97%)	20 of 30 (67%)
Means and standard deviations	α = 1.0040 ± 0.0009	α = 1.0008 ± 0.0014	α = 1.0039 ± 0.0011
	β = 0.0001 ± 0.0001	β = 0.0028 ± 0.0035	β = −0.0003 ± 0.0007
	γ = −19.43 ± 2.67	γ = −20.45 ± 1.06	γ = −22.61 ± 0.69
	δ = 1.104 ± 0.047	δ = 1.0009 ± 0.0032	δ = 1.0110 ± 0.0492
	ϕ = 0 ± 0	ϕ = −1.575 ± 0.026	ϕ = −2.6730 ± 0.1149
Residual functions	f(θ,ω)={−0.2080, −0.29(ω+1.45), −0.05ω(ω+5.48), −0.24(ω+0.98), −0.26(ω+1.4), …}	f(θ,ω)={1.98sin(θ), cos(1.35θ), −0.204( ω−2.57), sin(−4.02θ), cos(−2.11θ), …}	f(θ,ω)={cos(1.21θ), cos(−5.05θ), 0, sin(θ), cos(1.12θ), …}

A control algorithm (which fully samples the data set) and the full algorithm (which samples the data set using automated probing) were both presented with the same behavior from three hidden physical systems (SI Fig. 6 a–c). The systems differ only in the rotation of the box (angle offset), which changes the relationship between the variables (angle and angular velocity). Each system was perturbed to produce time series data (SI Fig. 6 d–f). Both algorithms were run against each data set 30 times; each trial was conducted for 2 min. For each trial, the model with lowest error against all observed system data was output. These resulting models were often found to fit a general form. The mean parameter estimation for the full algorithm is also shown, along with the residual functions many of the models possessed.