Table 6.

Root mean square prediction error by polynomial regression and the method of discovering differential equations, as well as the discovered differential equation, at each noise level. The predictions by our algorithm have much less error than the predictions by polynomial regressions.

noise (%)	root mean square prediction error by polynomial regression of degree 4	root mean square prediction error by discovering differential equations
0	7.3052	0.0001
1	14.4525	0.0479
2	21.6053	0.2083
4	19.7245	0.2369
10	73.0118	1.0310
noise (%)	discovered differential equation
0	dy/dx = 2.000( ± 0.005) + 1.000( ± 0.001)x − 1.000( ± 0.001)y
1	dy/dx = 1.997( ± 0.221) + 1.015( ± 0.051)x − 1.000( ± 0.027)y
2	dy/dx = 1.772( ± 0.430) + 1.076( ± 0.099)x − 0.972( ± 0.053)y
4	dy/dx = 2.625( ± 0.649) + 0.897( ± 0.152)x − 1.097( ± 0.081)y
10	dy/dx =  − 0.921( ± 0.774) + 1.426( ± 0.220)x − 0.622( ± 0.096)y