M1 Macbook Pro: 8 Core CPU and 8 Core GPU

Model parameters: BF_try_time=60, BF_ops_file_type="7ops.txt", polyfit_deg=3, NN_epochs=500

Normal: s = s0 + v0 * t - 0.5 * a * t**2

Buckingham π: -0.5π₂π₃² + π₃ + 1

Noise Level: 0 | Equation: 0.000000000000+(x0+(((x3*(x1+(x1-(x3*x2)))))/((0+1)+1)))

Noise Level: 0.01 | Equation: 0.000055730232+(x0+(((x3*(x1+(x1-(x3*x2)))))/((0+1)+1)))

Noise Level: 0.05 | Equation: sqrt(0.000452721013*(x2+(x1+x1)))

Noise Level: 0.1 | Equation: asin(1.839030019605+(-sqrt((x3+1))))

where x0 is s0, x1 is v0, x2 is a, & x3 is t

Noise Level: 0 | Equation: 0.500000000000*(((x1*(((x1*(-x0))+1)+1))+1)+1)

Noise Level: 0.01 | Equation: 0.500320478519*((x1+((x1*((x1*(-x0))+1))+1))+1)

Noise Level: 0.05 | Equation: 0.501244731139*(((x1*(((x1*(-x0))+1)+1))+1)+1)

Noise Level: 0.1 | Equation: 0.493612853809*(((x1*(((x1*(-x0))+1)+1))+1)+1)

where x0 is π₂ & x1 is π₃

Model parameters: BF_try_time=100, BF_ops_file_type="7ops.txt", polyfit_deg=3, NN_epochs=500

Normal: vt = ((2 * m * g) ** 0.5)/ (((rho * Cd * A) ** 0.5))

Buckingham π: π₃ = 2π₂/(π₁)²

Noise Level: 0 | Equation: 1.414213562373 * sqrt((x1*(x0/(x4*(x3*x2)))))

where x0 is m, x1 is g, x2 is rho, x3 is Cd, & x4 is A

Noise Level: 0 | Equation: 0.000000000000+((x1+x1)/(x0*x0))

where x0 is π₁ & x1 is π₂

Model parameters: BF_try_time=100, BF_ops_file_type="7ops.txt", polyfit_deg=3, NN_epochs=500

Normal: gPE = (gc * m1 * m2) * ((1/r1) - (1/r2))

Buckingham π: π₁ = (π₃π₂ - 1)/π₃

Noise Level: 0 | Equation: 0.000000000000+((((x3)(-1)-(x2)(-1)))(-((x0x1)*x4)))

where x0 is m1, x1 is m2, x2 is r1, x3 is r2, & x4 is gc

Noise Level: 0 | Equation: 0.000000000000+((x0/x1)+(-x0))

where x0 is π₂ & x1 is π₃

Model parameters: BF_try_time=100, BF_ops_file_type="7ops.txt", polyfit_deg=3, NN_epochs=500

Normal: pf = f * (l/d) * ((p * v * v)/2)

Buckingham π: π₁ = π₃π₂/2

Noise Level: 0 | Equation: asin(-1.489095888275+sqrt(sqrt(x1)))

where x0 is f, x1 is d, x2 is l, x3 is p, & x4 is v

Noise Level: 0 | Equation: 0.500000000000*(x1*x0)

where x0 is π₂ & x1 is π₃

Model parameters: BF_try_time=100, BF_ops_file_type="7ops.txt", polyfit_deg=3, NN_epochs=500

Normal: n = n0 * (np.e ** (((-m * g * x))/((kb * temp))))

Buckingham π: π₂ = π₃e^(-1/π₁)

Noise Level: 0 | Equation: asin(-0.000000578547+((((x4/((x1*x2)x3)))x5)((((x4/((x1x2)x3)))x5)((((x4/((x1x2)x3)))x5)((((x4/((x1x2)*x3)))*x5)*x0)))))

where x0 is n0, x1 is m, x2 is g, x3 is x, x4 is kb, & x5 is temp

Noise Level: 0 | Equation: asin(-0.000000613685+(x1*(x0*(x0*(x0*x0)))))

where x0 is π₁ & x1 is π₃

Model parameters: BF_try_time=100, BF_ops_file_type="7ops.txt", polyfit_deg=3, NN_epochs=500

Normal: f = (-g * m1 * m2) / (r * r)

Buckingham π: π₁ = -π₂

Noise Level: 0 | Equation: 0.000000000000+((x1x2)(x0/(x3*(-x3))))

where x0 is g, x1 is m1, x2 is m2, & x3 is r

Noise Level: 0 | Equation: -1.000000000000*(0+1)*x0

where x0 is π₂

It can be clearly noted that Buckingham π implementations tends to be 3-6 times faster than traditional methods. The solution time is consistently smaller than the normal equation data generated files, which tends to grow with more noise. Though both do struggle to represent the true parent equation at various noise tolerances, Buckingham π has significantly less error. It is also interesting to note that while the Buckingham π files also do possess high error, that's only because they are being compared to the dataset that has noise. When comparing them to the orginal they are actually were close to each other and represent the parent equation very well.

Although the parent equation in this example is considerably more complex leading to larger solution times for both, Buckingham π implementations allow for less error and to be marginally faster than the normal data files. With the same notes from the previous example, it can be added that the computational power of the machine will obviously further underline the difference in the times and errors between the two.

Same results as expected.

This was interesting as both the normal and Buckingham π equations couldn't model the expected results even when the data was correct. This could be a plausible limit of the codebase to accurately represent equations that contain exponents.

If you compare with, build on, or use aspects of the AI Feynman work, please cite the following:

@article{udrescu2020ai,
  title={AI Feynman: A physics-inspired method for symbolic regression},
  author={Udrescu, Silviu-Marian and Tegmark, Max},
  journal={Science Advances},
  volume={6},
  number={16},
  pages={eaay2631},
  year={2020},
  publisher={American Association for the Advancement of Science}
}

@article{udrescu2020ai,
  title={AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity},
  author={Udrescu, Silviu-Marian and Tan, Andrew and Feng, Jiahai and Neto, Orisvaldo and Wu, Tailin and Tegmark, Max},
  journal={arXiv preprint arXiv:2006.10782},
  year={2020}
}
```c