Feynman equations - python

Provides a convenient way to use the Feynman equations in python code.

Generate datasets and utilize equations

from Feynman.Functions import Feynman12 

inputSize = 10000

# (1) generate pandas datasets without noise

df = Feynman12.generate_df()

# (2) specify size of the uniform input range

df = Feynman12.generate_df(size = inputSize)

# (3) generate pandas datasets with noise
# pandas DataFrame ['q2','Ef','F']

df = Feynman12.generate_df(size = inputSize, noise_level=0.3)

# (4) generate datasets with noise and original target without noise
# pandas DataFrame ['q2','Ef','F','F_without_noise']

df = Feynman12.generate_df(size = inputSize, noise_level=0.3, include_original_target=True)

# (5) use the functions to calculate equations directly

inputSize = 10000
X = np.random.uniform([1.0,1.0], [5.0,5.0], (inputSize,2))
q2 = X[:,0] 
Ef = X[:,1] 
# f: q2*Ef
f1 = Feynman12.calculate(q2,Ef)

# (6) use the JSON representation
#     e.g. you want to iterate over all functions in code

# index one specific function
jsonArr = np.array(FunctionsJson)
json = jsonArr[[row['EquationName']== 'Feynman12' for row in FunctionsJson]][0]

# get executable python code
eq = eval(json['Formula_Lambda'])
f2 = [eq(row) for row in X]

# both produce same result
(f1==f2).all() #-> true

without noise	with noise
pairplot without noise	pairplot with noise
linechart without noise, sorted by [ 'Ef','q2', ]	linechart with noise, sorted by [ 'Ef','q2', ]
linechart without noise, sorted by [ 'q2','Ef', ]	linechart with noise, sorted by [ 'q2','Ef', ]

Information about function shapes in standard input space

Additional information about the function shape is provided by analyzing the value ranges of partial derivates of each Feynman function (whereever the partial derivatives can be symbolically calculated):

Problem	Variable	Input Space
Feynman 12	q2	[1.0,5.0]
Feynman 12	Ef	[1.0,5.0]

Problem	Derived by	Order	Derivative	Monotonicity
Feynman 12	q2	1	Ef	increasing
Feynman 12	q2	2	0	constant
Feynman 12	Ef	1	q2	increasing
Feynman 12	Ef	2	0	constant

The results of this analysis are provided in the file Feynman/Constraints.py. Calculation of the shape properties is implemented in create_files/sample_constraints.py.

Equation Sources

FeynmanEquations generate/src/FeynmanEquations.csv and generate/src/BonusEquations.csv were retrieved on 01.04.2022 from https://space.mit.edu/home/tegmark/aifeynman.html. The code in generate_functions.py parses these two files and generates the python code of Feynman/Functions.py.

Changes from the original:

The following entries had mismatches between the specified number of variables and the actual number of specified variables with their input ranges. A new download from the original source might therefore not match the csv provided in this repository.

#1
from: II.37.1,83,E_n,mom*(1+chi)*B,6,mom,1,5,B,1,5,chi,1,5,,,,,,,,,,,,,,,,,,,,,
to:   II.37.1,83,E_n,mom*(1+chi)*B,3,mom,1,5,B,1,5,chi,1,5,,,,,,,,,,,,,,,,,,,,,

#2
from: I.18.12,22,tau,r*F*sin(theta),2,r,1,5,F,1,5,theta,0,5,,,,,,,,,,,,,,,,,,,,,
to:   I.18.12,22,tau,r*F*sin(theta),3,r,1,5,F,1,5,theta,0,5,,,,,,,,,,,,,,,,,,,,,

#3
from: I.18.14,23,L,m*r*v*sin(theta),3,m,1,5,r,1,5,v,1,5,theta,1,5,,,,,,,,,,,,,,,,,,
to:   I.18.14,23,L,m*r*v*sin(theta),4,m,1,5,r,1,5,v,1,5,theta,1,5,,,,,,,,,,,,,,,,,,

#4
from: I.38.12,39,r,4*pi*epsilon*(h/(2*pi))**2/(m*q**2),3,m,1,5,q,1,5,h,1,5,epsilon,1,5,,,,,,,,,,,,,,,,,,
to:   I.38.12,39,r,4*pi*epsilon*(h/(2*pi))**2/(m*q**2),4,m,1,5,q,1,5,h,1,5,epsilon,1,5,,,,,,,,,,,,,,,,,,

#5
from: III.10.19,91,E_n,mom*sqrt(Bx**2+By**2+Bz**2),3,mom,1,5,Bx,1,5,By,1,5,Bz,1,5,,,,,,,,,,,,,,,,,,
to:   III.10.19,91,E_n,mom*sqrt(Bx**2+By**2+Bz**2),4,mom,1,5,Bx,1,5,By,1,5,Bz,1,5,,,,,,,,,,,,,,,,,,

#6
from: III.19.51,99,E_n,-m*q**4/(2*(4*pi*epsilon)**2*(h/(2*pi))**2)*(1/n**2),4,m,1,5,q,1,5,h,1,5,n,1,5,epsilon,1,5,,,,,,,,,,,,,,,
to:   III.19.51,99,E_n,-m*q**4/(2*(4*pi*epsilon)**2*(h/(2*pi))**2)*(1/n**2),5,m,1,5,q,1,5,h,1,5,n,1,5,epsilon,1,5,,,,,,,,,,,,,,,

#7
from: test_12,12,2.11 Jackson,12,F,q/(4*pi*epsilon*y**2)*(4*pi*epsilon*Volt*d-q*d*y**3/(y**2-d**2)**2),4,q,1,5,y,1,3,Volt,1,5,d,4,6,epsilon,1,5,,,,,,,,,,,,,,,
to:   test_12,12,2.11 Jackson,12,F,q/(4*pi*epsilon*y**2)*(4*pi*epsilon*Volt*d-q*d*y**3/(y**2-d**2)**2),5,q,1,5,y,1,3,Volt,1,5,d,4,6,epsilon,1,5,,,,,,,,,,,,,,,

#8
from: test_13,13,3.45 Jackson,13,Volt,1/(4*pi*epsilon)*q/sqrt(r**2+d**2-2*r*d*cos(alpha)),4,q,1,5,r,1,3,d,4,6,alpha,0,6,epsilon,1,5,,,,,,,,,,,,,,,
to:   test_13,13,3.45 Jackson,13,Volt,1/(4*pi*epsilon)*q/sqrt(r**2+d**2-2*r*d*cos(alpha)),5,q,1,5,r,1,3,d,4,6,alpha,0,6,epsilon,1,5,,,,,,,,,,,,,,,

#9
test_18,18,15.2.1 Weinberg,18,rho_0,3/(8*pi*G)*(c**2*k_f/r**2+H_G**2),4,G,1,5,k_f,1,5,r,1,5,H_G,1,5,c,1,5,,,,,,,,,,,,,,,
test_18,18,15.2.1 Weinberg,18,rho_0,3/(8*pi*G)*(c**2*k_f/r**2+H_G**2),5,G,1,5,k_f,1,5,r,1,5,H_G,1,5,c,1,5,,,,,,,,,,,,,,,

#10
test_19,19,15.2.2 Weinberg,19,pr,-1/(8*pi*G)*(c**4*k_f/r**2+H_G**2*c**2*(1-2*alpha)),5,G,1,5,k_f,1,5,r,1,5,H_G,1,5,alpha,1,5,c,1,5,,,,,,,,,,,,
test_19,19,15.2.2 Weinberg,19,pr,-1/(8*pi*G)*(c**4*k_f/r**2+H_G**2*c**2*(1-2*alpha)),6,G,1,5,k_f,1,5,r,1,5,H_G,1,5,alpha,1,5,c,1,5,,,,,,,,,,,,