Fitting "Phase" Transitions

[TiptoC01]

Hi Miles

I have been playing about with some real experimental data that hints at a continuous transition from one linear regime to another continuous linear regime. It is kind of reminisent of buckling data. The two regimes are not obvious and may just be my wishful thinking. However, I found pysr to really struggle to give good candidates for these sort of trends.

I have made a toy version below with simulated data listed below, and after tweaking with the operators complexities got something that seems reasonable. But I can't help feel I have almost put the answer I want in. Am I approaching this correctly?

The shift in the trends in the real data is much more subtle than this toy data ...

Carl

import numpy as np
import matplotlib.pyplot as plt
import sympy as sym
from pysr import PySRRegressor 

r = np.linspace(0,7,100)
eps = 0.01
out = (r+1) - np.sqrt(eps + 0.25 * (r-3)**2) 

yerr = np.ones(r.shape)*0.01  
w = 1.0/yerr**2  # Weighting by the inverse of the error squared


model = PySRRegressor(
    maxsize=30,
    niterations=100,
    binary_operators=["*","+"],
    unary_operators=["sqrt","square"],
    complexity_of_variables=1,
    complexity_of_constants=1,
    complexity_of_operators={"sqrt":1,"square":1,"*":2,"+":2},
    elementwise_loss="(prediction, target, weight) -> weight * (prediction - target)^2",
    model_selection="best"
)

model.fit(r.reshape(-1, 1), out,weights=w)

real_label = r'$(r+1) - \sqrt{\epsilon + 0.25 * (r-3)^2}$'

eq = model.sympy().simplify()

label = r'$'+sym.latex(eq)+'$'


plt.errorbar(r, out, yerr=yerr, fmt='.', label=real_label,capsize=2)
plt.plot(r, model.predict(r.reshape(-1, 1)),'-',label=label)
plt.legend()

plt.show()

[MilesCranmer]

I guess the issue here is that the remaining error is very tiny and remaining complexity is very large. Can you give it a cube operator maybe? Or even "^"?

Also try https://ai.damtp.cam.ac.uk/pysr/tuning/