Symbolic Regression (PySR)

Overview

Discover symbolic equations that fit data using PySR — a multi-population evolutionary algorithm with a Julia backend. PySR searches over the space of mathematical expressions to find equations that balance accuracy and simplicity.

When to Use

• Discovering governing equations from experimental data
• Finding conservation laws or invariants
• Replacing black-box ML models with interpretable formulas
• Dimensional analysis-constrained equation search
• Validating theoretical predictions against data

Do NOT Use When

• You already know the functional form (use physics-fitting instead)
• Data is high-dimensional (> 5-6 input variables)
• You need a time-series model (use sindy for dynamical systems)

Installation

PySR requires Julia (auto-installed on first run):

pip install pysr
# First import will install Julia ~2 min
python -c "import pysr; pysr.install()"

Core Workflows

1. Basic Equation Discovery

import numpy as np
from pysr import PySRRegressor

# Generate data from a known law (for testing)
np.random.seed(42)
X = np.random.randn(100, 2)  # 2 input variables
x1, x2 = X[:, 0], X[:, 1]
y = 2.5 * np.sin(x1) + x2**2  # true equation
y += 0.1 * np.random.randn(100)  # noise

# Fit
model = PySRRegressor(
    niterations=40,
    binary_operators=["+", "-", "*", "/"],
    unary_operators=["sin", "cos", "exp", "sqrt", "abs"],
    populations=20,
    population_size=50,
    maxsize=25,          # max expression complexity
    parsimony=0.0032,    # penalty for complexity
    progress=True,
)
model.fit(X, y)

# Results: Pareto front of accuracy vs complexity
print(model)
print(f"\nBest equation: {model.sympy()}")
print(f"Score (accuracy/complexity): {model.score(X, y):.4f}")

2. Physics-Constrained Search (Dimensional Analysis)

# Tell PySR about physical dimensions to constrain the search
# Example: Kepler's 3rd law from orbital data
# Variables: a (semi-major axis [m]), M (central mass [kg]), T (period [s])

import numpy as np
from pysr import PySRRegressor

# Training data: known planets
# T^2 = (4pi^2 / GM) * a^3
from scipy import constants as const
G = const.G
M_sun = 1.989e30

a_data = np.array([57.9, 108.2, 149.6, 227.9, 778.6]) * 1e9  # meters
T_data = np.array([88, 224.7, 365.2, 687, 4331]) * 86400       # seconds

X = np.column_stack([a_data, np.full_like(a_data, M_sun), np.full_like(a_data, G)])

model = PySRRegressor(
    niterations=50,
    binary_operators=["+", "-", "*", "/", "^"],
    unary_operators=["sqrt", "square", "cube"],
    populations=30,
    maxsize=20,
    parsimony=0.005,
    # Dimensional constraints (optional but powerful)
    # variable_names=["a", "M", "G"],
)
model.fit(X, T_data**2)  # Fit T^2 as function of a, M, G

print("Discovered equation for T²:")
print(model.sympy())
# Should find something proportional to a^3 / (G * M)

3. Custom Operators for Physics

model = PySRRegressor(
    niterations=40,
    binary_operators=["+", "-", "*", "/"],
    unary_operators=[
        "sin", "cos",
        "exp", "log",
        "sqrt", "square",
        "abs",
    ],
    # Extra operators defined in Julia
    extra_sympy_mappings={
        "inv": lambda x: 1/x,
    },
    # Constraints on operator nesting
    nested_constraints={
        "sin": {"sin": 0, "cos": 0},   # no sin(sin(x))
        "cos": {"sin": 0, "cos": 0},
        "exp": {"exp": 0, "log": 0},
        "log": {"exp": 0, "log": 0},
    },
    maxsize=30,
    parsimony=0.003,
)

4. Multi-Output Regression

# Discover multiple equations simultaneously
# Example: find both components of a 2D force field

X = np.random.randn(200, 2)
F_x = -X[:, 0] / (X[:, 0]**2 + X[:, 1]**2)**1.5
F_y = -X[:, 1] / (X[:, 0]**2 + X[:, 1]**2)**1.5

# Fit each component separately
model_Fx = PySRRegressor(niterations=40, binary_operators=["+", "-", "*", "/"],
                          unary_operators=["sqrt", "square", "inv(x)=1/x"],
                          maxsize=20)
model_Fy = PySRRegressor(niterations=40, binary_operators=["+", "-", "*", "/"],
                          unary_operators=["sqrt", "square", "inv(x)=1/x"],
                          maxsize=20)

model_Fx.fit(X, F_x)
model_Fy.fit(X, F_y)

print(f"F_x = {model_Fx.sympy()}")
print(f"F_y = {model_Fy.sympy()}")

5. Analyzing the Pareto Front

# The Pareto front shows the tradeoff between accuracy and complexity
import matplotlib.pyplot as plt

equations = model.equations_  # DataFrame of all equations

fig, ax = plt.subplots(figsize=(10, 6))
ax.scatter(equations['complexity'], equations['loss'], c='steelblue', s=20)

# Highlight Pareto-optimal equations
pareto = equations[equations['score'] > 0]
ax.scatter(pareto['complexity'], pareto['loss'], c='red', s=50,
           zorder=5, label='Pareto front')

ax.set_xlabel('Complexity', fontsize=13)
ax.set_ylabel('Loss (MSE)', fontsize=13)
ax.set_yscale('log')
ax.set_title('Accuracy vs Complexity Tradeoff')
ax.legend()
plt.savefig('pareto_front.png', dpi=150, bbox_inches='tight')

# Print top equations from Pareto front
print("\nPareto-optimal equations:")
for _, row in pareto.iterrows():
    print(f"  Complexity {int(row['complexity']):2d}: {row['equation']:<40s}  loss={row['loss']:.6f}")

Key Parameters

Parameter	Default	Description
niterations	40	Number of evolutionary iterations (more = better but slower)
populations	15	Number of independent populations (parallelism)
population_size	33	Size of each population
maxsize	20	Maximum expression tree size (complexity limit)
parsimony	0.0032	Penalty for complexity (higher = simpler equations)
binary_operators	["+","-","*","/"]	Allowed binary operations
unary_operators	[]	Allowed unary operations (sin, cos, exp, etc.)
nested_constraints	{}	Prevent operator nesting (sin(sin(x)))
deterministic	False	Set True for reproducibility (slower)
procs	auto	Number of parallel processes

Tips for Physics Problems

1. Start simple: Begin with ["+", "-", "*", "/"] only, add sin/cos/exp if needed
2. Use parsimony: Physics equations are usually simple. Set parsimony=0.005-0.01
3. Constrain nesting: Prevent sin(sin(x)) and exp(exp(x)) with nested_constraints
4. Normalize data: Scale variables to O(1) for better convergence
5. Provide enough data: 100-1000 points is typical; more helps with noise
6. Check the Pareto front: The "best" equation isn't always the most complex one
7. Validate discovered equations: Test on held-out data and check dimensional consistency

Troubleshooting

Symptom	Fix
Julia installation fails	Run python -c "import pysr; pysr.install()" manually
Very slow	Reduce maxsize, populations, or niterations
Only finds constants	Data may be too noisy; increase niterations or clean data
Overly complex equations	Increase parsimony (e.g., 0.01)
Missing the true equation	Add the needed operators (e.g., sin if periodicity expected)
Inconsistent results	Set deterministic=True and random_state=42