Revealing Nature’s Laws: How Symbolic Regression Fits In#

Why Does Symbolic Regression Matter Now?#

In an age dominated by large-scale machine learning, interpretability has become a focal point. Regulatory frameworks like GDPR require explanations behind automated decisions, and many scientific disciplines desire models that can be understood and validated in the light of known theories. Symbolic regression’s ability to produce explicit formulas satisfies the thirst for interpretable modeling.

Representation of Solutions#

One of the most popular ways to represent equations is as syntax trees. For example, the expression:

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f(x, y) = x + 2 * y

could be represented as a tree:

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      (+)
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     /   \
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    x    (*)
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        /   \
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       2     y

Each node is either a function (like +, -, *, /, sin, cos, etc.) or a terminal (like variables x and y, or constants). This representation allows an evolutionary algorithm to mutate, crossover, and evaluate expressions with relative ease.

Genetic Operators and Evolution#

Crossover: Two parent trees get partial subtrees swapped to create offspring.

Mutation: A subtree in an individual solution is replaced randomly, potentially altering an operator or a terminal.

These operators mimic the idea of biological evolution:

• Crossover is akin to reproduction between parent organisms passing genetic material to offspring.
• Mutation introduces new genetic material, possibly giving the population access to new and better solutions.

Over many generations, the population ideally converges—or at least moves—toward better solutions.

Fitness Functions and Selection#

A fitness function is used to measure how well each candidate expression explains or fits the data. This fitness is often the mean squared error (MSE) or another relevant metric. Solutions that perform better (i.e., have lower MSE) are more likely to be chosen to reproduce and form the next generation. Various selection methods (e.g., tournament selection, roulette-wheel selection) can govern how the best or diverse solutions are chosen.

Symbolic regression also includes a mechanism to penalize complexity, commonly referred to as parsimony pressure. This helps avoid bloated expressions that overfit. Balancing accuracy with simplicity is key for interpretability and generalization.

Using gplearn#

Below is a minimal working example:

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import numpy as np
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from gplearn.genetic import SymbolicRegressor
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import matplotlib.pyplot as plt
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# Step 1: Generate Synthetic Data
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# Our true function: f(x) = x^2 + 3*x + 2
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np.random.seed(42)
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X = np.linspace(-10, 10, 200).reshape(-1, 1)
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y_true = X.ravel()**2 + 3*X.ravel() + 2
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# Add some noise
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y_noisy = y_true + np.random.normal(scale=5, size=y_true.shape)
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# Step 2: Configure Symbolic Regressor
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est = SymbolicRegressor(
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    population_size=500,
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    generations=20,
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    tournament_size=20,
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    stopping_criteria=0.01,
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    function_set=['add', 'sub', 'mul', 'div'],
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    parsimony_coefficient=0.01,
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    max_samples=0.9,
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    verbose=1,
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    random_state=42
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)
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# Step 3: Train the Model
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est.fit(X, y_noisy)
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# Step 4: Evaluate & Visualize
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y_pred = est.predict(X)
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print("Best individual formula:", est._program)
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mse = np.mean((y_pred - y_true) ** 2)
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print("MSE on true function:", mse)
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# Plot the results
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plt.scatter(X, y_noisy, label='Noisy Data', alpha=0.6)
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plt.plot(X, y_true, 'g-', label='True Function')
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plt.plot(X, y_pred, 'r--', label='Symbolic Regressor', linewidth=2)
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plt.legend()
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plt.show()

Interpreting and Evaluating Results#

Symbolic regression might produce an expression very close to x^2 + 3x + 2, perhaps with minor structural differences due to noise or random variation in the search. Even if the discovered expression differs, the final MSE can confirm how close it is in terms of predictive performance.

Controlling Model Complexity Through Parsimony#

One of the biggest challenges in symbolic regression is overfitting. Large expression trees can fit training data perfectly but fail to generalize. Parsimony pressure is the standard technique to control bloat. By penalizing solutions with too many nodes, we favor simpler solutions. Mathematically, one might augment the fitness function:

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fitness_individual = MSE + λ * size(individual)

where size(individual) is the number of nodes in the tree (or another measure of complexity), and λ is a small coefficient controlling complexity.

An alternative or supplementary approach is lexicographic parsimony pressure, where solutions of lower complexity automatically win in ties of performance.

Hybrid Approaches and Neural Methods#

Neural networks excel at handling very high-dimensional data, but they remain black boxes. Researchers have proposed hybrid methods that:

• Use a neural network to generate features or partial representations.
• Feed these representations into a symbolic regressor (e.g., a genetic program or a separate symbolic module).

One emerging theme is deep symbolic regression, where a neural network might propose symbolic building blocks in a reinforcement learning loop. Another is bridging the representational power of neural networks with the interpretability of symbolic equations—though research here is ongoing and not without challenges.

Domain Constraints and Physical Consistency#

In scientific fields, not all functional forms are physically valid. For example, if modeling a velocity function in physics, you might insist that certain positivity conditions remain true, or that certain transformations correlate with known physical laws (e.g., conservation of energy). Symbolic regression can incorporate such domain knowledge by:

• Restricting allowed operators or sub-expressions.
• Encoding constraints in the fitness function or using penalization.
• Guiding the algorithm with partial known relationships, ensuring discovered solutions remain physically consistent.

Physics and Biology#

In physics, a famous success story includes the works applying symbolic regression to motion data to discover laws akin to Newton’s second law or Kepler’s laws of planetary motion. In biophysics, symbolic formulas have been discovered to describe the regulatory mechanisms in gene expression or metabolic pathways.

Finance and Economics#

Symbolic regression can discover formulas linking market variables, interest rates, or trading volumes to predict asset prices—though caution is warranted. Financial data is notoriously noisy and high-dimensional, so robust search strategies and domain knowledge constraints become paramount.

Engineering and Control Systems#

The design or tuning of control systems can benefit from interpretable relationships. For instance, if a system’s dynamics can be approximated by a symbolic expression, engineers can manipulate that expression for better performance or stability. Symbolic controllers, in some advanced work, might yield explicit control laws that are verifiably stable under certain conditions.

Modern Genetic Programming Libraries#

Several libraries have emerged (or matured) in Python, Julia, R, and other ecosystems:

• gplearn (Python): A straightforward solution with a scikit-learn-like interface.
• PySR (Julia/Python): High-performance symbolic regression with GPU acceleration and user-friendly syntax.
• DEAP (Python): A framework for evolutionary algorithms, including symbolic regression.
• Eureqa/InsightfulAI: A closed-source commercial solution specializing in symbolic regression.

Choosing a library depends on performance needs, programming language preference, and the ease of integrating domain constraints.

Scaling to High-Dimensional Data#

A critical challenge arises when you have dozens, hundreds, or thousands of features. The search space becomes enormous. Strategies to cope include:

• Dimensionality reduction: Using principal component analysis (PCA) or autoencoders to reduce dimensionality before symbolic regression.
• Feature selection: Evolving solutions that only select subsets of input variables.
• Parallelization and GPU acceleration: Speeding up evaluations, which can be computationally expensive if fitness evaluations are large or repeated.

Continual Learning and Interpretability#

Researchers are looking at how symbolic regression can evolve solutions incrementally, learning from streams of data without losing previously found patterns. This approach can be especially helpful in fields where data arrives continuously, but interpretability is critical.

Discovering Kepler’s Third Law#

Kepler’s third law states that the square of a planet’s orbital period (T) is proportional to the cube of the semi-major axis (a) of its orbit around the sun. Simplified equation:

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T^2 �?a^3

Suppose you collect data on various planets: you measure T (period) and a (semi-major axis). Symbolic regression might output a relationship such as:

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T^2 = 4π^2/GM * a^3

(or a simplified version ignoring constants). The insight is that the discovered law is deeply tied to universal constants in gravitational physics (G, M for the mass of the sun). While a naive approach might not exactly yield that constant ratio, it can approximate it from the data.

Coupled Pendulum Dynamics#

A coupled pendulum system can have complex oscillatory behaviors with energy transfer between two pendulums. By recording angles and angular velocities over time, symbolic regression might uncover approximate equations of motion. This approach is more challenging due to multi-variable, time-series data, but the symbolic expression found can highlight the interplay between the pendulums�?angles, gravitational constants, and coupling stiffness.