From Raw Data to Revealed Equations: A Journey with Symbolic Regression#

1. The Search Space of Equations#

Symbolic regression typically works with a collection of mathematical operations, variables, and constants to create candidate expressions. Common operations include addition, subtraction, multiplication, and division, but can also extend to exponentiation, trigonometric functions, logarithms, and more. By exploring combinations of these operations, symbolic regression can, in principle, discover a vast array of functional forms.

2. Representation of Candidate Solutions#

Internally, symbolic regression systems encode mathematical expressions in a structure like a tree. Each node of the tree corresponds to an operator (e.g., +, �? ×, ÷) or a function (log, exp, sin, etc.), and the leaf nodes correspond to variables or constants. This tree-based representation dictates how the expression is constructed.

For instance, an expression like:

f(x, y) = (x² + 3 �?y) ÷ sin(x)

can be encoded as a tree with “÷�?as the root, having two children: the numerator (x² + 3y) and the denominator (sin(x)).

3. Fitness Function#

To evaluate how well a candidate expression fits the data, symbolic regression uses a fitness function—often a measure like mean squared error (MSE) or mean absolute error (MAE) between the expression’s predictions and the actual data values. Lower fitness metric values indicate expressions that better capture the underlying relation of the data. Fitness evaluation guides the search process toward better solutions, driving the generation of new candidate formulas.

1. Predefined Models vs. Discovery#

Traditional regression methods (linear, polynomial, logistic, etc.) require you to specify a form—like y = α + βx + …—before fitting. Symbolic regression starts with a blank slate, spontaneously generating and testing possible expressions that relate inputs to outputs. This shift from fitting known forms to searching for the forms themselves is the essence of symbolic regression.

2. Interpretability#

Symbolic regression is naturally interpretable, yielding closed-form equations. Meanwhile, many popular regression techniques can turn into “models�?that are not obviously interpreted if their complexity is high. The equation discovered by symbolic regression provides, at a glance, insight into which variables matter most, how they combine, and whether the relationship is linear, exponential, sinusoidal, etc.

3. Complexity and Overfitting#

In traditional regression, we manage overfitting by restricting polynomial degree or applying regularization. Symbolic regression addresses over-complexity through various strategies such as parsimony pressure (encouraging simpler expressions), multi-objective optimization, or domain constraints on functions.

Step 1: Generate Data in Python#

Below is a small snippet that creates noisy data from this equation:

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import numpy as np
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# Set random seed for reproducibility
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np.random.seed(42)
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# Generate data
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X = np.linspace(0, 10, 50)
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y_true = 2 * X + 1
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noise = np.random.normal(0, 1, size=X.shape)
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y_noisy = y_true + noise
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data = np.column_stack([X, y_noisy])

Here we have 50 data points where each row contains [x, y].

Step 2: Initialize a Symbolic Regression Tool#

Several Python packages allow symbolic regression. For this brief example, let’s assume we use a fictitious library called “SymbolicGP.�?(Later, we’ll discuss actual libraries.) A basic usage might look like this:

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from symbolic_gp import SymbolicRegressor
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# Initialize a symbolic regressor with basic operators
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regressor = SymbolicRegressor(operators=['+', '-', '*', '/'],
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                              max_depth=5,
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                              population_size=200,
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                              generations=50)
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# Fit the regressor
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regressor.fit(X.reshape(-1,1), y_noisy)
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# Retrieve the discovered formula
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formula = regressor.get_best_formula()
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print("Discovered formula:", formula)

With sufficient training data and an appropriate fitness function, the discovered formula might be close to �?x + 1�?plus a small constant to account for the noise. In practice, if you run symbolic regression on actual noisy data, you might see something like �?2.05 + x) + 0.97x,�?which simplifies to approximately 2.02x + 2.05. The library itself might auto-simplify it to something more elegant, depending on its simplification module.

Step 3: Visualize Results#

Symbolic regression is particularly illustrative when you plot the discovered formula against the real data points:

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import matplotlib.pyplot as plt
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y_pred = regressor.predict(X.reshape(-1,1))
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plt.scatter(X, y_noisy, label='Data')
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plt.plot(X, y_pred, color='red', label='Symbolic Regression Fit')
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plt.xlabel('X')
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plt.ylabel('Y')
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plt.legend()
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plt.show()

You’ll see how closely the symbolic formula aligns with the data.

1. Cleaning and Preprocessing#

Just like with any modeling approach, raw or unclean data can derail the performance of symbolic regression. Handle outliers, missing values, and inconsistent scaling among variables. While symbolic regression is somewhat resilient to different data ranges, normalizing or standardizing your features might still be beneficial, depending on the tool.

2. Variable Selection#

Symbolic regression can attempt to use all the variables in different ways. However, having too many irrelevant features may overwhelm the search process. Initial correlation checks, domain knowledge, or feature selection techniques can reduce the candidate variable space, helping the evolutionary search converge on simpler, more interpretable expressions.

3. Handling Noisy Data#

Real data often contains measurement noise, which can lead symbolic regression astray or push it toward overfitting. Strategies to mitigate this include:

• Limiting maximum expression depth.
• Using parsimony pressure or multi-objective optimization (balancing accuracy and simplicity).
• Using cross-validation or hold-out sets to gauge out-of-sample performance.

4. Validation and Testing#

Even though symbolic regression attempts to find a formula that fits the training data, you still need robust validation approaches. Split your dataset into training, validation, and test sets. Observe whether the discovered expressions generalize well to new data. Overly complex expressions might fit the training set nicely but fail on the test set.

1. Multi-Objective Optimization#

In many scenarios, you might care about more than just the error metric. For instance, you want a formula with high accuracy and low complexity. This leads to multi-objective optimization, typically employing methods like Pareto fronts. Points on the Pareto front represent different trade-offs between two or more objectives—e.g., accuracy vs. interpretability.

2. Constraints and Domain Knowledge#

Sometimes you already have partial knowledge about the final formula. For example, you might know the model should be dimensionally consistent, or that certain physical variables must appear together (like mass and gravitational acceleration). Incorporating domain constraints can prune the search space, leading to more plausible and faster-to-obtain solutions.

3. Hybrid Methods#

Symbolic regression can be hybridized with other methods. For instance, you can embed neural network transformations or use partial symbolic transformations combined with standard regression. Another approach is to use a learned neural representation as a starting point for symbolic regression, hoping to convert the neural network’s approximate relationships into a closed-form formula for interpretability.

4. Parallelism and GPU Acceleration#

Evaluating candidate solutions can be computationally expensive. Parallelism, either with multi-core CPUs or via GPU acceleration, speeds up the evolutionary process. Libraries like SymbolicRegression.jl (the core behind PySR) exploit parallel computing to handle large-scale symbolic regression tasks.

1. Hyperparameters#

Symbolic regression has numerous hyperparameters that strongly influence performance and results:

• Population size
• Number of generations
• Maximum tree depth
• Types of operators allowed
• Mutation rate
• Crossover rate

Tuning these hyperparameters can significantly affect runtime and the majesty (or monstrosity) of the discovered equation.

2. Stopping Criteria#

When does the search stop? Common criteria include:

• Reaching a specified fitness threshold (e.g., below a certain error).
• Exceeding a maximum number of generations or function evaluations.
• Lack of improvement over a certain number of generations (“early stopping�?.

3. Parsimony Principles#

Expressions can grow unwieldy, turning from a neat polynomial into a baroque tapestry of nested functions. Parsimony pressure (or complexity penalty) helps the algorithm prefer simpler expressions at the same accuracy level. Some frameworks implement a cost function that penalizes the number of nodes in an expression, guiding the evolutionary process to remain succinct.

4. Post-Processing and Simplification#

After an expression is found, symbolic manipulation libraries (e.g., Sympy in Python) can simplify it. Automated simplification might combine like terms, reduce fractions, or reorder operators to produce more comprehensible formulas.

1. Dynamic Function Sets#

Professional-level symbolic regression often includes dynamic or domain-driven function sets. For example, if you suspect a process has strong frequency components, you explicitly include sine and cosine functions. If exponential decay is likely, you include exp. You can also omit irrelevant functions to tighten the search space.

2. Ensemble Symbolic Methods#

An advanced extension of symbolic regression is to create an ensemble of discovered expressions. Each expression might capture certain aspects of the data. By combining them—either by voting or weighting—one might achieve more robust predictions, though interpretability can decrease. Nevertheless, the ensemble approach can sometimes highlight recurring elements across solutions, offering insights into reliable structural components.

3. Hybrid with Physics-Informed Neural Networks#

In certain scientific contexts, you can combine symbolic regression with physics-informed neural networks (PINNs). The network is guided by known physical laws but can also discover novel functional relationships that fit the data better. Then, a symbolic regression layer attempts to represent the discovered patterns in closed-form expressions. This domain-driven approach ensures the final model is consistent with fundamental principles while improving interpretability.

4. Automated Dimensional Analysis#

When dealing with physical data (mass, length, time, etc.), dimensional consistency is crucial. A sophisticated approach is to automatically preserve dimensional homogeneity by only combining variables in dimensionally consistent ways. This can dramatically shrink the search space and improve the validity of the output equations.

5. Uncertainty Quantification#

Understanding the confidence intervals or uncertainty in the discovered equations is another frontier. Some frameworks can generate a distribution of plausible expressions. By examining how frequently certain structural elements appear (such as polynomial terms, exponents, or certain variable interactions), you gain insight into which parts of the formula are most robust.

Step 1: Installing gplearn#

If you haven’t already:

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pip install gplearn

Step 2: Generating Data#

Let’s suppose we have a more complex function:

y = x² + 2x + sin(x)

with added noise for realism:

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import numpy as np
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np.random.seed(42)
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X = np.linspace(-5, 5, 100).reshape(-1,1)
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y_true = X[:, 0]**2 + 2*X[:, 0] + np.sin(X[:, 0])
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noise = np.random.normal(scale=0.2, size=X.shape[0])
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y_noisy = y_true + noise

Step 3: Fitting the Model#

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from gplearn.genetic import SymbolicRegressor
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est = SymbolicRegressor(population_size=5000,
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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', 'sin'],
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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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est.fit(X, y_noisy)

Step 4: Retrieve the Best Program#

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print("Best program:", est._program)
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y_pred = est.predict(X)

The result might look like:

Best program: add(add(mul(X, X), mul(2.0, X)), sin(X))

We can see that the discovered formula is x² + 2x + sin(x), which matches our original function. Minor discrepancies may arise due to random noise or limited precision, but as a demonstration, it shows the power of symbolic regression.

Step 5: Evaluate and Visualize#

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import matplotlib.pyplot as plt
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plt.scatter(X, y_noisy, color='black', label='Data')
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plt.plot(X, y_true, color='green', label='True function')
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plt.plot(X, y_pred, color='red', linestyle='--', label='Symbolic Fit')
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plt.legend()
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plt.xlabel('X')
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plt.ylabel('Y')
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plt.title('Symbolic Regression with gplearn')
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plt.show()

The red dash line should closely trace the green line, validating the approach.