Symbolic Equation Discovery

Discover interpretable scientific equations from data using LLM-guided evolutionary search.

Input

• $0 — Dataset description, variable names, and physical context

References

• LLM-SR patterns (prompts, evolution, sampling): ~/.claude/skills/symbolic-equation/references/llmsr-patterns.md

Workflow (from LLM-SR)

Step 1: Define Problem Specification

Create a specification with:

1. Input variables: Physical quantities with types (e.g., x: np.ndarray, v: np.ndarray)
2. Output variable: Target quantity to predict
3. Evaluation function: Fitness metric (typically negative MSE with parameter optimization)
4. Physical context: Domain knowledge to guide equation discovery

# Example specification
@equation.evolve
def equation(x: np.ndarray, v: np.ndarray, params: np.ndarray) -> np.ndarray:
    """Describe the acceleration of a damped nonlinear oscillator."""
    return params[0] * x

Step 2: Initialize Multi-Island Buffer

• Create N islands (default: 10) for population diversity
• Each island maintains independent clusters of equations
• Clusters group equations by performance signature

Step 3: Evolutionary Search Loop

Repeat until convergence or max samples:

1. Select island: Random island selection
2. Build prompt: Sample top equations from clusters (softmax-weighted by score)
3. LLM proposes: Generate new equation as improved version
4. Evaluate: Execute on test data, compute fitness score
5. Register: Add to island's cluster if valid

Step 4: Prompt Construction

Present previous equations as versioned sequence:

def equation_v0(x, v, params):
    """Initial version."""
    return params[0] * x

def equation_v1(x, v, params):
    """Improved version of equation_v0."""
    return params[0] * x + params[1] * v

def equation_v2(x, v, params):
    """Improved version of equation_v1."""
    # LLM completes this

Step 5: Island Reset (Diversity Maintenance)

Periodically (default: every 4 hours):

1. Sort islands by best score
2. Reset bottom 50% of islands
3. Seed each reset island with best equation from a surviving island
4. Restart cluster sampling temperature

Step 6: Extract Best Equations

After search completes:

1. Collect best equation from each island
2. Rank by fitness score
3. Simplify if possible (algebraic simplification)
4. Report with physical interpretation

Cluster Sampling

Temperature-scheduled softmax over cluster scores:

temperature = T_init * (1 - (num_programs % period) / period)
probabilities = softmax(cluster_scores / temperature)

• Higher temperature → more exploration
• Lower temperature → more exploitation of best clusters
• Within clusters: shorter programs are preferred (Occam's razor)

Rules

• Equations must use only standard mathematical operations
• Parameter optimization via scipy BFGS or Adam
• Fitness = negative MSE (higher is better)
• Timeout protection for equation evaluation
• No recursive equations allowed
• Physical interpretability is preferred over pure fit

Related Skills

• Upstream: [data-analysis](../data-analysis/), [math-reasoning](../math-reasoning/)
• Downstream: [paper-writing-section](../paper-writing-section/)
• See also: [algorithm-design](../algorithm-design/)