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:

• Input variables: Physical quantities with types (e.g., x: np.ndarray, v: np.ndarray)

• Output variable: Target quantity to predict

• Evaluation function: Fitness metric (typically negative MSE with parameter optimization)

• 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:

• Select island: Random island selection

• Build prompt: Sample top equations from clusters (softmax-weighted by score)

• LLM proposes: Generate new equation as improved version

• Evaluate: Execute on test data, compute fitness score

• 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):

• Sort islands by best score

• Reset bottom 50% of islands

• Seed each reset island with best equation from a surviving island

• Restart cluster sampling temperature

Step 6: Extract Best Equations

After search completes:

• Collect best equation from each island

• Rank by fitness score

• Simplify if possible (algebraic simplification)

• 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, math-reasoning

• Downstream: paper-writing-section

• See also: algorithm-design