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symbolic-equation

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Symbolic Equation Discovery

符号方程发现

Discover interpretable scientific equations from data using LLM-guided evolutionary search.
使用LLM引导的进化搜索从数据中发现可解释的科学方程。

Input

输入

  • $0
    — Dataset description, variable names, and physical context
  • $0
    — 数据集描述、变量名称和物理背景

References

参考文献

  • LLM-SR patterns (prompts, evolution, sampling): 
    ~/.claude/skills/symbolic-equation/references/llmsr-patterns.md
  • LLM-SR模式（提示词、进化、采样）：
    ~/.claude/skills/symbolic-equation/references/llmsr-patterns.md

Workflow (from LLM-SR)

工作流程（源自LLM-SR）

Step 1: Define Problem Specification

步骤1：定义问题规范

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
python
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创建包含以下内容的规范：
  1. 输入变量：带类型的物理量（例如 
    x: np.ndarray
    , 
    v: np.ndarray
    ）
  2. 输出变量：需要预测的目标量
  3. 评估函数：适应度指标（通常是带参数优化的负均方误差）
  4. 物理背景：用于指导方程发现的领域知识
python
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Example specification

示例规范

@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
undefined
@equation.evolve def equation(x: np.ndarray, v: np.ndarray, params: np.ndarray) -> np.ndarray: """描述阻尼非线性振荡器的加速度。""" return params[0] * x
undefined

Step 2: Initialize Multi-Island Buffer

步骤2：初始化多岛屿缓冲区

  • Create N islands (default: 10) for population diversity
  • Each island maintains independent clusters of equations
  • Clusters group equations by performance signature
  • 创建N个岛屿（默认值：10）以保证种群多样性
  • 每个岛屿维护独立的方程聚类
  • 聚类根据性能特征对方程进行分组

Step 3: Evolutionary Search Loop

步骤3：进化搜索循环

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
重复执行直至收敛或达到最大样本数：
  1. 选择岛屿：随机选择岛屿
  2. 构建提示词：从聚类中采样顶级方程（按分数进行softmax加权）
  3. LLM生成：生成作为改进版本的新方程
  4. 评估：在测试数据上执行，计算适应度分数
  5. 注册：如果有效则添加到岛屿的聚类中

Step 4: Prompt Construction

步骤4：提示词构建

Present previous equations as versioned sequence:
python
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
将先前的方程作为版本化序列呈现：
python
def equation_v0(x, v, params):
    """初始版本。"""
    return params[0] * x

def equation_v1(x, v, params):
    """equation_v0的改进版本。"""
    return params[0] * x + params[1] * v

def equation_v2(x, v, params):
    """equation_v1的改进版本。"""
    # LLM完成此部分

Step 5: Island Reset (Diversity Maintenance)

步骤5：岛屿重置（多样性维护）

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
定期执行（默认：每4小时）：
  1. 按最佳分数对岛屿排序
  2. 重置排名后50%的岛屿
  3. 用存活岛屿中的最佳方程为每个重置岛屿播种
  4. 重启聚类采样温度

Step 6: Extract Best Equations

步骤6：提取最佳方程

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
搜索完成后：
  1. 收集每个岛屿的最佳方程
  2. 按适应度分数排名
  3. 尽可能简化（代数简化）
  4. 结合物理解释进行报告

Cluster Sampling

聚类采样

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)
基于温度调度的softmax聚类分数：
temperature = T_init * (1 - (num_programs % period) / period)
probabilities = softmax(cluster_scores / temperature)
  • 温度越高 → 探索性越强
  • 温度越低 → 对最佳聚类的利用性越强
  • 聚类内部：优先选择更简短的方程（奥卡姆剃刀原则）

Rules

规则

  • 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
  • 方程只能使用标准数学运算
  • 通过scipy BFGS或Adam进行参数优化
  • 适应度 = 负均方误差（分数越高越好）
  • 方程评估的超时保护
  • 不允许递归方程
  • 优先考虑物理解释性而非纯拟合效果

Related Skills

相关技能

  • Upstream: data-analysis, math-reasoning
  • Downstream: paper-writing-section
  • See also: algorithm-design
  • 上游：数据分析, 数学推理
  • 下游：论文写作章节
  • 另见：算法设计