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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、变量名称以及物理上下文

References

参考资料

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

Workflow (from LLM-SR)

工作流（来自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
undefined
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（通常为带参数优化的负均方误差）
  4. 物理上下文：用于指导方程发现的领域知识
python
undefined

Example specification

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
undefined
@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

Step 2: Initialize Multi-Island Buffer

步骤2：初始化多岛缓冲区

  • 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：10个）以保证种群多样性
  • 每个岛屿维护独立的方程聚类
  • 聚类会按照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
重复执行直到收敛或达到最大采样数：
  1. 选择岛屿：随机选取岛屿
  2. 构建提示词：从聚类中采样顶级方程（按分数的softmax加权）
  3. LLM生成提议：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：如果有效则添加到对应岛屿的聚类中

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
将历史方程展示为带版本的序列：
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

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小时一次）：
  1. 按最佳分数对岛屿排序
  2. 重置排名后50%的岛屿
  3. 用存活岛屿的最佳方程为每个重置的岛屿做种子初始化
  4. 重置聚类采样温度

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. 按适应度分数排序
  3. 尽可能进行代数化简
  4. 附带物理解释输出结果

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)
对聚类分数使用带温度调度的softmax：
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 → 对最优聚类的利用程度越高
  • 聚类内部：更偏好更简短的程序（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
  • 方程仅可使用标准数学运算
  • 通过scipy BFGS或Adam进行参数优化
  • 适应度 = 负均方误差（值越高越好）
  • 方程评估设有超时保护
  • 不允许使用递归方程
  • 相比纯拟合结果，优先选择具备物理可解释性的结果

Related Skills

相关技能

  • Upstream: data-analysis, math-reasoning
  • Downstream: paper-writing-section
  • See also: algorithm-design
  • 上游：data-analysis、math-reasoning
  • 下游：paper-writing-section
  • 另请参见：algorithm-design