科学方程发现Agent
1. 引言
科学方程是自然规律的精炼表达。从牛顿第二定律 到爱因斯坦的质能方程 ，方程发现是科学进步的核心驱动力。然而，传统方程发现依赖人类科学家的灵感和直觉，是一个耗时且困难的过程。
科学方程发现Agent结合符号回归、深度学习和物理约束，力图自动化这一过程12。本节介绍这一领域的最新进展。
本文档为 科学Agent基础 的进阶内容。
2. 符号回归基础
2.1 问题定义
符号回归旨在从数据中自动发现数学表达式。形式化定义：
给定数据集 ，寻找符号表达式 使得：
其中 由基本操作（）和变量、常数构成。
2.2 传统方法
2.2.1 遗传编程 (GP)
class GeneticProgramming:
def __init__(self, population_size=1000, generations=100):
self.pop_size = population_size
self.generations = generations
def evolve(self, X, y):
# 初始化种群
population = self.initialize_population()
for gen in range(self.generations):
# 评估适应度
fitness = [self.fitness(ind, X, y) for ind in population]
# 选择
parents = self.selection(population, fitness)
# 交叉
offspring = self.crossover(parents)
# 变异
offspring = self.mutate(offspring)
# 替代
population = self.replacement(population, offspring)
# 精英保留
best = self.get_best(population, fitness)
if self.is_perfect(best):
return best
return self.get_best(population)
def fitness(self, individual, X, y):
# 计算均方误差
pred = individual.evaluate(X)
mse = np.mean((pred - y) ** 2)
# 复杂度惩罚
complexity = individual.complexity()
# AIC-like 准则
return -mse - 0.01 * complexity2.2.2 Eureqa算法
Eureqa使用基于搜索的符号回归：
- 公式长度：鼓励简洁公式
- 预测误差：最小化残差
- 稀疏搜索：优先探索有前途的方向
2.3 深度学习方法
2.3.1 神经网络辅助
class NeuralSymbolicRegression:
def __init__(self):
self.nn = NeuralNetwork()
self.symbolic_layer = SymbolicExpressionLayer()
def forward(self, x):
# 神经网络预测
nn_pred = self.nn(x)
# 提取符号表达式
symbolic_expr = self.symbolic_layer.extract(nn_pred)
return symbolic_expr3. PhysX: 物理引导的LLM Agent
3.1 核心思想
PhysX是一个物理引导的LLM Agent，专门用于科学方程发现1。核心思想是：
- 物理约束集成：利用物理定律约束搜索空间
- 量纲分析：利用物理量纲减少候选空间
- 先验知识利用：整合物理直觉和领域知识
3.2 系统架构
┌─────────────────────────────────────────────────────────────┐
│ PhysX Agent │
├─────────────────────────────────────────────────────────────┤
│ │
│ ┌───────────────┐ ┌───────────────┐ ┌─────────────┐ │
│ │ Physics │───▶│ LLM │───▶│ Verifier │ │
│ │ Knowledge │ │ Generator │ │ (Dimensional│ │
│ │ Base │ │ │ │ Analysis) │ │
│ └───────────────┘ └───────────────┘ └─────────────┘ │
│ │ │ │ │
│ │ │ │ │
│ ▼ ▼ ▼ │
│ ┌─────────────────────────────────────────────────────────┐│
│ │ Physics-Guided Search ││
│ │ - Unit constraints - Conservation laws ││
│ │ - Dimensional analysis - Symmetry ││
│ └─────────────────────────────────────────────────────────┘│
└─────────────────────────────────────────────────────────────┘
3.3 量纲分析模块
量纲分析是PhysX的核心组件：
class DimensionalAnalyzer:
# 基础量纲
BASE_UNITS = {
'mass': 'M',
'length': 'L',
'time': 'T',
'temperature': 'Θ',
'current': 'I'
}
def analyze(self, expr: str) -> str:
"""返回表达式的量纲"""
# 解析表达式
tree = self.parse(expr)
# 递归计算量纲
dim = self.compute_dimension(tree)
return dim
def check_consistency(self, equation: str) -> bool:
"""检查方程两边量纲是否一致"""
left, right = equation.split('=')
left_dim = self.analyze(left)
right_dim = self.analyze(right)
return left_dim == right_dim
def generate_candidates(self, variables: List[dict]) -> List[str]:
"""基于量纲生成候选方程"""
# 提取变量量纲
dims = [v['dimension'] for v in variables]
# 构建量纲方程
target_dim = '?' # 待发现的目标量纲
# 求解量纲方程
solutions = self.solve_dimension_equation(dims, target_dim)
# 生成候选表达式
candidates = []
for sol in solutions:
expr = self.build_expression(sol, variables)
candidates.append(expr)
return candidates3.4 LLM生成器
class LLMGenerator:
def __init__(self, llm):
self.llm = llm
def generate(self, context: dict) -> List[str]:
prompt = f"""
给定以下物理情境，生成可能的方程形式：
变量：{context['variables']}
已知关系：{context['known_relations']}
物理约束：{context['constraints']}
生成5个候选方程，要求：
1. 符合量纲一致
2. 物理意义合理
3. 包含必要的物理常数
"""
response = self.llm.generate(prompt)
# 解析方程
equations = self.parse_equations(response)
return equations
def refine(self, candidate: str, feedback: dict) -> str:
"""根据反馈精炼方程"""
prompt = f"""
给定候选方程：{candidate}
反馈：{feedback}
请修正方程使其更符合物理规律。
"""
refined = self.llm.generate(prompt)
return refined3.5 验证器
class EquationVerifier:
def verify(self, equation: str, data: np.ndarray) -> dict:
result = {
'dimensional': self.check_dimensional(equation),
'statistical': self.check_statistical(equation, data),
'physical': self.check_physical_constraints(equation),
'score': 0.0
}
# 综合评分
result['score'] = (
0.4 * result['dimensional'] +
0.3 * result['statistical'] +
0.3 * result['physical']
)
return result
def check_dimensional(self, equation: str) -> float:
"""检查量纲一致性"""
analyzer = DimensionalAnalyzer()
return 1.0 if analyzer.check_consistency(equation) else 0.0
def check_statistical(self, equation: str, data: np.ndarray) -> float:
"""检查统计显著性"""
expr = sympify(equation)
# 计算预测
pred = self.evaluate(expr, data)
# 计算R²
r2 = self.calculate_r2(data['target'], pred)
# 计算AIC
aic = self.calculate_aic(data['target'], pred, complexity(expr))
# 综合评分
return 0.5 * r2 + 0.5 * min(1.0, 100 / aic)
def check_physical_constraints(self, equation: str) -> float:
"""检查物理约束"""
# 能量守恒、单调性等
constraints = [
self.check_energy_conservation,
self.check_monotonicity,
self.check_symmetry
]
scores = [c(equation) for c in constraints]
return np.mean(scores)4. SR-Scientist: Agentic AI方程发现
4.1 Agent架构
SR-Scientist采用Agent架构进行方程发现2：
┌─────────────────────────────────────────────────────────────┐
│ SR-Scientist Agent │
├─────────────────────────────────────────────────────────────┤
│ │
│ ┌─────────┐ ┌─────────┐ ┌─────────┐ ┌─────────┐ │
│ │ Planner │──▶│Searcher │──▶│ Tester │──▶│ Refiner │ │
│ └─────────┘ └─────────┘ └─────────┘ └─────────┘ │
│ │ │ │
│ │ Feedback Loop │ │
│ └──────────────────────────────────────────────────┘ │
│ │
└─────────────────────────────────────────────────────────────┘
4.2 规划器
class SRPlanner:
def plan(self, task: dict) -> SearchPlan:
# 分析任务
task_type = self.classify_task(task)
# 确定搜索策略
if task_type == 'mechanical':
search_strategy = 'physics_guided'
elif task_type == 'biological':
search_strategy = 'data_driven'
else:
search_strategy = 'hybrid'
# 规划搜索路径
plan = {
'strategy': search_strategy,
'max_iterations': 100,
'refinement_rounds': 5,
'tool_usage': self.select_tools(task)
}
return plan4.3 搜索器
class SRSearcher:
def search(self, plan: SearchPlan, data: np.ndarray) -> List[Candidate]:
candidates = []
for iteration in range(plan['max_iterations']):
# 生成候选
new_candidates = self.generate_candidates(plan, data)
# 评估候选
for cand in new_candidates:
score = self.evaluate(cand, data)
cand['score'] = score
if score > self.threshold:
candidates.append(cand)
# 更新搜索策略
plan = self.update_plan(plan, candidates)
if self.has_converged(candidates):
break
return candidates
def generate_candidates(self, plan: Plan, data: np.ndarray) -> List[str]:
if plan['strategy'] == 'physics_guided':
return self.physics_guided_search(plan, data)
elif plan['strategy'] == 'data_driven':
return self.data_driven_search(plan, data)
else:
return self.hybrid_search(plan, data)4.4 测试器
class SRTester:
def test(self, candidate: str, data: np.ndarray,
ground_truth: str = None) -> TestResult:
result = {
'fit_quality': self.test_fit(candidate, data),
'simplicity': self.test_simplicity(candidate),
'robustness': self.test_robustness(candidate, data),
'generalization': self.test_generalization(candidate, data)
}
if ground_truth:
result['correctness'] = self.test_correctness(
candidate, ground_truth
)
result['pass'] = all([
result['fit_quality'] > 0.9,
result['robustness'] > 0.8,
result['generalization'] > 0.7
])
return result
def test_fit(self, candidate: str, data: np.ndarray) -> float:
"""测试拟合质量"""
expr = sympify(candidate)
pred = self.evaluate(expr, data)
r2 = self.calculate_r2(data['target'], pred)
return r2
def test_robustness(self, candidate: str, data: np.ndarray) -> float:
"""测试对噪声的鲁棒性"""
noise_levels = [0.01, 0.05, 0.1, 0.2]
scores = []
for noise in noise_levels:
noisy_data = self.add_noise(data, noise)
pred = self.evaluate(sympify(candidate), noisy_data)
score = self.calculate_r2(noisy_data['target'], pred)
scores.append(score)
return np.mean(scores)
def test_generalization(self, candidate: str, data: np.ndarray) -> float:
"""测试泛化能力"""
train_data, test_data = self.split_data(data, ratio=0.8)
expr = sympify(candidate)
train_pred = self.evaluate(expr, train_data)
test_pred = self.evaluate(expr, test_data)
train_r2 = self.calculate_r2(train_data['target'], train_pred)
test_r2 = self.calculate_r2(test_data['target'], test_pred)
# 泛化gap
gap = abs(train_r2 - test_r2)
# 考虑gap的评分
return test_r2 - 0.1 * gap4.5 精炼器
class SRRefiner:
def refine(self, candidate: str, test_result: TestResult,
feedback: dict) -> str:
"""根据测试结果和反馈精炼方程"""
# 识别问题
issues = self.identify_issues(test_result, feedback)
# 制定精炼策略
if 'overfitting' in issues:
candidate = self.simplify(candidate)
elif 'underfitting' in issues:
candidate = self.add_complexity(candidate)
elif 'numerical_issue' in issues:
candidate = self.rescale(candidate)
return candidate
def simplify(self, expr: str) -> str:
"""简化表达式"""
sympy_expr = sympify(expr)
simplified = sympy_expr.simplify()
return str(simplified)
def add_complexity(self, expr: str) -> str:
"""增加表达式复杂度"""
# 添加交互项、高阶项等
pass5. AlphaEvolve风格的方法
5.1 演化-验证框架
AlphaEvolve结合了演化搜索和机器学习验证：
class AlphaEvolve:
def __init__(self, llm, verifier):
self.llm = llm
self.verifier = verifier
self.population = []
def evolve(self, task: dict, max_iterations: int = 1000):
# 初始化
self.initialize(task)
for iteration in range(max_iterations):
# 1. LLM生成
candidates = self.llm.generate(
context=self.get_context(),
n=10
)
# 2. 验证筛选
validated = []
for cand in candidates:
result = self.verifier.verify(cand, task['data'])
if result['score'] > self.threshold:
validated.append((cand, result))
# 3. 排序选择
validated.sort(key=lambda x: x[1]['score'], reverse=True)
# 4. 精英保留
self.population = validated[:self.population_size]
# 5. 反馈LLM
if validated:
self.feedback(validated[0])
# 检查终止条件
if self.is_solved(validated):
return validated[0]
return self.get_best()6. 物理约束集成
6.1 守恒律约束
class ConservationConstraint:
def __init__(self, conservation_type):
self.type = conservation_type # energy, momentum, charge, etc.
def check(self, equation: str, data: np.ndarray) -> float:
if self.type == 'energy':
return self.check_energy_conservation(equation, data)
elif self.type == 'momentum':
return self.check_momentum_conservation(equation, data)
def check_energy_conservation(self, equation: str,
data: np.ndarray) -> float:
"""检查能量守恒"""
# 计算能量变化
# ...
return conservation_score6.2 对称性约束
class SymmetryConstraint:
def check(self, equation: str, symmetry_type: str) -> bool:
if symmetry_type == 'translation':
return self.check_translation_invariance(equation)
elif symmetry_type == 'rotation':
return self.check_rotation_invariance(equation)
elif symmetry_type == 'scale':
return self.check_scale_invariance(equation)
def check_scale_invariance(self, equation: str) -> bool:
"""检查尺度不变性"""
# 对于正确发现的物理定律
# 缩放变量应导致方程形式不变
pass7. 评估与基准
7.1 Feynman方程发现基准
| 方程 | 复杂度 | 发现成功率(PhysX) | 发现成功率(SR-Scientist) |
|---|---|---|---|
| 低 | 95% | 92% | |
| 中 | 88% | 85% | |
| 高 | 72% | 68% | |
| 很高 | 45% | 52% |
7.2 真实数据评估
| 数据集 | 物理领域 | 发现率 | 准确率 |
|---|---|---|---|
| pendulum | 力学 | 85% | 0.92 |
| spring_mass | 力学 | 82% | 0.89 |
| circuit | 电磁 | 68% | 0.84 |
| chemical | 化学 | 55% | 0.78 |
8. 局限性与未来方向
8.1 当前局限性
- 长方程困难：复杂物理定律的方程难以发现
- 常数估计：物理常数估计不准确
- 多尺度问题：跨尺度物理规律发现困难
- 噪声敏感：实验数据噪声影响发现
8.2 未来方向
- 多物理场集成：处理耦合物理系统
- 不确定性量化：估计发现方程的不确定性
- 交互式发现：人类-机器协作方程发现
- 跨学科迁移：不同领域方程发现知识迁移