数据规模差异大时SciPy curve_fit无法稳定拟合AMM模型参数

我们使用scipy.optimize.curve_fit拟合自动做市商（AMM）模型的参数X_eff和Y_eff，模型公式为：

y_out = (Y_eff * x_in) / (X_eff + x_in)

当输入数据点(x_in, y_out)的规模差异显著，或大尺度数据点存在微小变动时，拟合得到的X_eff和Y_eff值会表现出极强的敏感性。

复现代码（无注释版） #

import numpy as np
from scipy.optimize import curve_fit

def swap_model(x_in, X_eff, Y_eff):
    denominator = X_eff + x_in
    if np.isscalar(denominator):
        return (Y_eff * x_in) / (denominator if not np.isclose(denominator, 0) else 1e-9)
    else:
        denominator_copy = np.copy(denominator)
        denominator_copy[np.isclose(denominator_copy, 0)] = 1e-9
        return (Y_eff * x_in) / denominator_copy

bounds = ([1e-6, 1e-6], [np.inf, np.inf])

x_test_value = 6507.3125

print("--- Case 1: Poor Fit ---")
x_data_bad = np.array([1, 2, 39000])
y_data_bad = np.array([8423.724335, 16846.531751, 319765062.090722])
y_actual_for_bad_case_test = 54311422.15982

try:
    popt_bad, _ = curve_fit(swap_model, x_data_bad, y_data_bad, bounds=bounds, maxfev=5000)
    X_eff_bad, Y_eff_bad = popt_bad
    print(f"Fitted (Bad): X_eff = {X_eff_bad:.4f}, Y_eff = {Y_eff_bad:.4f}")
    y_pred_bad = swap_model(x_test_value, X_eff_bad, Y_eff_bad)
    error_bad = abs(y_actual_for_bad_case_test - y_pred_bad) / y_actual_for_bad_case_test * 100
    print(f"Predicted for {x_test_value:.4f}: {y_pred_bad:.4f}, Actual: {y_actual_for_bad_case_test:.4f}, Error: {error_bad:.2f}%")
except RuntimeError as e: print(f"RuntimeError: {e}")
except Exception as e: print(f"Other exception in bad fit: {e}")


print("\n--- Case 2: Better Fit ---")
x_data_good = np.array([1, 2, 32000])
y_data_good = np.array([8430.169107, 16859.464909, 264727159.854379])
y_actual_for_good_case_test = 54549399.392518

try:
    popt_good, _ = curve_fit(swap_model, x_data_good, y_data_good, bounds=bounds, maxfev=5000)
    X_eff_good, Y_eff_good = popt_good
    print(f"Fitted (Good): X_eff = {X_eff_good:,.4f}, Y_eff = {Y_eff_good:,.4f}")
    y_pred_good = swap_model(x_test_value, X_eff_good, Y_eff_good)
    error_good = abs(y_actual_for_good_case_test - y_pred_good) / y_actual_for_good_case_test * 100
    print(f"Predicted for {x_test_value:.4f}: {y_pred_good:.4f}, Actual: {y_actual_for_good_case_test:.4f}, Error: {error_good:.2f}%")
except RuntimeError as e: print(f"RuntimeError: {e}")
except Exception as e: print(f"Other exception in good fit: {e}")

复现代码（带注释版） #

# BUG DEMONSTRATION

import numpy as np
from scipy.optimize import curve_fit

def swap_model(x_in, X_eff, Y_eff):
    denominator = X_eff + x_in
    # 避免除零，不过参数边界设置已经能起到一定作用
    if np.isscalar(denominator):
        return (Y_eff * x_in) / (denominator if not np.isclose(denominator, 0) else 1e-9)
    else: # 处理数组输入
        denominator_copy = np.copy(denominator) # 复制数组避免修改原输入
        denominator_copy[np.isclose(denominator_copy, 0)] = 1e-9
        return (Y_eff * x_in) / denominator_copy

bounds = ([1e-6, 1e-6], [np.inf, np.inf]) # 参数必须为正数

# 用于预测测试的输入值
x_test_value = 6507.3125

print("--- Case 1: Poor Fit ---")
x_data_bad = np.array([1, 2, 39000])
y_data_bad = np.array([8423.724335, 16846.531751, 319765062.090722])
# 该场景下x_test_value对应的实际输出值
y_actual_for_bad_case_test = 54311422.15982

try:
    popt_bad, _ = curve_fit(swap_model, x_data_bad, y_data_bad, bounds=bounds, maxfev=5000)
    X_eff_bad, Y_eff_bad = popt_bad
    print(f"Fitted (Bad): X_eff = {X_eff_bad:.4f}, Y_eff = {Y_eff_bad:.4f}")
    y_pred_bad = swap_model(x_test_value, X_eff_bad, Y_eff_bad)
    error_bad = abs(y_actual_for_bad_case_test - y_pred_bad) / y_actual_for_bad_case_test * 100
    print(f"Predicted for {x_test_value:.4f}: {y_pred_bad:.4f}, Actual: {y_actual_for_bad_case_test:.4f}, Error: {error_bad:.2f}%")
except RuntimeError as e: print(f"RuntimeError: {e}")
except Exception as e: print(f"Other exception in bad fit: {e}")


print("\n--- Case 2: Better Fit ---")
x_data_good = np.array([1, 2, 32000])
y_data_good = np.array([8430.169107, 16859.464909, 264727159.854379])
# 该场景下x_test_value对应的实际输出值
y_actual_for_good_case_test = 54549399.392518

try:
    popt_good, _ = curve_fit(swap_model, x_data_good, y_data_good, bounds=bounds, maxfev=5000)
    X_eff_good, Y_eff_good = popt_good
    print(f"Fitted (Good): X_eff = {X_eff_good:,.4f}, Y_eff = {Y_eff_good:,.4f}") # 添加逗号提升可读性
    y_pred_good = swap_model(x_test_value, X_eff_good, Y_eff_good)
    error_good = abs(y_actual_for_good_case_test - y_pred_good) / y_actual_for_good_case_test * 100
    print(f"Predicted for {x_test_value:.4f}: {y_pred_good:.4f}, Actual: {y_actual_for_good_case_test:.4f}, Error: {error_good:.2f}%")
except RuntimeError as e: print(f"RuntimeError: {e}")
except Exception as e: print(f"Other exception in good fit: {e}")

拟合效果对比 #

• 有效拟合：
  
• 无效拟合：
  

内容的提问来源于stack exchange，提问作者notajoke