from scipy.integrate import solve_ivp import matplotlib.pyplot as plt # # 全局拓扑与模型参数 # alpha_k 0.5 # 独立征候通道压制系数 beta_m 0.3 # 连坐机制压制系数 m_crit 5.0 # 僵死相变临界阈值 # ODE动力学参数真实系统Delta_true, xi_L delta 0.1 epsilon 0.05 zeta_base 0.02 # 基础外部冲击响应系数僵死时衰减 alpha_xi 0.3 beta_xi 0.1 gamma_xi 0.01 # 对冲效率衰减函数 theta(xi_L) def theta(xi_L): return 1.0 / (1.0 0.1 * xi_L) # # 时变监督强度 phi_t(t) 【已嵌入73窦宪胜利归入明章脉冲】 # def phi_t(t, phi_00.1): pulses [ {A: 1.0, t: 41, sigma: 10}, # 光武中兴 {A: 0.8, t: 72, sigma: 8}, # 明章之治包含73年窦宪破匈奴正向治理增益 {A: 0.5, t: 99, sigma: 5}, # 和帝亲政 {A: 0.3, t: 152, sigma: 3}, # 桓帝初年整顿 ] phi phi_0 for p in pulses: phi p[A] * np.exp(-((t - p[t])**2) / (2 * p[sigma]**2)) # 党锢之祸阶梯衰减 if t 169: phi - 0.7 elif t 166: phi - 0.4 if t 184: phi 0.2 # 黄巾起义后短期强化监督 return max(phi, 0.01) # # Gamma(t)稀疏外部冲击脉冲羌乱、黄巾73事件移入phi_t # def Gamma(t): shocks [ {t: 107, I: 3.0}, # 羌乱爆发第一次 {t: 140, I: 2.0}, # 羌乱再起 {t: 184, I: 5.0}, # 黄巾起义 ] g 0.0 for s in shocks: g s[I] * np.exp(-((t - s[t])**2) / 2) return g # # sigma^* 拓扑公式 # def compute_sigma_star(sigma_0, kernel_ratio, k_eff, m_eff): suppression 1.0 / (1.0 alpha_k * k_eff beta_m * m_eff) sigma_star sigma_0 * kernel_ratio * suppression sigma_star np.clip(sigma_star, 0, sigma_0) return sigma_star # # ODE系统调用时变phi_t(t) # def system(t, y, sigma_star, m_eff): Delta_true, xi_L y phi phi_t(t) if m_eff m_crit: zeta zeta_base * 0.1 else: zeta zeta_base dDelta_dt delta * xi_L - epsilon * Delta_true * phi zeta * Gamma(t) dxi_dt alpha_xi * Delta_true * (1 - theta(xi_L)) - beta_xi * xi_L * phi gamma_xi * xi_L**2 return [dDelta_dt, dxi_dt] # # 热力图批量仿真函数 # def run_heatmap(sigma_0, m_eff_fixed): k_eff_list np.linspace(0, 6, 30) kernel_ratio_list np.linspace(0, 1, 30) K, KR np.meshgrid(k_eff_list, kernel_ratio_list) xi_final_grid np.zeros_like(K) status_grid np.zeros_like(K) t_span (25, 220) y0 [0.05, 0.0] t_eval np.linspace(*t_span, 400) for i, kr in enumerate(kernel_ratio_list): for j, ke in enumerate(k_eff_list): sigma_star compute_sigma_star(sigma_0, kr, ke, m_eff_fixed) sol solve_ivp(system, t_span, y0, t_evalt_eval, args(sigma_star, m_eff_fixed)) xi_end sol.y[1][-1] xi_final_grid[i,j] xi_end if m_eff_fixed m_crit: status_grid[i,j] 2 elif xi_end 10: status_grid[i,j] 1 else: status_grid[i,j] 0 return K, KR, xi_final_grid, status_grid # # 【修正】东汉各朝代坐标表k_eff, kernel_ratio, label, sigma0_per_dynasty, year # eastern_han_coords [ (2.5, 0.40, 光武, 0.4, 25), (3.0, 0.33, 明章, 0.5, 57), (3.5, 0.29, 和帝, 0.6, 88), (2.0, 0.38, 安帝, 0.7, 106), (1.5, 0.44, 顺帝, 0.8, 125), (0.8, 0.50, 桓帝, 0.9, 146), (0.3, 0.58, 灵帝, 1.0, 168), ] # # 【修正】参数扫描配置sigma_0 扩展为 7 个值每个朝代一个 # sigma_0_values [0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0] m_set [ (m_eff 0 (无连坐), 0.0), (fm_eff m_crit0.1 {m_crit0.1}, m_crit 0.1) ] # # 图1热力图 东汉坐标点14张子图7个sigma_0 × 2组m_eff # fig1, axes1 plt.subplots(len(m_set), len(sigma_0_values), figsize(24, 8)) for row_idx, (m_label, m_val) in enumerate(m_set): for col_idx, s0 in enumerate(sigma_0_values): K, KR, xi_grid, stat_grid run_heatmap(s0, m_val) ax axes1[row_idx, col_idx] contour ax.contourf(K, KR, xi_grid, levels20, cmapviridis) ax.contour(K, KR, xi_grid, levels[10], colorsred, linestyles--) ax.set_title(f$\sigma_0{s0}$ | {m_label}, fontsize10) ax.set_xlabel($k_{eff}$) ax.set_ylabel(kernel ratio) ax.grid(alpha0.2) # 打点仅绘制匹配当前子图 sigma_0 的朝代 for (ke, kr, name, dyn_s0, year) in eastern_han_coords: if np.isclose(dyn_s0, s0, atol0.01): ax.plot(ke, kr, markero, colorwhite, markersize8, markeredgecolorblack) ax.text(ke0.1, kr0.03, name, colorwhite, fontsize9, fontweightbold, bboxdict(boxstyleround,pad0.3, facecolorblack, alpha0.7)) plt.tight_layout() plt.savefig(topology_heatmap_7dynasties.png, dpi150, bbox_inchestight) plt.show() # # 图2东汉轨迹图按时间顺序连接各朝代点 # # 对每个朝代用其 (k_eff, kernel_ratio, sigma_0) 代入 ODE 计算 xi_L 终值 dynasty_years [] dynasty_xi_final [] dynasty_names [] for (ke, kr, name, dyn_s0, year) in eastern_han_coords: # 使用 m_eff0无连坐作为基准 sigma_star compute_sigma_star(dyn_s0, kr, ke, m_eff_fixed0.0) sol solve_ivp(system, (25, 220), [0.05, 0.0], t_evalnp.linspace(25, 220, 400), args(sigma_star, 0.0)) xi_end sol.y[1][-1] dynasty_years.append(year) dynasty_xi_final.append(xi_end) dynasty_names.append(name) fig2, ax2 plt.subplots(figsize(12, 6)) ax2.plot(dynasty_years, dynasty_xi_final, b-o, linewidth2, markersize8, label东汉轨迹) ax2.axhline(y10, colorred, linestyle--, linewidth2, label稳态边界 ξ_L10) ax2.fill_between(dynasty_years, 0, 10, alpha0.1, colorgreen, label稳态区) ax2.fill_between(dynasty_years, 10, max(dynasty_xi_final)*1.1, alpha0.1, colorred, label发散区) # 标注朝代名称 for i, name in enumerate(dynasty_names): ax2.annotate(name, (dynasty_years[i], dynasty_xi_final[i]), textcoordsoffset points, xytext(0, 12), fontsize10, fontweightbold, bboxdict(boxstyleround,pad0.3, facecoloryellow, alpha0.8)) ax2.set_xlabel(年份, fontsize12) ax2.set_ylabel(ξ_L 终值, fontsize12) ax2.set_title(东汉王朝熵债演化轨迹m_eff0, fontsize14) ax2.legend() ax2.grid(alpha0.3) plt.tight_layout() plt.savefig(eastern_han_trajectory.png, dpi150, bbox_inchestight) plt.show() # # 图3sigma_0 与 xi_L 终值的关系 # fig3, ax3 plt.subplots(figsize(10, 6)) sigma_0_list [item[3] for item in eastern_han_coords] ax3.plot(sigma_0_list, dynasty_xi_final, r-s, linewidth2, markersize8) ax3.axhline(y10, colorred, linestyle--, linewidth2, label稳态边界 ξ_L10) for i, name in enumerate(dynasty_names): ax3.annotate(name, (sigma_0_list[i], dynasty_xi_final[i]), textcoordsoffset points, xytext(8, 8), fontsize10, fontweightbold) ax3.set_xlabel($\sigma_0$基准扭曲度, fontsize12) ax3.set_ylabel(ξ_L 终值, fontsize12) ax3.set_title(基准扭曲度与熵债终值的关系, fontsize14) ax3.legend() ax3.grid(alpha0.3) plt.tight_layout() plt.savefig(sigma0_vs_xiL.png, dpi150, bbox_inchestight) plt.show() # # 输出汇总 # print(*80) print(仿真完成) print(图1热力图7个sigma_0 × 2组m_eff 14张子图) print(图2东汉轨迹图按时间顺序连接) print(图3sigma_0 与 xi_L 终值关系图) print(*80) print(东汉朝代坐标与xi_L终值) print(f{朝代:6} {年份:6} {k_eff:8} {kernel_ratio:12} {sigma_0:8} {xi_L终值:10}) print(-*60) for i, (ke, kr, name, dyn_s0, year) in enumerate(eastern_han_coords): print(f{name:6} {year:6} {ke:8} {kr:12} {dyn_s0:8} {dynasty_xi_final[i]:10.4f}) print(*80)优化策略描述分段线性化将非线性项拆分为多个线性区间通过分段函数近似非线性行为。例如在电力市场建模中将非线性成本函数分段处理以提升求解效率。二进制扩展法引入二进制变量对非线性项进行离散逼近适用于处理乘积项如 $x \cdot y$。此方法在能源调度等场景中被广泛应用但需权衡精度与计算代价。McCormick 包络通过构建非线性项的上下界包络将其转化为线性约束适用于处理乘积项或分数项。该方法在混合整数非线性规划中具有较高的适用性。非线性项引入在模型中直接保留非线性项通过高精度数值求解器如solve_ivp进行求解。适用于非线性程度较低或计算资源充足的情况。迭代法处理对于强非线性问题采用迭代法逐步逼近解。例如在热网建模中通过温度固定和迭代求解实现非线性项的收敛。随机效应调整在统计模型中通过引入随机效应或非线性项调整模型偏差提高拟合效果。例如在混合效应模型中通过非线性项优化模型性能。变量变换对非线性项进行变量替换如对数变换、指数变换使其更接近线性关系。例如在Logistic回归中通过变量变换优化模型校准度。以上优化策略可根据具体模型特点和计算需求选择使用以提升模型的稳定性、准确性和求解效率。参考来源电力市场两级协同优化与可再生能源消纳模型R语言混合效应模型诊断实战如何在4步内发现并修正模型偏差SPSSAU实战如何用Hosmer-Lemeshow检验优化你的二元Logistic回归模型优化模型线性化避坑指南二进制扩展法处理x*y项什么时候用什么时候慎用多区域综合能源系统热网建模与Matlab运行优化复现实践
