benchmarks/formula_summary.md

# /data/yiming/real-sr/benchmarks Ground-Truth Formula 整理

- 统计范围：`benchmarks/*/*/metadata.yaml` 中非空 `ground_truth` 条目
- 不计入：`ground_truth: []` 的 reference-only 公式、`formulas/fit_*.py` 拟合脚本、未列入 metadata 的草稿公式
- 任务数：22
- 公式数：38

## 任务列表

- `astrophysics/cranmer2020_dm_halo`：2 个公式
- `astrophysics/desi_dr1`：2 个公式
- `astrophysics/exoplanet_mass_radius`：2 个公式
- `astrophysics/exoplanet_mass_radius_lowmass`：1 个公式
- `astrophysics/gwtc4_bbh_massfunction`：3 个公式
- `astrophysics/kepler_third_law`：1 个公式
- `astrophysics/pantheon_plus_hubble`：1 个公式
- `astrophysics/sparc_tully_fisher_rar`：2 个公式
- `astrophysics/wadekar2023_sz_flux_mass`：1 个公式
- `battery_degradation/attia_closed_loop`：1 个公式
- `battery_degradation/severson_124cell`：3 个公式
- `dynamical_systems/hudson_bay_lynx_hare`：1 个公式
- `dynamical_systems/ibm_double_pendulum`：3 个公式
- `ecology_epi/bjornstad_measles_tsir`：1 个公式
- `ecology_epi/tycho_measles_multicountry`：1 个公式
- `industry_techspec/cec_pv_module`：3 个公式
- `industry_techspec/duramat_pv_degradation`：3 个公式
- `industry_techspec/iea_15mw_turbine`：1 个公式
- `materials/estm_thermoelectric`：1 个公式
- `materials/sisso_perovskite_tau`：1 个公式
- `materials/weng_perovskite_oer`：3 个公式
- `soft_sensors/uci_ccpp`：1 个公式

## `astrophysics/cranmer2020_dm_halo`

- 任务标题：Dark matter halo overdensity from GN + SR distillation (Cranmer et al. 2020)
- 任务背景：Quijote fiducial cosmology, realisation 0, snapshot 4 (z = 0), 215,854 FoF halos in a 1000 Mpc/h box. Each halo is annotated with its smoothed overdensity delta_i (Gaussian smoothing radius 20 Mpc/h on the CIC CDM density field) and with the variable-length...
- Formula `cranmer2020_halo`
- 表达式：`\hat\delta_i = C_1 + \frac{e_i}{C_2 + C_3 M_i},\; e_i = \sum_{j \neq i,\, |r_i - r_j| < 50} \frac{C_4 + M_j}{C_5 + (C_6\,|r_i-r_j|)^{C_7}}`
- 使用输入：`['M_i', 'abs_v_i', 'nbr_M', 'nbr_abs_dv', 'nbr_abs_dr']`
- 拟合结果：nmse=未提供, r2=未提供, rmse=未提供, kendall_tau=未提供, mape=未提供, mae=0.0882
- Formula `cranmer2020_halo_no_mass`
- 表达式：`\hat\delta_i = C_1 + \frac{e_i\,|v_i|}{C_2 + C_3 e_i},\; e_i = \sum_{j \neq i,\, |r_i - r_j| < 50} \frac{C_4 + |v_i - v_j|}{C_5 + (C_6\,|r_i-r_j|)^{C_7}}`
- 使用输入：`['M_i', 'abs_v_i', 'nbr_M', 'nbr_abs_dv', 'nbr_abs_dr']`
- 拟合结果：nmse=未提供, r2=未提供, rmse=未提供, kendall_tau=未提供, mape=未提供, mae=0.12

## `astrophysics/desi_dr1`

- 任务标题：DESI DR1 baryon acoustic oscillation distance-redshift relation (flat ΛCDM and w0waCDM)
- 任务背景：DESI DR1 BAO consensus measurements (Table 1 of Adame et al. 2025, JCAP 02:021). These are the post-processed distance-ratio scalars extracted from BAO fits to the DR1 spectroscopic clustering catalogues (~6 million galaxies / quasars + Lyα forest) and are ...
- Formula `flat_lcdm_bao`
- 表达式：`\begin{aligned} y(z,\text{obs}) &= \begin{cases} D_M(z)/r_d & \text{if obs\_is\_DM} = 1 \\ D_H(z)/r_d & \text{if obs\_is\_DH} = 1 \\ D_V(z)/r_d & \text{otherwise} \end{cases} \\ D_M(z) &= \frac{c}{H_0}\int_0^z \frac{dz'}{E(z')}, \quad D_H(z) = \frac{c}{H(z)} = \frac{c}{H_0 E(z)} \\ D_V(z) &= \left[z\, D_M(z)^2\, D_H(z)\right]^{1/3} \\ E(z) &= \sqrt{\Omega_m(1+z)^3 + (1-\Omega_m)} \end{aligned}`
- 使用输入：`['z_eff', 'obs_is_DM', 'obs_is_DH']`
- 拟合结果：nmse=0.004636, r2=0.995364, rmse=0.57591, kendall_tau=0.969697, mape=0.02
- Formula `w0wa_cdm_bao`
- 表达式：`\begin{aligned} E(z) &= \sqrt{\Omega_m(1+z)^3 + (1-\Omega_m)\,(1+z)^{3(1+w_0+w_a)}\, \exp\!\left(-\frac{3 w_a z}{1+z}\right)} \\ w(a) &= w_0 + w_a(1 - a), \quad a = 1/(1+z) \end{aligned}`
- 使用输入：`['z_eff', 'obs_is_DM', 'obs_is_DH']`
- 拟合结果：nmse=0.003456, r2=0.996544, rmse=0.497298, kendall_tau=0.969697, mape=0.019223

## `astrophysics/exoplanet_mass_radius`

- 任务标题：Exoplanet mass-radius relation, full range (Chen-Kipping 2017 / Bashi 2017, NASA Exoplanet Archive)
- 任务背景：NASA Exoplanet Archive "Planetary Systems Composite Parameters" table (pscomppars), queried via the TAP service. pscomppars assembles a single best-available value per parameter per confirmed exoplanet, drawing the most accurate value for each parameter acr...
- Formula `chen_kipping_2017`
- 表达式：`R(M) = \begin{cases} C^{(1)}\,M^{S^{(1)}} & M \le T^{(1)} \\ C^{(2)}\,M^{S^{(2)}} & T^{(1)} < M \le T^{(2)} \\ C^{(3)}\,M^{S^{(3)}} & T^{(2)} < M \le T^{(3)} \\ C^{(4)}\,M^{S^{(4)}} & M > T^{(3)} \end{cases},\quad C^{(1)}=1.008,\ S^{(1,2,3,4)}=(0.279,0.589,-0.044,0.881),\ T^{(1,2,3)}=(2.04\,M_\oplus,\,0.414\,M_J,\,0.0800\,M_\odot)`
- 使用输入：`['pl_bmasse']`
- 拟合结果：nmse=0.2533, r2=0.7467, rmse=2.8899, kendall_tau=0.7212, mape=0.2664, log10_rmse=0.1606, log10_r2=0.8284
- Formula `bashi_2017`
- 表达式：`R(M) = \begin{cases} 0.852\,M^{0.55} & M < 124\,M_\oplus \\ 11.52\,M^{0.01} & M \ge 124\,M_\oplus \end{cases}`
- 使用输入：`['pl_bmasse']`
- 拟合结果：nmse=0.2646, r2=0.7354, rmse=2.9535, kendall_tau=0.5129, mape=0.2759, log10_rmse=0.1591, log10_r2=0.8316

## `astrophysics/exoplanet_mass_radius_lowmass`

- 任务标题：Exoplanet mass-radius relation below 120 M_earth (Otegi 2020, NASA Exoplanet Archive)
- 任务背景：NASA Exoplanet Archive "Planetary Systems Composite Parameters" table (pscomppars), queried via the TAP service and filtered to the Otegi, Bouchy & Helled 2020 low-mass domain: pl_rade IS NOT NULL, pl_bmasse IS NOT NULL, pl_bmasse < 120 M_earth, pl_dens IS ...
- Formula `otegi_2020`
- 表达式：`R = \begin{cases} 1.03\,M^{0.29} & \rho > 3.3\ \mathrm{g\,cm^{-3}}\ (\mathrm{rocky}) \\ 0.70\,M^{0.63} & \rho \le 3.3\ \mathrm{g\,cm^{-3}}\ (\mathrm{volatile\text{-}rich}) \end{cases}`
- 使用输入：`['pl_bmasse', 'pl_dens']`
- 拟合结果：nmse=0.1388, r2=0.8612, rmse=1.2093, kendall_tau=0.7911, mape=0.1765, log10_rmse=0.1314, log10_r2=0.8216, N_evaluated=1482

## `astrophysics/gwtc4_bbh_massfunction`

- 任务标题：GWTC-4.0 binary-black-hole primary-mass distribution (Power Law + Peak and variants)
- 任务背景：GWOSC cumulative GWTC event-level summary (O1 + O2 + O3 + O4a compact-binary candidates)
- Formula `plpp`
- 表达式：`p(m_1 \mid \Lambda) = (1 - \lambda_{\mathrm{peak}})\,\mathcal{P}(m_1; -\alpha, m_{\min}, m_{\max})\,S(m_1; m_{\min}, \delta_m) + \lambda_{\mathrm{peak}}\,\mathcal{N}(m_1; \mu_{\mathrm{peak}}, \sigma_{\mathrm{peak}}^2)\,S(m_1; m_{\min}, \delta_m)\,\Theta(m_{\max} - m_1)`
- 使用输入：`['mass_1_source']`
- 拟合结果：nmse=0, r2=1, rmse=0, kendall_tau=1, mape=0, mean_log_likelihood=-6.00481, kendall_tau_vs_m1=-0.392757, mae_vs_uniform=2.66194, n_scored=161, rmse_vs_uniform=3.46636
- Formula `powerlaw`
- 表达式：`p(m_1 \mid \Lambda) \propto m_1^{-\alpha}\,S(m_1; m_{\min}, \delta_m)\,\Theta(m_{\max} - m_1)`
- 使用输入：`['mass_1_source']`
- 拟合结果：nmse=0, r2=1, rmse=0, kendall_tau=1, mape=0, mean_log_likelihood=-5.7675, kendall_tau_vs_m1=-0.99952, mae_vs_uniform=2.02633, n_scored=159, rmse_vs_uniform=2.24746
- Formula `broken_powerlaw`
- 表达式：`p(m_1 \mid \Lambda) \propto \begin{cases} (m_1/m_{\mathrm{break}})^{-\alpha_1} & m_{\min} \le m_1 < m_{\mathrm{break}} \\ (m_1/m_{\mathrm{break}})^{-\alpha_2} & m_{\mathrm{break}} \le m_1 < m_{\max} \end{cases} \cdot S(m_1; m_{\min}, \delta_m)`
- 使用输入：`['mass_1_source']`
- 拟合结果：nmse=0, r2=1, rmse=0, kendall_tau=1, mape=0, mean_log_likelihood=-4.91324, kendall_tau_vs_m1=-0.999848, mae_vs_uniform=1.16589, n_scored=163, rmse_vs_uniform=1.60752

## `astrophysics/kepler_third_law`

- 任务标题：Kepler's third law, 8-planet solar-system benchmark (P^2 ∝ a^3)
- 任务背景：JPL/SSD "Approximate Positions of the Planets", Table 1 (J2000, valid 1800-2050 AD); orbital period derived as P=36000/Ldot from the mean-longitude rate.
- Formula `kepler_third`
- 表达式：`P^2 = \frac{4\pi^2}{G\,M_\odot}\,a^3 \;\Longleftrightarrow\; P[\mathrm{yr}] = a[\mathrm{AU}]^{3/2}`
- 使用输入：`['a_au']`
- 拟合结果：nmse=4.9142e-07, r2=1, rmse=0.0387414, kendall_tau=1, mape=0.00020867

## `astrophysics/pantheon_plus_hubble`

- 任务标题：Pantheon+/SH0ES Type Ia supernova Hubble diagram (flat ΛCDM μ(z))
- 任务背景：Pantheon+ & SH0ES joint SN Ia distance-modulus compilation (1701 SNe, 0.001 < z_HD < 2.3)
- Formula `lcdm_exact`
- 表达式：`\mu(z) = 5\,\log_{10}\!\left[(1+z_{\mathrm{HEL}})\,\frac{c}{H_0}\int_0^{z_{\mathrm{HD}}}\frac{dz'}{\sqrt{\Omega_m(1+z')^3 + (1-\Omega_m)}}\right] + 25`
- 使用输入：`['z_HD', 'z_HEL']`
- 拟合结果：nmse=0.002647, r2=0.997353, rmse=0.173572, kendall_tau=0.965903, mape=0.00331

## `astrophysics/sparc_tully_fisher_rar`

- 任务标题：SPARC radial-acceleration and baryonic Tully-Fisher relations (Lelli+2016, McGaugh+2016, Lelli+2019)
- 任务背景：Spitzer Photometry and Accurate Rotation Curves (SPARC) database: 175 disk galaxies with homogeneous 3.6 micron near-infrared photometry (tracing stellar mass) plus HI (21 cm) and occasionally H-alpha rotation curves drawn from three decades of literature. ...
- Formula `rar_mcgaugh2016`
- 表达式：`g_{\rm obs} = \dfrac{g_{\rm bar}}{1 - \exp\!\left(-\sqrt{g_{\rm bar}/g_\dagger}\right)}`
- 使用输入：`['g_bar']`
- 拟合结果：nmse=0.3793, r2=0.6207, rmse=4.048e-10, kendall_tau=0.7853, mape=0.4264, log10_rmse_full=0.1989, log10_rmse_paper_cut=0.1426
- Formula `btf_lelli2019`
- 表达式：`\log_{10}(M_{\rm bar}/M_\odot) = s\,\log_{10}(V_{\rm flat}/\mathrm{km\,s^{-1}}) + I,\quad s = 3.85,\; I = 1.99`
- 使用输入：`['V_flat']`
- 拟合结果：nmse=0.6673, r2=0.3327, rmse=4.839e10, kendall_tau=0.8207, mape=0.4604, log10_rmse=0.2413, log10_r2=0.9183

## `astrophysics/wadekar2023_sz_flux_mass`

- 任务标题：Sunyaev-Zel'dovich flux-mass (Y-M) relation for low-mass halos (Wadekar et al. 2023)
- 任务背景：CAMELS IllustrisTNG LH + CAMELS SIMBA LH halo catalogs at z = 0.27 (integrated SZ flux, gas / stellar / total mass within R_200c and R_200c/2, reprocessed from the radial profile files released with Wadekar et al. 2023).
- Formula `wadekar2023_main`
- 表达式：`Y = C \, \frac{M_{200c}^{5/3}}{\Omega_m \, \left(1 + M_{\star}(r < R_{200c}/2) / M_{\mathrm{gas}}(r < R_{200c}/2)\right)}`
- 使用输入：`['M_200c', 'M_gas_half', 'M_star_half', 'Omega_m']`
- 拟合结果：nmse=0.138763, r2=0.861237, rmse=5.74527e-08, kendall_tau=0.748846, mape=0.561405

## `battery_degradation/attia_closed_loop`

- 任务标题：Attia 2020 closed-loop 233-cell A123 LFP fast-charge benchmark (Nature 578:397)
- 任务背景：Closed-loop battery cycling dataset released with Attia et al. 2020 (https://data.matr.io/1/, Stanford-TRI Battery Informatics Lab). 233 A123 APR18650M1A LFP/graphite 18650 cells, cycled with 224 distinct 6-step 10-minute fast-charge protocols (specified by...
- Formula `attia_si_table1_early_predictor`
- 表达式：`\log_{10}(\mathrm{cycle\_life}) = \beta_0 + \sum_{k=1}^{5} w_k \cdot \frac{x_k - \mu_k}{\sigma_k}`
- 使用输入：`['Q_cycle_2', 'Q_max_minus_Q_cycle_2', 'log10_min_dQ_100_10', 'log10_var_dQ_100_10', 'log10_skew_dQ_100_10']`
- 拟合结果：nmse=0.844766, r2=0.155234, rmse=159.418, kendall_tau=0.706417, mape=0.182549

## `battery_degradation/severson_124cell`

- 任务标题：Severson 124-cell A123 LFP cycle-life benchmark (Nature Energy 2019)
- 任务背景：MIT-Stanford-TRI battery dataset released with Severson et al. 2019 (https://data.matr.io/1/projects/5c48dd2bc625d700019f3204). 124 A123 APR18650M1A LFP/graphite 18650 cells, cycled with 72 distinct fast-charge protocols at 30 degC environmental control. We...
- Formula `severson_variance`
- 表达式：`\log_{10}(\mathrm{cycle\_life}) = \beta_0 + \beta_1 \cdot \frac{\log_{10}|\mathrm{var}_V \Delta Q_{100-10}(V)| - \mu}{\sigma}`
- 使用输入：`['log10_var_dQ_100_10']`
- 拟合结果：nmse=0.157643, r2=0.842357, rmse=150.155, kendall_tau=0.724577, mape=0.134703
- Formula `severson_discharge`
- 表达式：`\log_{10}(\mathrm{cycle\_life}) = \beta_0 + \sum_{k=1}^{6} \beta_k \cdot z_k(x_k)`
- 使用输入：`['log10_min_dQ_100_10', 'log10_var_dQ_100_10', 'log10_skew_dQ_100_10', 'log10_kurt_dQ_100_10', 'Q_cycle_2', 'Q_max_minus_Q_cycle_2']`
- 拟合结果：nmse=0.134903, r2=0.865097, rmse=138.903, kendall_tau=0.774177, mape=0.117757
- Formula `severson_full`
- 表达式：`\log_{10}(\mathrm{cycle\_life}) = \beta_0 + \sum_{k=1}^{7} \beta_k \cdot z_k(x_k)`
- 使用输入：`['log10_var_dQ_100_10', 'log10_min_dQ_100_10', 'slope_Q_2_100', 'intercept_Q_2_100', 'Q_cycle_2', 'log10_abs_dQ_V2', 'Q_cycle_100_minus_2']`
- 拟合结果：nmse=0.247806, r2=0.752194, rmse=188.26, kendall_tau=0.785461, mape=0.1559

## `dynamical_systems/hudson_bay_lynx_hare`

- 任务标题：Hudson's Bay Company lynx/hare series — Lotka–Volterra hare derivative
- 任务背景：Paired HBC annual fur-return counts for Canada lynx and snowshoe hare, 1845–1935 (compiled by MacLulich 1937 and Elton & Nicholson 1942); filtered to years where both series are present, then augmented with centred finite differences (O(h^2)) of both series...
- Formula `lv_hare`
- 表达式：`\dot{H} = \alpha\, H - \beta\, H\, L`
- 使用输入：`['H_t', 'L_t']`
- 拟合结果：nmse=0.920852, r2=0.079148, rmse=15.9823, kendall_tau=0.344103, mape=1.03429

## `dynamical_systems/ibm_double_pendulum`

- 任务标题：IBM Double Pendulum Chaotic Dataset — Lagrangian / Hamiltonian / Euler–Lagrange equations of motion
- 任务背景：21 high-speed (400 fps, 480x480 px) videos of a real planar double pendulum recorded by IBM Research AI (Asseman, Kornuta, Ozcan 2018). IBM applied a five-times-upscaled template-matching tracker to each frame to extract pixel positions of three fiducial ma...
- Formula `eom_theta1`
- 表达式：`\ddot{\theta}_1 = \frac{-g(2 m_1 + m_2)\sin\theta_1 - m_2 g \sin(\theta_1 - 2\theta_2) - 2 \sin(\theta_1 - \theta_2)\, m_2 \bigl( \dot{\theta}_2^{\,2} l_2 + \dot{\theta}_1^{\,2} l_1 \cos(\theta_1 - \theta_2) \bigr)} {l_1\bigl(2 m_1 + m_2 - m_2 \cos(2(\theta_1 - \theta_2))\bigr)}`
- 使用输入：`['theta1', 'theta2', 'omega1', 'omega2']`
- 拟合结果：nmse=未提供, r2=未提供, rmse=未提供, kendall_tau=未提供, mape=未提供
- Formula `eom_theta2`
- 表达式：`\ddot{\theta}_2 = \frac{2 \sin(\theta_1 - \theta_2)\bigl( \dot{\theta}_1^{\,2} l_1 (m_1 + m_2) + g(m_1 + m_2)\cos\theta_1 + \dot{\theta}_2^{\,2} l_2 m_2 \cos(\theta_1 - \theta_2) \bigr)} {l_2\bigl(2 m_1 + m_2 - m_2 \cos(2(\theta_1 - \theta_2))\bigr)}`
- 使用输入：`['theta1', 'theta2', 'omega1', 'omega2']`
- 拟合结果：nmse=未提供, r2=未提供, rmse=未提供, kendall_tau=未提供, mape=未提供
- Formula `hamiltonian`
- 表达式：`H = \tfrac{1}{2} m_1 l_1^2 \dot{\theta}_1^{\,2} + \tfrac{1}{2} m_2 \bigl( l_1^2 \dot{\theta}_1^{\,2} + l_2^2 \dot{\theta}_2^{\,2} + 2 l_1 l_2 \dot{\theta}_1 \dot{\theta}_2 \cos(\theta_1 - \theta_2) \bigr) - m_1 g l_1 \cos\theta_1 - m_2 g (l_1 \cos\theta_1 + l_2 \cos\theta_2)`
- 使用输入：`['theta1', 'theta2', 'omega1', 'omega2']`
- 拟合结果：nmse=未提供, r2=未提供, rmse=未提供, kendall_tau=未提供, mape=未提供

## `ecology_epi/bjornstad_measles_tsir`

- 任务标题：UK measles 1944-1966 — TSIR (time-series SIR) recurrence
- 任务背景：UK measles case reports, 60 cities, 1944-1966, biweekly. Case counts compiled by the General Register Office; hosted as tab-delimited text (60measles.txt, 60cities.txt) on Ottar Bjornstad's Penn State Entomology lab page. These are the same underlying case-...
- Formula `tsir_bjornstad`
- 表达式：`\mathbb{E}[I_{t+1}] \;=\; \beta_{b(t)}\,\frac{S_t\,I_t^{\alpha}}{N_t}`
- 使用输入：`['cases_t', 'S_t', 'N_t', 'biweek']`
- 拟合结果：nmse=0.104157, r2=0.895843, rmse=183.106, kendall_tau=0.773908, mape=0.79776

## `ecology_epi/tycho_measles_multicountry`

- 任务标题：US state-level measles 1909-1962 (Project Tycho Level 2) — TSIR recurrence
- 任务背景：Project Tycho Level 2 version 1.1.0 — weekly reported measles case counts across 52 US states and territories, 1909-1962, compiled by the University of Pittsburgh Project Tycho team from state health- department notifiable-disease returns (the same primary ...
- Formula `bjornstad_tsir`
- 表达式：`\mathbb{E}[I_{t+1}] \;=\; \beta_{b(t)}\,\frac{S_t\,I_t^{\alpha}}{N_t}`
- 使用输入：`['cases_biweekly', 'S_t_reconstructed', 'population_denom', 'biweek']`
- 输出列：`cases_next_biweekly`
- 拟合结果：nmse=0.445855, r2=0.554145, rmse=631.17, kendall_tau=0.67351, mape=0.865189

## `industry_techspec/cec_pv_module`

- 任务标题：California Energy Commission certified PV module database (Sandia / CEC 5-parameter single-diode)
- 任务背景：California Energy Commission Solar Equipment List for PV modules, as snapshotted by NREL's SAM and redistributed inside the pvlib-python package (bundled file pvlib/data/sam-library-cec-modules-*.csv; accessed via pvlib.pvsystem.retrieve_sam("CECMod")). Eac...
- Formula `single_diode_pmp`
- 表达式：`P_{mp} = \max_V\; V \cdot I(V), \quad I(V) = I_L - I_0\!\left[\exp\!\left(\dfrac{V + I R_s}{a}\right) - 1\right] - \dfrac{V + I R_s}{R_{sh}}`
- 使用输入：`['I_L_ref', 'I_o_ref', 'R_s', 'R_sh_ref', 'a_ref']`
- 拟合结果：nmse=8.40382e-13, r2=1, rmse=5.39922e-05, kendall_tau=0.999851, mape=1.75054e-07
- Formula `power_from_imp_vmp`
- 表达式：`P_{mp} = I_{mp} \cdot V_{mp}`
- 使用输入：`['I_mp_ref', 'V_mp_ref']`
- 拟合结果：nmse=2.044e-31, r2=1, rmse=2.6629e-14, kendall_tau=0.999996, mape=5.52e-17
- Formula `efficiency_area_scaling`
- 表达式：`P_{mp,STC} = \bar{\eta} \cdot A_c \cdot G_{STC}, \quad G_{STC} = 1000\,\mathrm{W/m^2},\; \bar{\eta} \approx 0.154`
- 使用输入：`['A_c']`
- 拟合结果：nmse=0.334844, r2=0.665156, rmse=34.0811, kendall_tau=0.581581, mape=0.097689

## `industry_techspec/duramat_pv_degradation`

- 任务标题：DuraMAT PV Fleet performance-loss-rate database (Jordan et al. 2022)
- 任务背景：DuraMAT DataHub "Photovoltaic Fleet Degradation Insights -- Data" (2022-02-02 snapshot). File fleet_results_public.csv, 4915 individual inverter records from the DOE PV Fleet Performance Data Initiative, representing roughly 6-7% of the US installed PV capa...
- Formula `jordan_tech_median`
- 表达式：`\widehat{\mathrm{PLR}}(\mathrm{tech}) = m_{\mathrm{tech}},\; m \in \{m_{\mathrm{Al\text{-}BSF}}, m_{\mathrm{PERC}}, m_{\mathrm{n\text{-}SHJ}}, m_{\mathrm{n\text{-}IBC}}, m_{\mathrm{n\text{-}PERT}}\}`
- 使用输入：`['technology2']`
- 拟合结果：nmse=1.01341, r2=-0.013413, rmse=0.849647, kendall_tau=0.1326, mape=1.27987
- Formula `jordan_climate_dep`
- 表达式：`\widehat{\mathrm{PLR}}(z) = m_z,\; z \in \{T2,T3,T4,T5,T6\}`
- 使用输入：`['pv_climate_zone']`
- 拟合结果：nmse=1.05963, r2=-0.05963, rmse=0.868805, kendall_tau=0.057742, mape=1.14162
- Formula `jordan_mounting_median`
- 表达式：`\widehat{\mathrm{PLR}}(t, \tau) = \begin{cases} -0.68 & t = \mathrm{FALSE} \\ -0.76 & t = \mathrm{TRUE},\; \tau \in \{\mathrm{mono\text{-}Si}, \mathrm{multi\text{-}Si}, \mathrm{c\text{-}Si}\} \\ -0.61 & t = \mathrm{TRUE},\; \tau = \mathrm{CdTe} \\ -0.75 & \text{otherwise (fleet fallback)} \end{cases}`
- 使用输入：`['tracking', 'technology1']`
- 拟合结果：nmse=1.03683, r2=-0.036829, rmse=0.859407, kendall_tau=0.149275, mape=1.17184

## `industry_techspec/iea_15mw_turbine`

- 任务标题：IEA 15-MW offshore reference wind turbine rotor performance Cp(lambda, beta)
- 任务背景：ROSCO-toolbox rotor performance table Cp_Ct_Cq.IEA15MW.txt distributed with the IEA-15-240-RWT OpenFAST model (IEAWindTask37 repo, Apache-2.0, pinned commit 86d51c8a1ee65be4f3686087a5c443c0b57e5cfb, file header timestamp 2022-01-13). The table itself is pro...
- Formula `bem_cp_heuristic`
- 表达式：`C_p(\lambda, \beta) = c_1\!\left(\dfrac{c_2}{\lambda_i} - c_3\,\beta - c_4\right)\exp\!\left(-\dfrac{c_5}{\lambda_i}\right) + c_6\,\lambda, \quad \dfrac{1}{\lambda_i} = \dfrac{1}{\lambda + 0.08\,\beta} - \dfrac{0.035}{\beta^3 + 1}`
- 使用输入：`['lambda_tsr', 'beta_pitch_deg']`
- 拟合结果：nmse=0.038015, r2=0.961985, rmse=0.172365, kendall_tau=0.822591, mape=0.659892

## `materials/estm_thermoelectric`

- 任务标题：热电优值ZT定义式验证（Na & Chang 2022, ESTM数据库）
- 任务背景：ESTM（Experimental Symmetrized Thermoelectric Materials）数据库， 由 Na & Chang 2022 从 Starrydata2 / Citrination 文本挖掘数据筛选并标准化而来。 原始 Excel 文件托管于 GitHub 仓库 KRICT-DATA/SIMD。
- Formula `zt_definition`
- 表达式：`ZT = \frac{S^2 \, \sigma \, T}{\kappa}`
- 使用输入：`['S_uV_K', 'sigma_S_m', 'kappa_W_mK', 'T_K']`
- 拟合结果：nmse=0.00291, r2=0.99709, rmse=0.018746, kendall_tau=0.990714, mape=0.014818

## `materials/sisso_perovskite_tau`

- 任务标题：SISSO 钙钛矿容忍因子 τ（Bartel et al. 2019, Science Advances）
- 任务背景：576 ABX3 钙钛矿化合物（氧化物 + 卤化物），离子半径来自 Shannon 有效离子半径表， 稳定性实验标签来自文献汇编（DFT 辅助实验标注）。原始数据来自论文 companion GitHub 仓库 https://github.com/CJBartel/perovskite-stability 的 TableS1.csv。
- Formula `bartel2019_tau`
- 表达式：`\tau = \frac{r_X}{r_B} - n_A\!\left(n_A - \frac{r_A/r_B}{\ln(r_A/r_B)}\right)`
- 使用输入：`['r_A', 'r_B', 'r_X', 'n_A']`
- 拟合结果：nmse=0.011455, r2=0.988545, rmse=0.97515, kendall_tau=0.994285, mape=0.004456

## `materials/weng_perovskite_oer`

- 任务标题：钙钛矿氧化物 OER 催化活性的 μ/t 描述符（Weng et al. 2020, Nat. Commun.）
- 任务背景：Experimental V_RHE (voltage vs. reversible hydrogen electrode) of 23 ABO3 oxide perovskite electrocatalysts at 5 mA cm^-2 (iR-corrected, 0.1 M KOH, glassy-carbon RDE, 2016-2020 Toledo / Soochow / Hunan labs). 18 compounds are classical perovskites synthesis...
- Formula `weng_mu_over_t`
- 表达式：`V_{\mathrm{RHE}} \;=\; 1.554 \cdot \frac{\mu}{t} + 1.092, \quad \mu = \frac{r_B}{r_O}, \quad t = \frac{r_A + r_O}{\sqrt{2}(r_B + r_O)}.`
- 使用输入：`['r_A', 'r_B', 'r_O']`
- 拟合结果：nmse=0.516241, r2=0.483759, rmse=0.043138, kendall_tau=0.678699, mape=0.017764
- Formula `weng_tolerance_only`
- 表达式：`V_{\mathrm{RHE}} \;=\; \frac{1.751}{t}, \quad t = \frac{r_A + r_O}{\sqrt{2}(r_B + r_O)}.`
- 使用输入：`['r_A', 'r_B', 'r_O']`
- 拟合结果：nmse=0.518953, r2=0.481047, rmse=0.043252, kendall_tau=0.664192, mape=0.017115
- Formula `weng_mu_over_t_refit23`
- 表达式：`V_{\mathrm{RHE}} \;=\; 1.804 \cdot \frac{\mu}{t} + 1, \quad \mu = \frac{r_B}{r_O}, \quad t = \frac{r_A + r_O}{\sqrt{2}(r_B + r_O)}.`
- 使用输入：`['r_A', 'r_B', 'r_O']`
- 拟合结果：nmse=0.499831, r2=0.500169, rmse=0.042447, kendall_tau=0.678699, mape=0.017643

## `soft_sensors/uci_ccpp`

- 任务标题：Combined Cycle Power Plant full-load electrical output (Tufekci 2014, UCI dataset 294)
- 任务背景：Hourly-averaged SCADA readings from one CCGT (combined-cycle gas turbine) power plant operated at full load for six years (2006-2011). Released by the authors through the UCI Machine Learning Repository (dataset 294) as companion material to Tufekci 2014 an...
- Formula `tufekci_linear`
- 表达式：`EP = \beta_0 + \beta_{AT}\,AT + \beta_V\,V + \beta_{AP}\,AP + \beta_{RH}\,RH`
- 使用输入：`['AT', 'V', 'AP', 'RH']`
- 拟合结果：nmse=0.0713039, r2=0.928696, rmse=4.55713, kendall_tau=0.82977, mape=0.00799513