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Learning Generalized Residual Exchange-Correlation-Uncertain Functional for Density Functional Theory
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Proposes RBNet, a residual Bayesian neural network that predicts mean and variance of a residual exchange-correlation energy density added to a conventional XC functional (B3LYP or DM21), improving DFT accuracy on GMTKN55 subsets.
Problem
Systematic errors in exchange-correlation (XC) approximations in Kohn-Sham density functional theory, which limit accuracy even for hybrid functionals like B3LYP and learned functionals like DM21.
Approach
Model the XC energy density as a Gaussian (mean and variance) and learn only a residual on top of a conventional XC functional, with the mean scaled by tanh and the variance bounded by the predicted energy magnitude, to stabilize SCF convergence and avoid loss explosion.
Key results
- B3LYP+RBNet reaches RMSE 1.95 kcal/mol on W4-17 versus 6.05 for B3LYP-D3(BJ) (+67.77%).
- DM21+RBNet reaches RMSE 2.45 on W4-17 versus 3.41 for DM21 (+28.15%) and 2.86 on G21IP versus 9.97 for DM21 (+71.31%).
- Ablation: RBNet (RMSE 1.95 on W4-17) outperforms a ResNet without variance (RMSE 3.94) under the B3LYP baseline.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add RBNet to B3LYP | B3LYP · W4-17 | RMSE: 6.05 -> 1.95 | Helped |
| Add RBNet to B3LYP | B3LYP · G21EA | RMSE: 6.31 -> 1.67 | Helped |
| Add RBNet to B3LYP | B3LYP · G21IP | RMSE: 5.83 -> 2.60 | Helped |
| Add RBNet to DM21 | DM21 · W4-17 | RMSE: 3.41 -> 2.45 | Helped |
| Add RBNet to DM21 | DM21 · G21EA | RMSE: 2.66 -> 2.34 | Helped |
| Add RBNet to DM21 | DM21 · G21IP | RMSE: 9.97 -> 2.86 | Helped |
| Use y(r) features instead of x(r) (ablation) | B3LYP+RBNet · W4-17 | RMSE: 2.35 -> 1.95 | Helped |
| Use y(r) features instead of x(r) (ablation) | B3LYP+RBNet · G21EA | RMSE: 3.51 -> 2.82 | Helped |
| Use y(r) features instead of x(r) (ablation) | B3LYP+RBNet · G21IP | RMSE: 3.47 -> 2.54 | Helped |
| Add variance modeling with RBNet instead of ResNet without variance (ablation) | B3LYP · W4-17 | RMSE: 3.94 -> 1.95 | Helped |
| Add variance modeling with RBNet instead of ResNet without variance (ablation) | B3LYP · G21EA | RMSE: 3.27 -> 2.82 | Helped |
| Add variance modeling with RBNet instead of ResNet without variance (ablation) | B3LYP · G21IP | RMSE: 3.74 -> 2.54 | Helped |
Limitations
- Reported gains are relative to DFT baselines on small reaction-energy benchmarks; the paper does not report generalization beyond these subsets.
- Training epoch count and some implementation details are left unspecified in the text.
- Uses a proprietary single-atom dataset whose details are not released.
Experimental setup
Datasets: W4-17 G21EA G21IP GMTKN55 Atom (proprietary single-atom energy dataset)
Benchmarks: W4-17 G21EA G21IP
Metrics: RMSE MAE MAD
Baselines: SCAN-D3(BJ) M06-2X-D3(0) ωB97X-V MN15-D3(BJ) B3LYP-D3(BJ) DM21 B3LYP+ResNet
Builds on: DM21 B3LYP residual learning Kendall and Gal heteroscedastic uncertainty loss
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