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The paper addresses the challenge of extrapolating machine learning-based initial guesses for Self-Consistent Field (SCF) calculations to larger molecules, where previous matrix-prediction models have failed. They introduce Solver-Aligned Initialization Learning (SAIL), which differentiates through the SCF solver to directly optimize for convergence speed, using a new metric called Effective Relative Iteration Count (ERIC). SAIL demonstrates significant improvements in reducing ERIC and achieving wall-time speedups on larger molecules compared to existing methods, indicating its effectiveness in accelerating SCF convergence for out-of-distribution molecular sizes.
ML models can accurately predict quantum properties out-of-distribution, but still fail to accelerate SCF convergence – until now.
Machine learning methods that predict initial guesses from molecular geometry can reduce this cost, but matrix-prediction models fail when extrapolating to larger molecules, degrading rather than accelerating convergence [Liu et al. 2025]. We show that this failure is a supervision problem, not an extrapolation problem: models trained on ground-state targets fit those targets well out of distribution, yet produce initial guesses that slow convergence. Solver-Aligned Initialization Learning (SAIL) resolves this for both Hamiltonian and density matrix models by differentiating through the SCF solver end-to-end. We introduce the Effective Relative Iteration Count (ERIC), a correction to the commonly used RIC that accounts for hidden Fock-build overhead. On QM40, containing molecules up to 4$\times$ larger than the training distribution, SAIL reduces ERIC by 37% (PBE), 33% (SCAN), and 27% (B3LYP), more than doubling the previous state-of-the-art reduction on B3LYP (10%). On QMugs molecules 10$\times$ the training size, SAIL delivers a 1.25$\times$ wall-time speedup at the hybrid level of theory, extending ML SCF acceleration to large drug-like molecules.
