投影Hessian学习：用于准确机器学习互原子势能的快速曲率监督
Projected Hessian Learning: Fast Curvature Supervision for Accurate Machine-Learning Interatomic Potentials
AI总结：
PHL通过投影Hessian学习实现快速曲率监督，提升机器学习互原子势能的训练效率和精度。
摘要：
Hessian矩阵（二阶导数）编码了比能量和力更丰富的局部曲率信息。然而，使用完整的Hessian矩阵训练机器学习互原子势能（MLIPs）通常是不实际的，因为显式地形成和存储Hessian矩阵会带来二次成本和内存消耗。我们引入了投影Hessian学习（PHL），一种可扩展的二阶训练框架，它通过仅使用Hessian向量积（HVPs）来注入曲率信息。与构建Hessian矩阵不同，PHL在随机探测方向上投影曲率，并使用无偏的随机迹损失，具有有利的系统规模扩展性，从而在不增加二次内存增长的情况下实现曲率指导的训练。我们对一个化学多样化的反应物、产物、过渡态、内禀反应坐标和正常模式采样几何结构数据集进行了基准测试，这些数据集在omegaB97XD/6-31G(d)水平上计算。我们比较了能量-力训练（E-F）、两种基于HVP的方案（E-F-HVP使用one-hot或随机探测）以及完整的能量-力-Hessian训练（E-F-H）。在每个minibatch中使用随机探测时，两种HVP方案在能量、力和Hessian精度上与完整Hessian训练相匹配，同时为研究的小分子系统提供了>24倍的epoch加速。在固定探测模式下，每个分子一个HVP的情况下，随机投影在远离开平衡几何结构时始终优于单列探测。总体而言，PHL用力复杂度曲率训练取代了显式Hessian监督，保留了大部分二阶精度收益，同时可扩展到更大的、更复杂的分子系统。
英文摘要：
The Hessian matrix (second derivatives) encodes far richer local curvature of the potential energy surface than energies and forces alone. However, training machine-learning interatomic potentials (MLIPs) with full Hessians is often impractical because explicitly forming and storing Hessian matrices scales quadratically in cost and memory. We introduce Projected Hessian Learning (PHL), a scalable second-order training framework that injects curvature information using only Hessian-vector products (HVPs). Rather than constructing the Hessian, PHL projects curvature along stochastic probe directions and uses an unbiased stochastic trace-based loss with favorable system-size scaling, enabling curvature-informed training without quadratic memory growth. We benchmark PHL on a chemically diverse dataset of reactants, products, transition states, intrinsic reaction coordinates, and normal-mode sampled geometries computed at omegaB97XD/6-31G(d). We compare energy-force training (E-F), two HVP-based schemes (E-F-HVP with one-hot or randomized probes), and full energy-force-Hessian training (E-F-H). With randomized probes per minibatch, both HVP schemes match full-Hessian training in energy, force, and Hessian accuracy while delivering >24x epoch speedups for the small molecular systems studied. In a fixed-probe regime with one HVP per molecule, randomized projections consistently outperform one-column probing, especially for far-from-equilibrium geometries. Overall, PHL replaces explicit Hessian supervision with force-complexity curvature training, retaining most second-order accuracy gains while scaling to larger, more complex molecular systems.
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