# Self-consistent Gradient-like Eigen Decomposition in Solving Schr"odinger Equations

arXiv: https://arxiv.org/abs/2202.01388 · HTML: https://mlsanity.com/paper/2202.01388/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 28 references that are in the map.
- First posted on arXiv 3 Feb 2022.

## Builds on (most cited first, 7 of 7)

- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [EigenGame: PCA as a Nash Equilibrium](https://mlsanity.com/paper/2010.00554/) (2020)
- [Kohn-Sham equations as regularizer: building prior knowledge into machine-learned physics](https://mlsanity.com/paper/2009.08551/) (2020)
- [First Efficient Convergence for Streaming k-PCA: a Global, Gap-Free, and Near-Optimal Rate](https://mlsanity.com/paper/1607.07837/) (2016)
- [Learning the exchange-correlation functional from nature with fully differentiable density functional theory](https://mlsanity.com/paper/2102.04229/) (2021)
- [Exponentially convergent stochastic k-PCA without variance reduction](https://mlsanity.com/paper/1904.01750/) (2019)

## Abstract (arXiv)

The Schr"odinger equation is at the heart of modern quantum mechanics. Since exact solutions of the ground state are typically intractable, standard approaches approximate Schr"odinger equation as forms of nonlinear generalized eigenvalue problems F(V)V = SVLambda in which F(V), the matrix to be decomposed, is a function of its own top-k smallest eigenvectors V, leading to a "self-consistency problem". Traditional iterative methods heavily rely on high-quality initial guesses of V generated via domain-specific heuristics methods based on quantum mechanics. In this work, we eliminate such a need for domain-specific heuristics by presenting a novel framework, Self-consistent Gradient-like Eigen Decomposition (SCGLED) that regards F(V) as a special "online data generator", thus allows gradient-like eigendecomposition methods in streaming k-PCA to approach the self-consistency of the equation from scratch in an iterative way similar to online learning. With several critical numerical improvements, SCGLED is robust to initial guesses, free of quantum-mechanism-based heuristics designs, and neat in implementation. Our experiments show that it not only can simply replace traditional heuristics-based initial guess methods with large performance advantage (achieved averagely 25x more precise than the best baseline in similar wall time), but also is capable of finding highly precise solutions independently without any traditional iterative methods.