# Solving the electronic Schr"odinger equation for multiple nuclear geometries with weight-sharing deep neural networks

arXiv: https://arxiv.org/abs/2105.08351 · HTML: https://mlsanity.com/paper/2105.08351/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 49 references that are in the map.
- First posted on arXiv 18 May 2021.

## Builds on (most cited first, 11 of 11)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [A Survey on Deep Transfer Learning](https://mlsanity.com/paper/1808.01974/) (2018)
- [Machine Learning Force Fields](https://mlsanity.com/paper/2010.07067/) (2020)
- [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)
- [Unifying machine learning and quantum chemistry -- a deep neural network for molecular wavefunctions](https://mlsanity.com/paper/1906.10033/) (2019)
- [Machine learning for electronically excited states of molecules](https://mlsanity.com/paper/2007.05320/) (2020)
- [Better, Faster Fermionic Neural Networks](https://mlsanity.com/paper/2011.07125/) (2020)

## Built on by (most cited first, 24 of 24)

- [Ab-Initio Potential Energy Surfaces by Pairing GNNs with Neural Wave Functions](https://mlsanity.com/paper/2110.05064/) (2021)
- [Ab-initio quantum chemistry with neural-network wavefunctions](https://mlsanity.com/paper/2208.12590/) (2022)
- [Generalizing Neural Wave Functions](https://mlsanity.com/paper/2302.04168/) (2023)
- [Gold-standard solutions to the Schr"odinger equation using deep learning: How much physics do we need?](https://mlsanity.com/paper/2205.09438/) (2022)
- [Electronic excited states in deep variational Monte Carlo](https://mlsanity.com/paper/2203.09472/) (2022)
- [Sampling-free Inference for Ab-Initio Potential Energy Surface Networks](https://mlsanity.com/paper/2205.14962/) (2022)
- [An ab initio foundation model of wavefunctions that accurately describes chemical bond breaking](https://mlsanity.com/paper/2506.19960/) (2025)
- [Neural Wave Functions for Superfluids](https://mlsanity.com/paper/2305.06989/) (2023)
- [Pairing-based graph neural network for simulating quantum materials](https://mlsanity.com/paper/2311.02143/) (2023)
- [Variational Monte Carlo on a Budget -- Fine-tuning pre-trained Neural Wavefunctions](https://mlsanity.com/paper/2307.09337/) (2023)
- [Neural Pfaffians: Solving Many Many-Electron Schr"odinger Equations](https://mlsanity.com/paper/2405.14762/) (2024)
- [Deep Variational Free Energy Approach to Dense Hydrogen](https://mlsanity.com/paper/2209.06095/) (2022)
- [Towards a Foundation Model for Neural Network Wavefunctions](https://mlsanity.com/paper/2303.09949/) (2023)
- [Forward Laplacian: A New Computational Framework for Neural Network-based Variational Monte Carlo](https://mlsanity.com/paper/2307.08214/) (2023)
- [nabla2DFT: A Universal Quantum Chemistry Dataset of Drug-Like Molecules and a Benchmark for Neural Network Potentials](https://mlsanity.com/paper/2406.14347/) (2024)
- [Molecular-orbital-based Machine Learning for Open-shell and Multi-reference Systems with Kernel Addition Gaussian Process Regression](https://mlsanity.com/paper/2207.08317/) (2022)
- [Artificial intelligence for artificial materials: moir'e atom](https://mlsanity.com/paper/2303.08162/) (2023)
- [Sorting Out Quantum Monte Carlo](https://mlsanity.com/paper/2311.05598/) (2023)
- [Hamilton-Zero: A Neural Tensor-Network Foundation Model for Ground States of Arbitrary Quadratic Qubit Hamiltonians](https://mlsanity.com/paper/2608.11911/) (2026)
- [Supervised learning of random quantum circuits via scalable neural networks](https://mlsanity.com/paper/2206.10348/) (2022)
- [Predictive Scale-Bridging Simulations through Active Learning](https://mlsanity.com/paper/2209.09811/) (2022)
- [A Score-Based Model for Learning Neural Wavefunctions](https://mlsanity.com/paper/2305.16540/) (2023)
- [Representability-Aware Neural Networks for Reduced Density Matrices: Application to Fractional Chern Insulators](https://mlsanity.com/paper/2605.20326/) (2026)
- [NucleoNet: A Nuclear-Aware Determinant Ansatz for Multi-Geometry Neural Variational Monte Carlo](https://mlsanity.com/paper/2610.12985/) (2026)

## Abstract (arXiv)

Accurate numerical solutions for the Schr"odinger equation are of utmost importance in quantum chemistry. However, the computational cost of current high-accuracy methods scales poorly with the number of interacting particles. Combining Monte Carlo methods with unsupervised training of neural networks has recently been proposed as a promising approach to overcome the curse of dimensionality in this setting and to obtain accurate wavefunctions for individual molecules at a moderately scaling computational cost. These methods currently do not exploit the regularity exhibited by wavefunctions with respect to their molecular geometries. Inspired by recent successful applications of deep transfer learning in machine translation and computer vision tasks, we attempt to leverage this regularity by introducing a weight-sharing constraint when optimizing neural network-based models for different molecular geometries. That is, we restrict the optimization process such that up to 95 percent of weights in a neural network model are in fact equal across varying molecular geometries. We find that this technique can accelerate optimization when considering sets of nuclear geometries of the same molecule by an order of magnitude and that it opens a promising route towards pre-trained neural network wavefunctions that yield high accuracy even across different molecules.