# Inexact iterative numerical linear algebra for neural network-based spectral estimation and rare-event prediction

arXiv: https://arxiv.org/abs/2303.12534 · HTML: https://mlsanity.com/paper/2303.12534/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 72 references that are in the map.
- First posted on arXiv 22 Mar 2023.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics](https://mlsanity.com/paper/1710.11239/) (2017)
- [Solving for high dimensional committor functions using artificial neural networks](https://mlsanity.com/paper/1802.10275/) (2018)
- [Analog forecasting of extreme-causing weather patterns using deep learning](https://mlsanity.com/paper/1907.11617/) (2019)
- [Computing Committor Functions for the Study of Rare Events Using Deep Learning](https://mlsanity.com/paper/1906.06285/) (2019)
- [Batch Stationary Distribution Estimation](https://mlsanity.com/paper/2003.00722/) (2020)
- [Nonlinear Discovery of Slow Molecular Modes using State-Free Reversible VAMPnets](https://mlsanity.com/paper/1902.03336/) (2019)
- [Probabilistic forecasts of extreme heatwaves using convolutional neural networks in a regime of lack of data](https://mlsanity.com/paper/2208.00971/) (2022)
- [Krylov-Bellman boosting: Super-linear policy evaluation in general state spaces](https://mlsanity.com/paper/2210.11377/) (2022)

## Built on by (most cited first, 9 of 9)

- [A Comprehensive Survey on Rare Event Prediction](https://mlsanity.com/paper/2309.11356/) (2023)
- [The surprising efficiency of temporal difference learning for rare event prediction](https://mlsanity.com/paper/2405.17638/) (2024)
- [Using pretrained graph neural networks with token mixers as geometric featurizers for conformational dynamics](https://mlsanity.com/paper/2409.19838/) (2024)
- [Estimating Committor Functions via Deep Adaptive Sampling on Rare Transition Paths](https://mlsanity.com/paper/2501.15522/) (2025)
- [Pushing the Limits of All-Atom Geometric Graph Neural Networks: Pre-Training, Scaling and Zero-Shot Transfer](https://mlsanity.com/paper/2410.21683/) (2024)
- [Probing reaction channels via reinforcement learning](https://mlsanity.com/paper/2305.17531/) (2023)
- [An exact multiple-time-step variational formulation for the committor and the transition rate](https://mlsanity.com/paper/2509.03539/) (2025)
- [Hierarchical geometric deep learning enables scalable analysis of molecular dynamics](https://mlsanity.com/paper/2512.06520/) (2025)
- [Machine learning kinetics from molecular dynamics data](https://mlsanity.com/paper/2609.17736/) (2026)

## Abstract (arXiv)

Understanding dynamics in complex systems is challenging because there are many degrees of freedom, and those that are most important for describing events of interest are often not obvious. The leading eigenfunctions of the transition operator are useful for visualization, and they can provide an efficient basis for computing statistics such as the likelihood and average time of events (predictions). Here we develop inexact iterative linear algebra methods for computing these eigenfunctions (spectral estimation) and making predictions from a data set of short trajectories sampled at finite intervals. We demonstrate the methods on a low-dimensional model that facilitates visualization and a high-dimensional model of a biomolecular system. Implications for the prediction problem in reinforcement learning are discussed.