# Regression-based projection for learning Mori-Zwanzig operators

arXiv: https://arxiv.org/abs/2205.05135 · HTML: https://mlsanity.com/paper/2205.05135/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 57 references that are in the map.
- First posted on arXiv 10 May 2022.

## Builds on (most cited first, 10 of 10)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics](https://mlsanity.com/paper/1710.11239/) (2017)
- [Machine Learning for Prediction with Missing Dynamics](https://mlsanity.com/paper/1910.05861/) (2019)
- [Model Reduction with Memory and the Machine Learning of Dynamical Systems](https://mlsanity.com/paper/1808.04258/) (2018)
- [Kernel-based Prediction of Non-Markovian Time Series](https://mlsanity.com/paper/2007.04286/) (2020)

## Built on by (most cited first, 2 of 2)

- [Mori-Zwanzig latent space Koopman closure for nonlinear autoencoder](https://mlsanity.com/paper/2310.10745/) (2023)
- [Data-driven Mori-Zwanzig modeling of Lagrangian particle dynamics in turbulent flows](https://mlsanity.com/paper/2507.16058/) (2025)

## Abstract (arXiv)

We propose to adopt statistical regression as the projection operator to enable data-driven learning of the operators in the Mori--Zwanzig formalism. We present a principled method to extract the Markov and memory operators for any regression models. We show that the choice of linear regression results in a recently proposed data-driven learning algorithm based on Mori's projection operator, which is a higher-order approximate Koopman learning method. We show that more expressive nonlinear regression models naturally fill in the gap between the highly idealized and computationally efficient Mori's projection operator and the most optimal yet computationally infeasible Zwanzig's projection operator. We performed numerical experiments and extracted the operators for an array of regression-based projections, including linear, polynomial, spline, and neural-network-based regressions, showing a progressive improvement as the complexity of the regression model increased. Our proposition provides a general framework to extract memory-dependent corrections and can be readily applied to an array of data-driven learning methods for stationary dynamical systems in the literature.