# Physics Guided Machine Learning for Variational Multiscale Reduced Order Modeling

arXiv: https://arxiv.org/abs/2205.12419 · HTML: https://mlsanity.com/paper/2205.12419/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 88 references that are in the map.
- First posted on arXiv 25 May 2022.

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

- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [Digital Twin: Values, Challenges and Enablers](https://mlsanity.com/paper/1910.01719/) (2019)
- [A Probabilistic Graphical Model Foundation for Enabling Predictive Digital Twins at Scale](https://mlsanity.com/paper/2012.05841/) (2020)
- [Deep Learning with Long Short-Term Memory for Time Series Prediction](https://mlsanity.com/paper/1810.10161/) (2018)
- [Physics guided machine learning using simplified theories](https://mlsanity.com/paper/2012.13343/) (2020)
- [Neural Closure Models for Dynamical Systems](https://mlsanity.com/paper/2012.13869/) (2020)
- [Data-Driven POD-Galerkin Reduced Order Model for Turbulent Flows](https://mlsanity.com/paper/1907.09909/) (2019)
- [Data-driven Discovery of Closure Models](https://mlsanity.com/paper/1803.09318/) (2018)
- [Progressive construction of a parametric reduced-order model for PDE-constrained optimization](https://mlsanity.com/paper/1407.7618/) (2014)
- [Nonlinear proper orthogonal decomposition for convection-dominated flows](https://mlsanity.com/paper/2110.08295/) (2021)
- [The ROMES method for statistical modeling of reduced-order-model error](https://mlsanity.com/paper/1405.5170/) (2014)

## Built on by (most cited first, 4 of 4)

- [Parametric Taylor series based latent dynamics identification neural networks](https://mlsanity.com/paper/2410.04193/) (2024)
- [Interpretable Diagnostics and Adaptive Data Assimilation for Neural ODEs via Discrete Empirical Interpolation](https://mlsanity.com/paper/2510.21852/) (2025)
- [Variational multiscale reinforcement learning for discovering reduced order closure models of nonlinear spatiotemporal transport systems](https://mlsanity.com/paper/2207.12854/) (2022)
- [Uncertainty-aware Multi-fidelity Closure via Conditional Normalizing Flows](https://mlsanity.com/paper/2606.09857/) (2026)

## Abstract (arXiv)

We propose a new physics guided machine learning (PGML) paradigm that leverages the variational multiscale (VMS) framework and available data to dramatically increase the accuracy of reduced order models (ROMs) at a modest computational cost. The hierarchical structure of the ROM basis and the VMS framework enable a natural separation of the resolved and unresolved ROM spatial scales. Modern PGML algorithms are used to construct novel models for the interaction among the resolved and unresolved ROM scales. Specifically, the new framework builds ROM operators that are closest to the true interaction terms in the VMS framework. Finally, machine learning is used to reduce the projection error and further increase the ROM accuracy. Our numerical experiments for a two-dimensional vorticity transport problem show that the novel PGML-VMS-ROM paradigm maintains the low computational cost of current ROMs, while significantly increasing the ROM accuracy.