# Physics-aware, probabilistic model order reduction with guaranteed stability

arXiv: https://arxiv.org/abs/2101.05834 · HTML: https://mlsanity.com/paper/2101.05834/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 50 references that are in the map.
- First posted on arXiv 14 Jan 2021.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Deep Variational Bayes Filters: Unsupervised Learning of State Space Models from Raw Data](https://mlsanity.com/paper/1605.06432/) (2016)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability](https://mlsanity.com/paper/1906.03663/) (2019)
- [Equivariant Hamiltonian Flows](https://mlsanity.com/paper/1909.13739/) (2019)
- [Black box variational inference for state space models](https://mlsanity.com/paper/1511.07367/) (2015)
- [Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systems](https://mlsanity.com/paper/1912.12976/) (2019)
- [Model Reduction with Memory and the Machine Learning of Dynamical Systems](https://mlsanity.com/paper/1808.04258/) (2018)
- [Deep Generative Markov State Models](https://mlsanity.com/paper/1805.07601/) (2018)
- [Variational Koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations](https://mlsanity.com/paper/1610.06773/) (2016)
- [Physics-Aware Gaussian Processes in Remote Sensing](https://mlsanity.com/paper/2012.07986/) (2020)
- [A physics-aware, probabilistic machine learning framework for coarse-graining high-dimensional systems in the Small Data regime](https://mlsanity.com/paper/1902.03968/) (2019)
- [Machine Learning for Molecular Dynamics on Long Timescales](https://mlsanity.com/paper/1812.07669/) (2018)
- [Physics-constrained, data-driven discovery of coarse-grained dynamics](https://mlsanity.com/paper/1802.03824/) (2018)
- [Structured Black Box Variational Inference for Latent Time Series Models](https://mlsanity.com/paper/1707.01069/) (2017)

## Built on by (most cited first, 12 of 12)

- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [Semi-supervised Invertible Neural Operators for Bayesian Inverse Problems](https://mlsanity.com/paper/2209.02772/) (2022)
- [Simulate Time-integrated Coarse-grained Molecular Dynamics with Multi-Scale Graph Networks](https://mlsanity.com/paper/2204.10348/) (2022)
- [Generative Learning of the Solution of Parametric Partial Differential Equations Using Guided Diffusion Models and Virtual Observations](https://mlsanity.com/paper/2408.00157/) (2024)
- [Interpretable learning of effective dynamics for multiscale systems](https://mlsanity.com/paper/2309.05812/) (2023)
- [Generative Learning for Forecasting the Dynamics of Complex Systems](https://mlsanity.com/paper/2402.17157/) (2024)
- [From latent dynamics to meaningful representations](https://mlsanity.com/paper/2209.00905/) (2022)
- [Physics-enhanced Neural Networks in the Small Data Regime](https://mlsanity.com/paper/2111.10329/) (2021)
- [Interpretable reduced-order modeling with time-scale separation](https://mlsanity.com/paper/2303.02189/) (2023)
- [Closure Discovery for Coarse-Grained Partial Differential Equations Using Grid-based Reinforcement Learning](https://mlsanity.com/paper/2402.00972/) (2024)
- [Physics-aware, deep probabilistic modeling of multiscale dynamics in the Small Data regime](https://mlsanity.com/paper/2102.04269/) (2021)
- [Reinforcement Learning Closures for Underresolved Partial Differential Equations using Synthetic Data](https://mlsanity.com/paper/2505.11308/) (2025)

## Abstract (arXiv)

Given (small amounts of) time-series' data from a high-dimensional, fine-grained, multiscale dynamical system, we propose a generative framework for learning an effective, lower-dimensional, coarse-grained dynamical model that is predictive of the fine-grained system's long-term evolution but also of its behavior under different initial conditions. We target fine-grained models as they arise in physical applications (e.g. molecular dynamics, agent-based models), the dynamics of which are strongly non-stationary but their transition to equilibrium is governed by unknown slow processes which are largely inaccessible by brute-force simulations. Approaches based on domain knowledge heavily rely on physical insight in identifying temporally slow features and fail to enforce the long-term stability of the learned dynamics. On the other hand, purely statistical frameworks lack interpretability and rely on large amounts of expensive simulation data (long and multiple trajectories) as they cannot infuse domain knowledge. The generative framework proposed achieves the aforementioned desiderata by employing a flexible prior on the complex plane for the latent, slow processes, and an intermediate layer of physics-motivated latent variables that reduces reliance on data and imbues inductive bias. In contrast to existing schemes, it does not require the a priori definition of projection operators from the fine-grained description and addresses simultaneously the tasks of dimensionality reduction and model estimation. We demonstrate its efficacy and accuracy in multiscale physical systems of particle dynamics where probabilistic, long-term predictions of phenomena not contained in the training data are produced.