# Physics-aware, deep probabilistic modeling of multiscale dynamics in the Small Data regime

arXiv: https://arxiv.org/abs/2102.04269 · HTML: https://mlsanity.com/paper/2102.04269/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 24 references that are in the map.
- First posted on arXiv 8 Feb 2021.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observables](https://mlsanity.com/paper/2006.01789/) (2020)
- [Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systems](https://mlsanity.com/paper/1912.12976/) (2019)
- [Physics-aware, probabilistic model order reduction with guaranteed stability](https://mlsanity.com/paper/2101.05834/) (2021)
- [Enforcing constraints for interpolation and extrapolation in Generative Adversarial Networks](https://mlsanity.com/paper/1803.08182/) (2018)
- [Predictive Coarse-Graining](https://mlsanity.com/paper/1605.08301/) (2016)
- [Physics-constrained, data-driven discovery of coarse-grained dynamics](https://mlsanity.com/paper/1802.03824/) (2018)

## Built on by (most cited first, 1 of 1)

- [Predicting and explaining nonlinear material response using deep Physically Guided Neural Networks with Internal Variables](https://mlsanity.com/paper/2308.03915/) (2023)

## Abstract (arXiv)

The data-based discovery of effective, coarse-grained (CG) models of high-dimensional dynamical systems presents a unique challenge in computational physics and particularly in the context of multiscale problems. The present paper offers a probabilistic perspective that simultaneously identifies predictive, lower-dimensional coarse-grained (CG) variables as well as their dynamics. We make use of the expressive ability of deep neural networks in order to represent the right-hand side of the CG evolution law. Furthermore, we demonstrate how domain knowledge that is very often available in the form of physical constraints (e.g. conservation laws) can be incorporated with the novel concept of virtual observables. Such constraints, apart from leading to physically realistic predictions, can significantly reduce the requisite amount of training data which enables reducing the amount of required, computationally expensive multiscale simulations (Small Data regime). The proposed state-space model is trained using probabilistic inference tools and, in contrast to several other techniques, does not require the prescription of a fine-to-coarse (restriction) projection nor time-derivatives of the state variables. The formulation adopted is capable of quantifying the predictive uncertainty as well as of reconstructing the evolution of the full, fine-scale system which allows to select the quantities of interest a posteriori. We demonstrate the efficacy of the proposed framework in a high-dimensional system of moving particles.