# Generative Ensemble Regression: Learning Particle Dynamics from Observations of Ensembles with Physics-Informed Deep Generative Models

arXiv: https://arxiv.org/abs/2008.01915 · HTML: https://mlsanity.com/paper/2008.01915/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Aug 2020.

## Built on by (most cited first, 6 of 6)

- [Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning](https://mlsanity.com/paper/2106.09004/) (2021)
- [Learning the temporal evolution of multivariate densities via normalizing flows](https://mlsanity.com/paper/2107.13735/) (2021)
- [Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas](https://mlsanity.com/paper/2108.12570/) (2021)
- [Learning Effective SDEs from Brownian Dynamics Simulations of Colloidal Particles](https://mlsanity.com/paper/2205.00286/) (2022)
- [Measure-conditional Discriminator with Stationary Optimum for GANs and Statistical Distance Surrogates](https://mlsanity.com/paper/2101.06802/) (2021)
- [Weak Collocation Regression for Inferring Stochastic Dynamics with L'evy Noise](https://mlsanity.com/paper/2403.08292/) (2024)

## Abstract (arXiv)

We propose a new method for inferring the governing stochastic ordinary differential equations (SODEs) by observing particle ensembles at discrete and sparse time instants, i.e., multiple "snapshots". Particle coordinates at a single time instant, possibly noisy or truncated, are recorded in each snapshot but are unpaired across the snapshots. By training a physics-informed generative model that generates "fake" sample paths, we aim to fit the observed particle ensemble distributions with a curve in the probability measure space, which is induced from the inferred particle dynamics. We employ different metrics to quantify the differences between distributions, e.g., the sliced Wasserstein distances and the adversarial losses in generative adversarial networks (GANs). We refer to this method as generative "ensemble-regression" (GER), in analogy to the classic "point-regression", where we infer the dynamics by performing regression in the Euclidean space. We illustrate the GER by learning the drift and diffusion terms of particle ensembles governed by SODEs with Brownian motions and Levy processes up to 100 dimensions. We also discuss how to treat cases with noisy or truncated observations. Apart from systems consisting of independent particles, we also tackle nonlocal interacting particle systems with unknown interaction potential parameters by constructing a physics-informed loss function. Finally, we investigate scenarios of paired observations and discuss how to reduce the dimensionality in such cases by proving a convergence theorem that provides theoretical support.