Generative Ensemble Regression: Learning Particle Dynamics from Observations of Ensembles with Physics-informed Deep Generative Models
Name
21m1413018.pdf
Description
Published version
Size
2.64 MB
Format
Adobe PDF
Checksum (MD5)
66309f48a09889baabbb099700ccd0a6
Author(s)
Yang, Liu
Daskalakis, Constantinos
Karniadakis, George E
Date Issued
February 2022
Journal
SIAM Journal on Scientific Computing
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Version
Final published version
Abstract
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. 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 Lévy 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.
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DOI of Published Version
https://doi.org/10.1137/21M1413018
