Abstract Although the governing equations of many systems, when derived from first principles, may be viewed as known, it is often too expensive to numerically simulate all the interactions they describe. Therefore, researchers often seek simpler descriptions that describe complex phenomena without numerically resolving all the interacting components. Stochastic differential equations (SDEs) arise naturally as models in this context. The growth in data acquisition, both through experiment and through simulations, provides an opportunity for the systematic derivation of SDE models in many disciplines. However, inconsistencies between SDEs and real data at short time scales often cause problems, when standard statistical methodology is applied to parameter estimation. The incompatibility between SDEs and real data can be addressed by deriving sufficient statistics from the time-series data and learning parameters of SDEs based on these. Here, we study sufficient statistics computed from time averages, an approach that we demonstrate to lead to sufficient statistics on a variety of problems and that has the secondary benefit of obviating the need to match trajectories. Following this approach, we formulate the fitting of SDEs to sufficient statistics from real data as an inverse problem and demonstrate that this inverse problem can be solved by using ensemble Kalman inversion. Furthermore, we create a framework for non-parametric learning of drift and diffusion terms by introducing hierarchical, refinable parameterizations of unknown functions, using Gaussian process regression. We demonstrate the proposed methodology for the fitting of SDE models, first in a simulation study with a noisy Lorenz ’63 model, and then in other applications, including dimension reduction in deterministic chaotic systems arising in the atmospheric sciences, large-scale pattern modeling in climate dynamics and simplified models for key observables arising in molecular dynamics. The results confirm that the proposed methodology provides a robust and systematic approach to fitting SDE models to real data.
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Stochastic Bridges of Linear Systems
We consider particles obeying Langevin dynamics while being at known positions and having known velocities at the two end-points of a given interval. Their motion in phase space can be modeled as an Ornstein–Uhlenbeck process conditioned at the two end-points—a generalization of the Brownian bridge. Using standard ideas from stochastic optimal control we construct a stochastic differential equation (SDE) that generates such a bridge that agrees with the statistics of the conditioned process, as a degenerate diffusion. Higher order linear diffusions are also considered. In general, a time-varying drift is sufficient to modify the prior SDE and meet the end-point conditions. When the drift is obtained by solving a suitable differential Lyapunov equation, the SDE models correctly the statistics of the bridge. These types of models are relevant in controlling and modeling distribution of particles and the interpolation of density functions.
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- NSF-PAR ID:
- 10114173
- Date Published:
- Journal Name:
- IEEE Transactions on Automatic Control
- Volume:
- 61
- Issue:
- 2
- ISSN:
- 0018-9286
- Page Range / eLocation ID:
- 526-530
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TAC.2015.2440567
