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The understanding of chaotic systems is challenging not only for theoretical research but also for many important applications. Chaotic behavior is found in many nonlinear dynamical systems, such as those found in climate dynamics, weather, the stock market, and the space-time dynamics of virus spread. A reliable solution for these systems must handle their complex space-time dynamics and sensitive dependence on initial conditions. We develop a deep learning framework to push the time horizon at which reliable predictions can be made further into the future by better evaluating the consequences of local errors when modeling nonlinear systems. Our approach observes the future trajectories of initial errors at a time horizon to model the evolution of the loss to that point with two major components: 1) a recurrent architecture, Error Trajectory Tracing, that is designed to trace the trajectories of predictive errors through phase space, and 2) a training regime, Horizon Forcing, that pushes the model’s focus out to a predetermined time horizon. We validate our method on classic chaotic systems and real-world time series prediction tasks with chaotic characteristics, and show that our approach outperforms the current state-of-the-art methods.
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The effect of time resolution on the observed first passage times in diffusive dynamics
Single-molecule and single-particle tracking experiments are typically unable to resolve fine details of thermal motion at short timescales where trajectories are continuous. We show that, when a diffusive trajectory [Formula: see text] is sampled at finite time intervals δt, the resulting error in measuring the first passage time to a given domain can exceed the time resolution of the measurement by more than an order of magnitude. Such surprisingly large errors originate from the fact that the trajectory may enter and exit the domain while being unobserved, thereby lengthening the apparent first passage time by an amount that is larger than δt. Such systematic errors are particularly important in single-molecule studies of barrier crossing dynamics. We show that the correct first passage times, as well as other properties of the trajectories such as splitting probabilities, can be recovered via a stochastic algorithm that reintroduces unobserved first passage events probabilistically.
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- Award ID(s):
- 1955552
- PAR ID:
- 10412337
- Date Published:
- Journal Name:
- The Journal of Chemical Physics
- Volume:
- 158
- Issue:
- 11
- ISSN:
- 0021-9606
- Page Range / eLocation ID:
- 111101
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0142166
