Eulerian Methods for Visualizing Continuous Dynamical Systems using Lyapunov Exponents
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Bibliographic record
Abstract
We propose a new Eulerian numerical approach for constructing forward flow maps in continuous dynamical systems. The new algorithm improves the original formulation developed in [S. Leung, J. Comput. Phys., 230 (2011), pp. 3500--3524; S. Leung, Chaos, 23 (2013), 043132] so that the associated PDEs are solved forward in time and, therefore, the forward flow map can now be determined on the fly. Thanks to the simplicity of the implementations, we are now able to efficiently compute the unstable coherent structures in the flow based on quantities like the finite time Lyapunov exponent (FTLE), the finite size Lyapunov exponent (FSLE) and also a related infinitesimal size Lyapunov exponent (ISLE). When applied to the ISLE computations, the Eulerian method is particularly computationally efficient. For each separation factor $r$ in the definition of the ISLE, typical Lagrangian methods are required to shoot and monitor an individual set of ray trajectories. If the scale factor in the definition changes, these methods have to begin the computations all over again. The proposed Eulerian method, however, needs to extract only an isosurface of volumetric data for an individual value of $r$, which can be easily done using any well-developed efficient interpolation method or simply an isosurface extraction algorithm. Moreover, we provide a theoretical link between the FTLE and ISLE fields, which explains the similarity in these solutions observed in various applications.
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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.004 | 0.000 |
| Scholarly communication | 0.004 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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