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This article introduces an isometric manifold embedding data-driven paradigm designed to enable model-free simulations with noisy data sampled from a constitutive manifold. The proposed data-driven approach iterates between a global optimization problem that seeks admissible solutions for the balance principle and a local optimization problem that finds the closest point projection of the Euclidean space that isometrically embeds a nonlinear constitutive manifold. To de-noise the database, a geometric autoencoder is introduced such that the encoder first learns to create an approximated embedding that maps the underlying low-dimensional structure of the high-dimensional constitutive manifold onto a flattened manifold with less curvature. We then obtain the noise-free constitutive responses by projecting data onto a denoised latent space that is completely flat by assuming that the noise and the underlying constitutive signal are orthogonal to each other. Consequently, a projection from the conservative manifold onto this de-noised constitutive latent space enables us to complete the local optimization step of the data-driven paradigm. Finally, to decode the data expressed in the latent space without reintroducing noise, we impose a set of isometry constraints while training the autoencoder such that the nonlinear mapping from the latent space to the reconstructed constituent manifold is distance-preserving. Numerical examples are used to both validate the implementation and demonstrate the accuracy, robustness, and limitations of the proposed paradigm.
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Empirical mode modeling: A data-driven approach to recover and forecast nonlinear dynamics from noisy data
Abstract Data-driven, model-free analytics are natural choices for discovery and forecasting of complex, nonlinear systems. Methods that operate in the system state-space require either an explicit multidimensional state-space, or, one approximated from available observations. Since observational data are frequently sampled with noise, it is possible that noise can corrupt the state-space representation degrading analytical performance. Here, we evaluate the synthesis of empirical mode decomposition with empirical dynamic modeling, which we term empirical mode modeling, to increase the information content of state-space representations in the presence of noise. Evaluation of a mathematical, and, an ecologically important geophysical application across three different state-space representations suggests that empirical mode modeling may be a useful technique for data-driven, model-free, state-space analysis in the presence of noise.
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- PAR ID:
- 10331730
- Date Published:
- Journal Name:
- Nonlinear Dynamics
- Volume:
- 108
- Issue:
- 3
- ISSN:
- 0924-090X
- Page Range / eLocation ID:
- 2147 to 2160
- 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/10.1007/s11071-022-07311-y
