Data-driven discovery of coordinates and governing equations

By Kathleen P. Champion, Bethany Lusch, J. Nathan Kutz, Steven L. Brunton. Proceedings of the National Academy of Sciences of the United States of America, 2019

1,027 citations recorded by the source corpus.

Significance Governing equations are essential to the study of physical systems, providing models that can generalize to predict previously unseen behaviors. There are many systems of interest across disciplines where large quantities of data have been collected, but the underlying governing equations remain unknown. This work introduces an approach to discover governing models from data. The proposed method addresses a key limitation of prior approaches by simultaneously discovering coordinates that admit a parsimonious dynamical model. Developing parsimonious and interpretable governing models has the potential to transform our understanding of complex systems, including in neuroscience, biology, and climate science. The discovery of governing equations from scientific data has the potential to transform data-rich fields that lack well-characterized quantitative descriptions. Advances in sparse regression are currently enabling the tractable identification of both the structure and parameters of a nonlinear dynamical system from data. The resulting models have the fewest terms necessary to describe the dynamics, balancing model complexity with descriptive ability, and thus promoting interpretability and generalizability. This provides an algorithmic approach to Occam’s razor for model discovery. However, this approach fundamentally relies on an effective coordinate system in which the dynamics have a simple representation. In this work, we design a custom deep autoencoder network to discover a coordinate transformation into a reduced space where the dynamics may be sparsely represented. Thus, we simultaneously learn the governing equations and the associated coordinate system. We demonstrate this approach on several example high-dimensional systems with low-dimensional behavior. The resulting modeling framework combines the strengths of deep neural networks for flexible representation and sparse identification of nonlinear dynamics (SINDy) for parsimonious models. This method places the discovery of coordinates and models on an equal footing.

Subfields: Artificial Intelligence, Molecular Biology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3c9961153493370500020c81527b3548c96f81e0

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Citations in the corpus

Interpretable scientific discovery with symbolic regression: a review (2022, methodology): “SINDY-AE [25] Data-driven discovery of coordinates and governing equations 2019 Linear url”
Solving Irregular and Data-enriched Differential Equations using Deep Neural Networks (2019, background): “These types of approaches are emerging with increasing interest [54, 55, 56, 57, 58], and it is easy to see how such approaches naturally lend themselves to the DNN frameworks.”
Enhancing computational fluid dynamics with machine learning (2021, background): “The modeling approaches above may be combined with a deep autoencoder to uncover a low-dimensional latent space, as in the SINDy-autoencoder [125].”
Stiff-PINN: Physics-Informed Neural Network for Stiff Chemical Kinetics (2020, methodology): “[6] K.”
Deep learning of physical laws from scarce data (2020, methodology): “To generate the solution, the problem domain is meshed into 201 spatial grid points for x ∈ [ 5 , 5] and 101 time steps for t ∈ [0 , 10].”