Scale dependence in WSINDy from localized test functions

Characterize how localization of compactly supported test functions in the Weak form Sparse Identification of Nonlinear Dynamics (WSINDy) algorithm controls the spatial and temporal scales represented in the weak-form data and influences the discovered partial differential equations, including the dependence of identified models on the test function support parameters that define supp(ψ) and their role in investigating specific length and time scales in geophysical weather modeling.

Background

WSINDy learns governing partial differential equations from spatiotemporal data by integrating the data against localized, compactly supported test functions, which implicitly impose length and time scales via the support radii. The paper emphasizes that choosing the test function support hyperparameters enables coarse-graining and scale selection in the learned models, as illustrated earlier when increasing support selectively captures larger-scale features.

In the conclusion, the authors explicitly note that the aspect of scale selection via test function localization is not fully understood. They point to this as a promising direction for future research with potential implications for unified weather-to-climate modeling, highlighting the need for a principled understanding of how this localization affects model discovery and interpretation.

References

In particular, an interesting (and not fully-understood) aspect of using a weak form representation of the dynamics is the ability to investigate a particular range of length and time scales by appropriately localizing the corresponding test functions. We suspect that this is a fruitful direction for future research, potentially bearing upon questions of scale dependence in the setting of data-driven modeling.

— Learning Weather Models from Data with WSINDy (2501.00738 - Minor et al., 1 Jan 2025) in Section Conclusion