Using noisy or incomplete data to discover models of spatiotemporal dynamics.

Area of Science:

• Applied Mathematics
• Data Science
• Computational Physics

Background:

• Sparse regression is a powerful tool for uncovering dynamic system models from data.
• Traditional sparse regression methods struggle with noisy data, particularly when high-order derivatives are needed for nonlinear partial differential equations (PDEs).
• Accurate derivative estimation is crucial but often infeasible in the presence of noise.

Purpose of the Study:

• To develop a robust sparse regression approach for identifying complex dynamical systems from noisy data.
• To overcome the limitations of existing methods in handling high-order derivatives and latent variables in PDE discovery.
• To demonstrate the efficacy of the proposed method on challenging benchmark problems and real-world fluid dynamics.

Main Methods:

• A novel weak formulation of the sparse regression problem is proposed.
• This approach avoids direct computation of high-order derivatives, mitigating noise sensitivity.
• The method is applied to reconstruct nonlinear PDEs, including the Kuramoto-Sivashinsky equation, and models with unobserved latent variables.

Main Results:

• Accurate reconstruction of PDEs with high-order derivatives from significantly noisy data.
• Successful identification of PDE models involving latent variables, such as those in weakly turbulent fluid flow.
• Demonstrated robustness and generality of the weak formulation approach.

Conclusions:

• The weak formulation offers a general and effective solution for sparse regression in the presence of noisy data.
• This method significantly enhances the ability to discover complex dynamical systems, including those with unmeasured components.
• The approach holds promise for advancing scientific discovery in fields relying on data-driven modeling of complex dynamics.