Sparse learning of partial differential equations with structured dictionary matrix.

Area of Science:

• Applied Mathematics
• Scientific Computing
• Machine Learning

Background:

• Partial Differential Equations (PDEs) are fundamental in modeling complex systems.
• Identifying coefficients in PDEs, especially spatially varying ones, is a significant challenge.
• Existing methods may struggle with parsimonious representation of spatiotemporal dynamics.

Purpose of the Study:

• To develop a structured learning approach for identifying continuous PDE models.
• To handle both constant and spatially varying coefficients within PDE models.
• To ensure parsimonious representations of parametric spatiotemporal dynamics.

Main Methods:

• Formulating the PDE identification problem as an ℓ1/ℓ2-mixed optimization problem.
• Utilizing block-sparsity for parsimonious representations.
• Proposing an iterative reweighted ℓ1/ℓ2 algorithm to solve the optimization problem.
• Constructing structured random dictionary matrices for constant-coefficient PDEs.

Main Results:

• The proposed method effectively identifies PDE models, including those with spatial-varying coefficients.
• An iterative algorithm successfully solves the ℓ1/ℓ2-mixed optimization problem.
• Structured random dictionary matrices enable recovery conditions for Lasso schemes.
• Numerical examples demonstrate effectiveness, particularly with limited data.

Conclusions:

• The structured learning approach offers an effective strategy for PDE model identification.
• The method is robust and performs well even with sparse or limited measurements.
• This work advances the field of scientific machine learning for dynamical systems.