Sparse learning of partial differential equations with structured dictionary matrix

Abstract

This paper presents a "structured" learning approach for the identification of continuous partial differential equation (PDE) models with both constant and spatial-varying coefficients. The identification problem of parametric PDEs can be formulated as an ℓ 1 /ℓ 2 -mixed optimization problem by explicitly using block structures. Block-sparsity is used to ensure parsimonious representations of parametric spatiotemporal dynamics. An iterative reweighted ℓ 1 /ℓ 2 algorithm is proposed to solve the ℓ 1 /ℓ 2 -mixed optimization problem. In particular, the estimated values of varying coefficients are further used as data to identify functional forms of the coefficients. In addition, a new type of structured random dictionary matrix is constructed for the identification of constant-coefficient PDEs by introducing randomness into a bounded system of Legendre orthogonal polynomials. By exploring the restricted isometry properties of the structured random dictionary matrices, we derive a recovery condition that relates the number of samples to the sparsity and the probability of failure in the Lasso scheme. Numerical examples, such as the Schrödinger equation, the Fisher-Kolmogorov-Petrovsky-Piskunov equation, the Burger equation, and the Fisher equation, suggest that the proposed algorithm is fairly effective, especially when using a limited amount of measurements.