Group projected Subspace Pursuit for Identification of variable coefficient differential equations (GP-IDENT)

TL;DR

This paper introduces GP-IDENT, a robust algorithm for identifying space-time varying PDEs from noisy data using B-spline bases, group projected subspace pursuit, and a residual reduction criterion, outperforming existing methods.

Contribution

The paper presents a novel GPSP algorithm with a new model selection criterion for PDE identification with varying coefficients from noisy observations.

Findings

GP-IDENT accurately identifies PDE terms from noisy data.

The GPSP algorithm is more robust than existing methods in distinguishing correlated features.

Model selection via RR is effective under different noise levels.

Abstract

We propose an effective and robust algorithm for identifying partial differential equations (PDEs) with space-time varying coefficients from a single trajectory of noisy observations. Identifying unknown differential equations from noisy observations is a difficult task, and it is even more challenging with space and time varying coefficients in the PDE. The proposed algorithm, GP-IDENT, has three ingredients: (i) we use B-spline bases to express the unknown space and time varying coefficients, (ii) we propose Group Projected Subspace Pursuit (GPSP) to find a sequence of candidate PDEs with various levels of complexity, and (iii) we propose a new criterion for model selection using the Reduction in Residual (RR) to choose an optimal one among the pool of candidates. The new GPSP considers group projected subspaces which is more robust than existing methods in distinguishing correlated group features. We test GP-IDENT on a variety of PDEs and PDE systems, and compare it with the state-of-the-art parametric PDE identification algorithms under different settings to illustrate its outstanding performance. Our experiments show that GP-IDENT is effective in identifying the correct terms from a large dictionary and the model selection scheme is robust to noise.