  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.
