PDE Identification Using Noise Adaptive Differentiation in Strong Form (S-IDENT)

Abstract: We explore identifying partial differential equations (PDEs) from noisy observations of single time-space trajectories. Recent developments show the benefits of identifying PDEs in their weak forms. We investigate the use of differential Strong-form dictionaries for PDE IDENTification (S-IDENT), which enables finding more general linear and nonlinear PDEs. Building on an extensive exploration of integral-type denoised differentiation approaches, we propose to use Savitzky--Golay (SG) differentiation with an adaptive window length chosen based on Stein's Unbiased Risk Estimate (SURE). This offers a guaranteed order of accuracy while producing estimators with minimal variance. The identification process is further refined and stabilized through trimming and reduction-in-residual model selection. Numerical evidence shows that S-IDENT can successfully identify nonlinear PDEs at higher levels of noise than existing strong-form methods, while also yielding results comparable to weak-form approaches. We further verify the effectiveness of S-IDENT through comparisons with various strategies to approximate differential features. We provide numerical evidence that general differential-form dictionaries are larger and more ill-conditioned than those used for weak-form identification, yet S-IDENT does not significantly suffer from this combinatorial increase in dictionary size.