Table 1.

The regression results of some standard PDE identified structure in Brunton’s PDE-FIND (Brunton et al. 2016b)

Partial differential equation(PDE)	Form of PDE	Approximate error (noise, no noise)
Korteweg–de Vries (KdV)	zt + 6zzx + zxxx = 0	1%, 7%
Burgers Equation	zt + zzx − ϵzxx = 0	0.15%, 0.8%
Schrodinger Equation	${\mathrm{iz}}_{\mathrm{t}}+\frac{1}{2}{\mathrm{z}}_{\mathrm{xx}}-\frac{{\mathrm{x}}^2}{2}\mathrm{z}=0$	0.25%, 10%
Nonlinear Schrödinger equation(NLS)	${\mathrm{iz}}_{\mathrm{t}}+\frac{1}{2}{\mathrm{z}}_{\mathrm{xx}}+\left|{\mathrm{z}}^2\right|\mathrm{z}=0$	0.05%, 3%
Kuramoto–Sivashinsky (KS) equation	zt + zzx + zxx + zxxxx = 0	1.3%, 52%
Reaction Diffusion Equation	zt = 0.1∇2z + λ(A)z − ω(A)w wt = 0.1∇2w + λ(A)w + ω(A)z A2 = z2 + w2, ω = − βA2, λ = 1 − A2	0.02%, 3.8%
Navier-Stokes Equation	${\mathrm{z}}_{\mathrm{t}}+\left(\mathrm{u}.\nabla \right)\mathrm{z}=\frac{1}{Re}{\nabla }^2\mathrm{z}$	1%, 7%