Robust and optimal sparse regression for nonlinear PDE models

Abstract

This paper investigates how models of spatiotemporal dynamics in the form of nonlinear partial differential equations can be identified directly from noisy data using a combination of sparse regression and weak formulation. Using the 4th-order Kuramoto-Sivashinsky equation for illustration, we show how this approach can be optimized in the limits of low and high noise, achieving accuracy that is orders of magnitude better than what existing techniques allow. In particular, we derive the scaling relation between the accuracy of the model, the parameters of the weak formulation, and the properties of the data, such as its spatial and temporal resolution and the level of noise.

Using noisy or incomplete data to discover models of spatiotemporal dynamics

“…In conclusion, let us point out that we have made no attempt to optimize our approach here. Several options are available to make it even more robust and accurate [19]. As an example, the size of the integration domains Ω k could be varied relative to the size of the spatiotemporal domain on which the data are available.…”

“…The data was then subsampled on a coarser grid with spacing ∆x = ∆y = 0.1 and ∆t = 0.2302, and Gaussian random noise with variance σ was added to both components of the flow velocity u. The integrals in (19) were evaluated over domains Ω k of size H x = 11.2, H y = 14.4, and H t ≈ 34.5.…”

Robust learning from noisy, incomplete, high-dimensional experimental data via physically constrained symbolic regression

“…In particular, there is no need to separate out the terms such as ∂ t u, which are only present in equations governing temporal evolution. In their absence, the linear system that appears in symbolic regression can be solved using alternative approaches such as singular value decomposition 17 .…”

“…2. Since integration leads to a reduction of noise due to averaging 17 , the domains are chosen to be large in both spatial directions. Their spatial width 2H x × 2H y was chosen to be slightly smaller than the size L x × L y of the flow domain to avoid the regions near the side walls where PIV is noisier than in the bulk.…”

“…This approach was originally introduced in the context of ordinary differential equations 15,16 . In the context of PDE models, it was shown to be as general as prior approaches based on the strong form 6,7 and superior in terms of both its flexibility and robustness 17,18 .…”