Mesh-free sparse identification of nonlinear dynamics
This addresses the challenge of scientific modeling for researchers and engineers by enabling equation identification from messy data, though it is incremental as it builds on existing SINDy methods.
The paper tackles the problem of identifying governing equations from dynamical systems using arbitrary sensor placements and non-uniform data, proposing mesh-free SINDy, which achieves robust PDE discovery with up to 75% noise and as few as 100 samples, with training under one minute.
Identifying the governing equations of a dynamical system is one of the most important tasks for scientific modeling. However, this procedure often requires high-quality spatio-temporal data uniformly sampled on structured grids. In this paper, we propose mesh-free SINDy, a novel algorithm which leverages the power of neural network approximation as well as auto-differentiation to identify governing equations from arbitrary sensor placements and non-uniform temporal data sampling. We show that mesh-free SINDy is robust to high noise levels and limited data while remaining computationally efficient. In our implementation, the training procedure is straight-forward and nearly free of hyperparameter tuning, making mesh-free SINDy widely applicable to many scientific and engineering problems. In the experiments, we demonstrate its effectiveness on a series of PDEs including the Burgers' equation, the heat equation, the Korteweg-De Vries equation and the 2D advection-diffusion equation. We conduct detailed numerical experiments on all datasets, varying the noise levels and number of samples, and we also compare our approach to previous state-of-the-art methods. It is noteworthy that, even in high-noise and low-data scenarios, mesh-free SINDy demonstrates robust PDE discovery, achieving successful identification with up to 75% noise for the Burgers' equation using 5,000 samples and with as few as 100 samples and 1% noise. All of this is achieved within a training time of under one minute.