Mesh-free sparse identification of nonlinear dynamics

• URL: http://arxiv.org/abs/2505.16058v1
• Date: Wed, 21 May 2025 22:18:37 GMT
• Title: Mesh-free sparse identification of nonlinear dynamics
• Authors: Mars Liyao Gao, J. Nathan Kutz, Bernat Font,
• Abstract summary: We show that mesh-free SINDy is robust to high noise levels and limited data while remaining computationally efficient.<n>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.
• Score: 3.1484174280822845
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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