Mesh-free sparse identification of nonlinear dynamics

TL;DR

The paper introduces mesh-free SINDy, a PDE discovery framework that identifies governing equations from nonuniform, scarce data by using a neural-network surrogate and automatic differentiation to build a library of candidate terms, followed by sparse regression via ensemble SINDy. The training loop is decoupled from the regression, reducing hyperparameter tuning and computational cost, while bootstrapped ensembles provide uncertainty quantification. The method demonstrates robust discovery across Burgers', heat, KdV, and 2D advection-diffusion equations, including challenging high-noise, low-data scenarios, with training times under a minute. This mesh-free approach broadens PDE discovery to unstructured sensor data and offers practical impact for scientific modeling and engineering, delivering both accuracy and principled uncertainty estimates.

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.

Figures (13)

• Figure 1: Illustration of the architecture of mesh-free SINDy. From $N$ sensor measurements at random space-time points $x_i^{(n)}$, the neural network approximates the system state $u_j^{(n)}(x_i^{(n)})$. As shown in the 1-D example, the auto-differentiation framework produces $\partial_t u$ (also noted $u_t$), and a library of $K$ candidate terms composed of spatial or mixed derivatives which will be used in the sparse regression.
• Figure 2: Uncertainty quantification and epoch-wise evaluation of the discovered SINDy model for the Burgers' equation.
• Figure 3: Uncertainty quantification and epoch-wise evaluation of the discovered SINDy model for the heat equation.
• Figure 4: Uncertainty quantification and epoch-wise evaluation of the discovered SINDy model for the KdV equation.
• Figure 5: Uncertainty quantification and epoch-wise evaluation of the discovered SINDy model for the 2D advection-difusion equation.