Data-driven discovery of spatiotemporal dynamical systems with sparse interpretable neural networks

Existing approaches to data-driven model discovery of nonlinear dynamical systems are mainly sparse optimization, symbolic regression, and Kolmogorov-Arnold networks, but they all suffer from the “curse of dimensionality”, i.e., the number of candidate functions grows exponentially with the dimension. Spatiotemporal dynamical systems, when represented by coupled ordinary differential equations, often involve hundreds or even thousands dimensions. Discovering the high-dimensional velocity field using large datasets presents a formidable challenge. We develop a machine-learning framework that integrates an interpretable neural network incorporating the matrix formulation of sparse regression with a specially designed sparsity promoting pruning scheme. Utilizing five paradigmatic spatiotemporal dynamical systems, we demonstrate that our framework is capable of accurately finding the velocity field of more than 100 dimensions and extrapolating to generate the correct coherent structures from untrained data. We further validate the effectiveness of our framework on an empirical dataset from a triple pendulum experiment. Our framework can potentially be scaled up to systems with thousands of dimensions, rendering data-driven model discovery of large complex systems feasible.