Leaning Sparse Nonlinear Dynamical Systems via Frank-Wolfe Algorithm with Integral Strategy

Sparse identification of nonlinear dynamical systems is an important project, directly addressing the physics community’s long-standing goal of data-driven discovery. Although many effective methods have been developed to enhance the ability for data-driven discovery of governing equations, critical challenges, including derivative computation and optimization methods, demanding more robust theoretical foundations and innovative computational strategies. This paper proposes a novel sparse optimization strategy for learning nonlinear dynamics from noisy data. The strategy is based on the Frank-Wolfe algorithm (also known as the conditional gradient method), an iterative first-order optimization technique for constrained convex optimization. Unlike traditional methods that rely on projection, the algorithm minimizes a linear approximation of the objective function over the feasible set to determine the search direction, naturally yielding sparse solutions, which is consistent with prior work in sparse identification of nonlinear dynamics (SINDy). However, our approach differs fundamentally: while SINDy enforces sparsity through user-defined thresholding, the Frank-Wolfe algorithm inherently produces sparse solutions by minimizing a convex objective over a compact set. This distinction endows our method with greater scalability to large-scale dynamics and noisy data. To further mitigate instabilities in numerical differentiation, particularly under noisy conditions, we incorporate an integral form to replace derivative estimation. The proposed framework thus offers a robust, scalable, and interpretable alternative for data-driven modeling of complex systems. Numerical experiments demonstrate the effectiveness and superiority of our proposed method, which combines subsampling and data normalization, in identifying nonlinear dynamical systems.