Second-order consistency for learning chaotic dynamics via randomized Jacobian matching

To address the issue that short-term accuracy cannot guarantee correct long-term dynamics of chaotic systems, researchers proposed a model constraint method based on stochastic Jacobi matching. This method compares the Jacobi matrices of the real and learned vector fields under random perturbed inputs, and uses Taylor expansion to decompose the expected loss into the Jacobi mismatch term and the Hessian mismatch term scaled by noise variance. Thus, without explicitly constructing the computationally intensive $O(d^3)$ Hessian tensor, second-order consistency supervision is implicitly achieved with an $O(d^2)$ memory cost. Experiments show that in the Lorenz 63 system, this method can significantly reduce the invariant measure error and the mean square error of Lyapunov spectrum; in the coupled Lorenz 96 system, it remains accurate in maintaining the edge distribution even when the forcing force increases, whereas traditional first-order methods enter false high-amplitude regions.