Title: Mastering Chaotic Dynamics via Second-Order Geometric Constraints

Abstract: Deriving chaotic dynamical systems from observational data demands more than just short-term predictive precision; the resulting model must faithfully replicate the geometry of the attractor and its invariant statistical properties. While traditional supervision techniques such as trajectory (zero-order) and Jacobian (first-order) matching effectively constrain the values and tangent structures of the vector field, they fail to restrict how the field deviates from its tangent plane. Consequently, a model may accurately match values and tangents at specific states while diverging in curvature, leading to local accuracy but long-term drift toward spurious attractors and distorted statistical measures.

This study demonstrates that enforcing second-order consistency resolves these issues, although computing the full Hessian is computationally prohibitive in high-dimensional spaces. To address this, we introduce model-constrained randomized Jacobian matching, a technique that evaluates the Jacobians of both the true and learned vector fields at randomly perturbed inputs. Through Taylor expansion analysis, we reveal that the expected randomized Jacobian loss breaks down into a nominal Jacobian discrepancy plus a Hessian discrepancy scaled by noise variance. This approach implicitly enforces second-order consistency with a computational cost of $\mathcal{O}(d^2)$, avoiding the need to construct the $\mathcal{O}(d^3)$ Hessian tensor.

Because the method relies solely on Jacobian evaluations, it remains scalable to high-dimensional systems where explicit Hessian matching is infeasible. Numerical simulations validate the robustness of second-order approaches. In the Lorenz~63 system, first-order methods suffer from catastrophic outliers in Lyapunov exponents under minimal temporal supervision—errors that second-order methods eradicate while correctly recovering the attractor. In coupled Lorenz~96 scenarios, an out-of-distribution forcing sweep distinguishes the methodologies: while all methods align up to a forcing level of $F=16$, only second-order techniques maintain the correct invariant measure and Lyapunov spectrum beyond $F=18$. Across both systems, randomized Jacobian matching achieves performance comparable to explicit Hessian matching but at a significantly reduced computational cost.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC