Learning Chaotic Dynamics through Second-Order Geometric Supervision

2026-05-31

AI Summary

Learning chaotic dynamical systems by matching only trajectories and first-order derivatives often fails to preserve the geometric and statistical properties of attractors, leading to spurious dynamics. This work proposes a stochastic Jacobian matching method that implicitly enforces second-order geometric constraints at O(d²) computational complexity, without explicitly computing Hessian tensors. By comparing Jacobian matrices under random perturbations and leveraging Taylor expansion analysis, the approach captures essential curvature information of the underlying dynamics. Evaluated on the Lorenz-63 and coupled Lorenz-96 systems, the method significantly outperforms existing first-order approaches: it effectively eliminates anomalous Lyapunov exponents, accurately recovers both the attractor geometry and the invariant measure, and achieves high fidelity with low computational overhead.

Abstract

Learning chaotic dynamical systems from data requires more than short-term predictive accuracy: the learned model must preserve the attractor geometry and its invariant statistics. Trajectory (zero-order) and Jacobian (first-order) matching supervise the values and tangent structure of the vector field, but neither constrains how the field bends away from its tangent plane. A model can thus match values and tangents at the supervised states yet curve differently from the truth, remaining locally accurate while drifting toward spurious attractors and distorting long-time statistics. We show that enforcing second-order consistency mitigates these failures, but forming the full Hessian is prohibitive in high dimensions. We propose model-constrained randomized Jacobian matching, which compares the Jacobians of the true and learned vector fields at randomly perturbed inputs. A Taylor expansion shows that the expected randomized Jacobian loss decomposes into the nominal Jacobian mismatch plus a Hessian mismatch scaled by the noise variance, implicitly enforcing second-order consistency at $\mathcal{O}(d^2)$ cost without forming the $\mathcal{O}(d^3)$ Hessian tensor. Using only Jacobian evaluations, the method scales to high dimensions where explicit Hessian matching does not. Numerical experiments confirm that second-order methods are robust. For Lorenz~63, first-order methods produce catastrophic Lyapunov-exponent outliers under minimal temporal supervision, which second-order methods eliminate while recovering the correct attractor. For coupled Lorenz~96, an out-of-distribution forcing sweep separates the methods: all agree up to $F=16$, but beyond $F=18$ only second-order methods preserve the invariant measure and Lyapunov spectrum. On both systems, randomized Jacobian matching performs comparably to explicit Hessian matching at much lower cost.

Problem

• chaotic dynamics
• attractor geometry
• invariant statistics
• second-order consistency
• long-time prediction

Innovation

• second-order consistency
• randomized Jacobian matching
• chaotic dynamics
• Hessian approximation
• geometric supervision