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Fast differentiation of hyperbolic chaos

Angxiu Ni

TL;DR

This work tackles the challenge of computing the linear response of high-dimensional hyperbolic chaotic systems by deriving a fast, pointwise estimator for the unstable divergence that is suitable for Monte-Carlo evaluation. It introduces a blended approach that splits the derivative of the SRB measure into a shadowing contribution, computed via nonintrusive shadowing, and an unstable contribution expressed through a renormalized second-order tangent framework with a dominant term $p$, evolving under a single propagation rule. The resulting fast response algorithm achieves cost $O(Mu)$ per step and is demonstrated on a 21-dimensional solenoid map, where it attains enormous speedups over path-perturbation methods while remaining robust in high dimensions. This advances gradient-based optimization, uncertainty quantification, and sensitivity analysis for complex chaotic systems by enabling scalable linear-response computations. The work also provides a detailed geometric and algebraic toolkit for second-order tangents, volume ratios, and projection operators that underpin the estimator and algorithm.

Abstract

We derive and prove the `fast response' formula for the linear response, the parameter derivatives of long-time-averaged statistics, of hyperbolic deterministic chaotic systems. The expression is pointwisely defined so we can compute the linear response in high-dimensions via Monte-Carlo-type algorithms. It has two parts, where the shadowing contribution is computed by the nonintrusive shadowing algorithm. The unstable contribution is expressed by renormalized second-order tangent equations; importantly, it does not contain any distributional derivatives. The algorithm's cost is solving $u$, the unstable dimension, many first-order and second-order tangent equations along a long orbit; the main error is the sampling error of the orbit. We numerically demonstrate the algorithm on a 21-dimensional example, which is difficult for previous methods.

Fast differentiation of hyperbolic chaos

TL;DR

This work tackles the challenge of computing the linear response of high-dimensional hyperbolic chaotic systems by deriving a fast, pointwise estimator for the unstable divergence that is suitable for Monte-Carlo evaluation. It introduces a blended approach that splits the derivative of the SRB measure into a shadowing contribution, computed via nonintrusive shadowing, and an unstable contribution expressed through a renormalized second-order tangent framework with a dominant term , evolving under a single propagation rule. The resulting fast response algorithm achieves cost per step and is demonstrated on a 21-dimensional solenoid map, where it attains enormous speedups over path-perturbation methods while remaining robust in high dimensions. This advances gradient-based optimization, uncertainty quantification, and sensitivity analysis for complex chaotic systems by enabling scalable linear-response computations. The work also provides a detailed geometric and algebraic toolkit for second-order tangents, volume ratios, and projection operators that underpin the estimator and algorithm.

Abstract

We derive and prove the `fast response' formula for the linear response, the parameter derivatives of long-time-averaged statistics, of hyperbolic deterministic chaotic systems. The expression is pointwisely defined so we can compute the linear response in high-dimensions via Monte-Carlo-type algorithms. It has two parts, where the shadowing contribution is computed by the nonintrusive shadowing algorithm. The unstable contribution is expressed by renormalized second-order tangent equations; importantly, it does not contain any distributional derivatives. The algorithm's cost is solving , the unstable dimension, many first-order and second-order tangent equations along a long orbit; the main error is the sampling error of the orbit. We numerically demonstrate the algorithm on a 21-dimensional example, which is difficult for previous methods.

Paper Structure

This paper contains 30 sections, 23 theorems, 148 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main results of the paper
  3. Literature review
  4. Review of literature after this paper
  5. Preparations
  6. Hyperbolicity and notations
  7. Some previous linear response formulas
  8. Nonintrusive shadowing algorithm
  9. Removing distributional derivatives and Expanding the unstable divergence
  10. Integration by parts and measure change
  11. Transforming to the derivative of volume ratio
  12. Definitions of pi and related quantities
  13. Intuitive proof
  14. Rigorous proof
  15. Expanding the volume ratio
  16. ...and 15 more sections

Key Result

Theorem 1

Denote the SRB measure of $f$ on $K$ by $\rho$, assume that $f$ is parameterized by some scalar $\gamma$, and define Then the derivative of the SRB measure, for a fixed objective function $\Phi$, is given by: where $X:=\delta f\circ f^{-1}$, $X(\cdot)$ is to differentiate in the direction of $X$, and $\Phi_n = \Phi\circ f^n$.

Figures (7)

  • Figure 1: Definitions of projections.
  • Figure 2: Subscript convention for multiple segments.
  • Figure 3: The empirical measure of a orbit with default setting.
  • Figure 4: Effects of $A$. Left: derivatives from 30 independent computations for each $A$. Right: the sample standard deviation of the computed derivatives, where the dashed line is $A^{-0.5}$.
  • Figure 5: Effects of $W$. Left: derivatives computed by different $W$'s. Right: standard deviation of derivatives, where the dashed line is $0.005W^{0.5}$.
  • ...and 2 more figures

Theorems & Definitions (63)

  • Theorem 1: path-perturbation formula for the linear response
  • Remark
  • Lemma 1: expression for measure change
  • Remark
  • proof
  • Lemma 2: expression of $\nabla_{X^s} \eta_* e$
  • Remark
  • proof
  • Lemma 3
  • Remark
  • ...and 53 more