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Adjoint Sensitivities of Chaotic Flows without Adjoint Solvers: A Data-Driven Approach

Defne E. Ozan, Luca Magri

TL;DR

This work tackles the challenge of computing adjoint sensitivities for chaotic flows without implementing code-specific adjoint solvers. It introduces a parameter-aware Echo State Network (ESN) whose dynamics support a derived adjoint that estimates $\dfrac{d\mathcal{J}}{d\bm{p}}$ for long-time objectives. Demonstrated on the Lorenz63 system, the data-driven adjoint sensitivities closely match those obtained from the true system's ensemble adjoint, reproducing long-time averages like $\bar{z}$ across unseen regimes; a known bias relative to direct polynomial fits is observed. This approach enables solver-agnostic sensitivity analysis for chaotic systems and highlights future work on scaling to higher-dimensional models.

Abstract

In one calculation, adjoint sensitivity analysis provides the gradient of a quantity of interest with respect to all system's parameters. Conventionally, adjoint solvers need to be implemented by differentiating computational models, which can be a cumbersome task and is code-specific. To propose an adjoint solver that is not code-specific, we develop a data-driven strategy. We demonstrate its application on the computation of gradients of long-time averages of chaotic flows. First, we deploy a parameter-aware echo state network (ESN) to accurately forecast and simulate the dynamics of a dynamical system for a range of system's parameters. Second, we derive the adjoint of the parameter-aware ESN. Finally, we combine the parameter-aware ESN with its adjoint version to compute the sensitivities to the system parameters. We showcase the method on a prototypical chaotic system. Because adjoint sensitivities in chaotic regimes diverge for long integration times, we analyse the application of ensemble adjoint method to the ESN. We find that the adjoint sensitivities obtained from the ESN match closely with the original system. This work opens possibilities for sensitivity analysis without code-specific adjoint solvers.

Adjoint Sensitivities of Chaotic Flows without Adjoint Solvers: A Data-Driven Approach

TL;DR

This work tackles the challenge of computing adjoint sensitivities for chaotic flows without implementing code-specific adjoint solvers. It introduces a parameter-aware Echo State Network (ESN) whose dynamics support a derived adjoint that estimates for long-time objectives. Demonstrated on the Lorenz63 system, the data-driven adjoint sensitivities closely match those obtained from the true system's ensemble adjoint, reproducing long-time averages like across unseen regimes; a known bias relative to direct polynomial fits is observed. This approach enables solver-agnostic sensitivity analysis for chaotic systems and highlights future work on scaling to higher-dimensional models.

Abstract

In one calculation, adjoint sensitivity analysis provides the gradient of a quantity of interest with respect to all system's parameters. Conventionally, adjoint solvers need to be implemented by differentiating computational models, which can be a cumbersome task and is code-specific. To propose an adjoint solver that is not code-specific, we develop a data-driven strategy. We demonstrate its application on the computation of gradients of long-time averages of chaotic flows. First, we deploy a parameter-aware echo state network (ESN) to accurately forecast and simulate the dynamics of a dynamical system for a range of system's parameters. Second, we derive the adjoint of the parameter-aware ESN. Finally, we combine the parameter-aware ESN with its adjoint version to compute the sensitivities to the system parameters. We showcase the method on a prototypical chaotic system. Because adjoint sensitivities in chaotic regimes diverge for long integration times, we analyse the application of ensemble adjoint method to the ESN. We find that the adjoint sensitivities obtained from the ESN match closely with the original system. This work opens possibilities for sensitivity analysis without code-specific adjoint solvers.
Paper Structure (5 sections, 10 equations, 2 figures)

This paper contains 5 sections, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Parameter-aware Echo State Network
  3. Adjoint of Echo State Network
  4. Computation of data-driven chaotic sensitivities
  5. Conclusions and future directions

Figures (2)

  • Figure 1: Short-term prediction (a,c) and long-term inference of the statistics (b,d) of $z$ for two regimes with different Lyapunov times (LTs); (a,b) $(s = 10, r = 28, b = 8/3 \approx 2.667)$, and (c,d) $(s = 13, r = 52, b = 1.75)$. The statistics are calculated over 5000 LTs, after a washout stage where we repeatedly feed the same initial condition, and a transient time that is discarded. The parameter-aware echo state network can successfully infer the dynamics and long-term statistics of different chaotic regimes even when they were not seen during training.
  • Figure 2: Inference of sensitivities of $\bar{z}$ to parameters, $s$, $r$, and $b$. Top row (a,b,c) shows the change of $\bar{z}$ for varying $s$, $r$, and $b$, for the true system, echo state network (ESN), and a polynomial fit on the true system's values. Botton row (d,e,f) shows the derivative of $\bar{z}$ with respect to the respective parameter obtained by the ensemble adjoint method using the true system's adjoint, ESN's adjoint, and by differentiating the polynomial fit.