Solving the inverse Frobenius-Perron problem using stationary densities of dynamical systems with input perturbations

Nie, Xiaokai; Coca, Daniel; Birkin, Mark

doi:10.1016/j.cnsns.2020.105302

Cited by 8 publications

(6 citation statements)

References 28 publications

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“…The task of learning a dynamical system from an invariant measure has also been studied in the discrete-time setting under the inverse Frobenius-Perron problem [45,51,54,64]. The Frobenius-Perron operator, also known as the transfer operator, characterizes the time-evolution of an initial measure µ 0 according to some prespecified dynamical system.…”

Section: Prior Work On Learning Dynamics From Invariant Measuresmentioning

confidence: 99%

Learning Dynamical Systems From Invariant Measures

Botvinick-Greenhouse¹,

Martin²,

Yang³

2023

Preprint

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We extend the methodology in [65] to learn autonomous continuous-time dynamical systems from invariant measures. We assume that our data accurately describes the dynamics' asymptotic statistics but that the available time history of observations is insufficient for approximating the Lagrangian velocity. Therefore, invariant measures are treated as the inference data and velocity learning is reformulated as a data-fitting, PDE-constrained optimization problem in which the stationary distributional solution to the Fokker-Planck equation is used as a differentiable surrogate forward model. We consider velocity parameterizations based upon global polynomials, piecewise polynomials, and fully connected neural networks, as well as various objective functions to compare synthetic and reference invariant measures. We utilize the adjoint-state method together with the backpropagation technique to efficiently perform gradient-based parameter identification. Numerical results for the Van der Pol oscillator and Lorenz-63 system, together with real-world applications to Hall-effect thruster dynamics and temperature prediction, are presented to demonstrate the effectiveness of the proposed approach.

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Section: Prior Work On Learning Dynamics From Invariant Measuresmentioning

confidence: 99%

Learning Dynamical Systems From Invariant Measures

Botvinick-Greenhouse¹,

Martin²,

Yang³

2023

Preprint

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show abstract

“…. , T , are constructed to be PWC over Q and as specified in (23) and (24). The matrices Â and B, which possess as columns the order-zero vector representations of the density function estimates, are then constructed as specified in ( 25) and (26), respectively.…”

Section: B Solution Stepsmentioning

confidence: 99%

“…Whereas the solution presented here was extended in[16] to allow for the reconstruction of a subset of those continuous maps that possess a unique invariant density, in this paper we consider maps that need not be continuous. Solutions to more general formulations of IFPP-III (i.e., for stochastically perturbed systems and systems driven by external inputs) were also presented in[22]-[24]. A summary of these solutions may be found in[5].…”

mentioning

confidence: 99%

Reconstructing Semi-Markov Maps with Arbitrary Branch Monotonicity from Density Sequences

McDonald¹,

Wyk²

2022

Preprint

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This paper presents a novel solution to the inverse Frobenius-Perron problem of reconstructing an unknown one-dimensional ergodic map from causal sequences of density functions generated by the map. The original solution to this problem successfully reconstructs members of the canonical map class (i.e., a subset of the piecewise linear semi-Markov maps), provided all of the map's branches are monotonically increasing. The map is reconstructed from a matrix estimate of its Frobenius-Perron operator, which governs the evolution of density functions under the map. The proposed solution first constructs a higher-order matrix estimate of this operator. A member of the newly proposed class of generalized hat maps, a superset of the canonical maps, is then constructed from this matrix estimate. The proposed solution successfully distinguishes between increasing and decreasing branches and enlarges the class of maps that can be reconstructed to canonical maps with any subset of decreasing branches. When used to reconstruct any piecewise linear semi-Markov map, the proposed solution generates a map with consistent invariant density and power spectrum characteristics, regardless of the unknown map's canonicity or branch monotonicity. Numerical examples that illustrate these characteristics of the proposed solution are presented.

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“…, T , are constructed to be PWC over Q and as specified in (22) and (23). The matrices Â and B, which possess as columns the order-zero vector representations of the density function estimates, are then constructed as specified in (24) and (25), respectively.…”

Section: B Solution Stepsmentioning

confidence: 99%

“…Whereas the original solution was extended in[16] to allow for the reconstruction of a subset of those continuous maps that possess a unique invariant density, in this paper we consider maps that need not be continuous. Solutions to more general formulations of IFPP-III (i.e., for stochastically perturbed systems and systems driven by external inputs) were also presented in[22]-[24]. A summary of these solutions may be found in[5].…”

mentioning

confidence: 99%

Reconstructing Semi-Markov Maps with Arbitrary Branch Monotonicity from Density Sequences

McDonald¹,

Wyk²

2022

Preprint

View full text Add to dashboard Cite

This paper presents a novel solution to the inverse Frobenius-Perron problem of reconstructing an unknown nonlinear and ergodic map from causal sequences of probability density functions generated by the map. The original solution to this problem successfully reconstructs members of the canonical map class (i.e., a subset of the piecewise linear semi-Markov maps), provided all the map's branches are monotonically increasing. The original solution constructs a matrix estimate of the map's Frobenius-Perron operator, which governs the evolution of density functions under iteration of the map, from the density sequences. The one-dimensional map is reconstructed from this matrix. In contrast, the proposed solution constructs a higher-order matrix estimate of the Frobenius-Perron operator. A member of the newly proposed class of generalized hat maps, a superset of the canonical maps, is constructed from this matrix estimate. The proposed solution successfully distinguishes between increasing and decreasing map branches and enlarges the class of maps that can be successfully reconstructed to canonical maps with any subset of decreasing branches. When used to reconstruct any piecewise linear semi-Markov map, the proposed solution generates a map with consistent invariant density and power spectrum mode characteristics, regardless of the unknown map's canonicity or branch monotonicity. Numerical examples that illustrate the proposed solution's characteristics are presented.

show abstract

Solving the inverse Frobenius-Perron problem using stationary densities of dynamical systems with input perturbations

Cited by 8 publications

References 28 publications

Learning Dynamical Systems From Invariant Measures

Learning Dynamical Systems From Invariant Measures

Reconstructing Semi-Markov Maps with Arbitrary Branch Monotonicity from Density Sequences

Reconstructing Semi-Markov Maps with Arbitrary Branch Monotonicity from Density Sequences

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