Predicting Forced Responses of Probability Distributions via the Fluctuation-Dissipation Theorem and Generative Modeling
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| Format: | Preprint |
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2025
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| _version_ | 1866918131058868224 |
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| author | Giorgini, Ludovico T. Falasca, Fabrizio Souza, Andre N. |
| author_facet | Giorgini, Ludovico T. Falasca, Fabrizio Souza, Andre N. |
| contents | We present a novel and flexible data-driven framework for estimating the response of higher-order moments of nonlinear stochastic systems to small external perturbations. The classical Generalized Fluctuation--Dissipation Theorem (GFDT) links the unperturbed steady-state distribution to the system's linear response. While standard implementations relying on Gaussian approximations can predict the mean response, they often fail to capture changes in higher-order moments. To overcome this, we combine GFDT with score-based generative modeling to estimate the system's score function directly from data. We demonstrate the framework's versatility by employing two complementary score estimation techniques tailored to the system's characteristics: (i) a clustering-based algorithm (KGMM) for systems with low-dimensional effective dynamics, and (ii) a denoising score matching method implemented with a U-Net architecture for high-dimensional, spatially-extended systems where reduced-order modeling is not feasible. Our method is validated on several stochastic models relevant to climate dynamics: three reduced-order models of increasing complexity and a 2D Navier--Stokes model representing a turbulent flow with a localized perturbation. In all cases, the approach accurately captures strongly nonlinear and non-Gaussian features of the system's response, significantly outperforming traditional Gaussian approximations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_13333 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Predicting Forced Responses of Probability Distributions via the Fluctuation-Dissipation Theorem and Generative Modeling Giorgini, Ludovico T. Falasca, Fabrizio Souza, Andre N. Machine Learning Chaotic Dynamics We present a novel and flexible data-driven framework for estimating the response of higher-order moments of nonlinear stochastic systems to small external perturbations. The classical Generalized Fluctuation--Dissipation Theorem (GFDT) links the unperturbed steady-state distribution to the system's linear response. While standard implementations relying on Gaussian approximations can predict the mean response, they often fail to capture changes in higher-order moments. To overcome this, we combine GFDT with score-based generative modeling to estimate the system's score function directly from data. We demonstrate the framework's versatility by employing two complementary score estimation techniques tailored to the system's characteristics: (i) a clustering-based algorithm (KGMM) for systems with low-dimensional effective dynamics, and (ii) a denoising score matching method implemented with a U-Net architecture for high-dimensional, spatially-extended systems where reduced-order modeling is not feasible. Our method is validated on several stochastic models relevant to climate dynamics: three reduced-order models of increasing complexity and a 2D Navier--Stokes model representing a turbulent flow with a localized perturbation. In all cases, the approach accurately captures strongly nonlinear and non-Gaussian features of the system's response, significantly outperforming traditional Gaussian approximations. |
| title | Predicting Forced Responses of Probability Distributions via the Fluctuation-Dissipation Theorem and Generative Modeling |
| topic | Machine Learning Chaotic Dynamics |
| url | https://arxiv.org/abs/2504.13333 |