arXiv subjects

Andre N. Souza

Publications and source records attributed to Andre N. Souza.

10 recordsLinked to original sources

arXiv2026

Statistical Parameter Calibration via the Generalized Fluctuation Dissipation Theorem and Generative Modeling

We introduce a response-theoretic framework that recasts parameter calibration of ergodic stochastic differential equations as a fluctuation-dissipation problem. Our central result is that the full Jacobian of any stationary observable with respect to drift and diffusion parameters admits an exact linear-response representation as a time-correlation integral evaluated along unperturbed dynamics alone, without perturbed simulations, adjoint derivations, or tangent-linear models. The key idea is to interpret infinitesimal parameter variations as causal perturbations of the dynamics, thereby bringing the Generalized Fluctuation-Dissipation Theorem to inverse problems. The resulting kernels couple observables to the score of the invariant density, for which modern score-estimation methods provide practical non-Gaussian estimators. We validate the framework across a hierarchy of models, from analytically tractable processes to stochastic parameterization in the chaotic Lorenz-96 system, and show that a single baseline trajectory can recover parameter sensitivities and calibration updates with accuracy comparable to finite-difference approaches. More broadly, the framework opens a new route to model calibration, statistical inverse problems, and uncertainty quantification when the quantities of interest are long-time statistics of complex dynamical systems.

nlin.CD↗

arXiv2025

KGMM: A K-means Clustering Approach to Gaussian Mixture Modeling for Score Function Estimation

We propose a hybrid method for accurately estimating the score function, i.e., the gradient of the log steady-state density, using a Gaussian Mixture Model (GMM) in conjunction with a bisecting K-means clustering step. Our approach, which we call KGMM, offers a systematic way to combine statistical density estimation with a neural-network-based interpolation of the score, leveraging the strengths of both. We demonstrate its ability to accurately reconstruct the long-time statistical properties of several paradigmatic systems, including potential systems, chaotic Lorenz-type models, and the Kuramoto-Sivashinsky equation. Numerical experiments show that KGMM yields robust estimates of the score function, even for small values of the covariance amplitude in the GMM, where standard GMM methods tend to fail because of noise amplification. We compare the performance of KGMM against the conventional Denoising Score Matching (DSM) approach, demonstrating that KGMM achieves more faithful reconstruction of the steady-state distribution for low-dimensional systems at a fraction of the computational cost. These accurate estimates allow us to build effective stochastic reduced-order models that reproduce the invariant measures of the target dynamics.

nlin.CD↗

arXiv2025

Surface to Seafloor: A Generative AI Framework for Decoding the Ocean Interior State

Understanding subsurface ocean dynamics is essential for quantifying oceanic heat and mass transport, but direct observations at depth remain sparse due to logistical and technological constraints. In contrast, satellite missions provide rich surface datasets-such as sea surface height, temperature, and salinity-that offer indirect but potentially powerful constraints on the ocean interior. Here, we present a probabilistic framework based on score-based diffusion models to reconstruct three-dimensional subsurface velocity and buoyancy fields, including the energetic ocean eddy field, from surface observations. Using a 15-level primitive equation simulation of an idealized double-gyre system, we evaluate the skill of the model in inferring the mean circulation and the mesoscale variability at depth under varying levels of surface information. We find that the generative model successfully recovers key dynamical structures and provides physically meaningful uncertainty estimates, with predictive skill diminishing systematically as the surface resolution decreases or the inference depth increases. These results demonstrate the potential of generative approaches for ocean state estimation and uncertainty quantification, particularly in regimes where traditional deterministic methods are underconstrained or ill-posed.

physics.geo-ph↗

arXiv2025

Predicting Forced Responses of Probability Distributions via the Fluctuation-Dissipation Theorem and Generative Modeling

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.

stat.ML↗

arXiv2024

A Modified Bisecting K-Means for Approximating Transfer Operators: Application to the Lorenz Equations

We investigate the convergence behavior of the extended dynamic mode decomposition for constructing a discretization of the continuity equation associated with the Lorenz equations using a nonlinear dictionary of over 1,000,000 terms. The primary objective is to analyze the resulting operator by varying the number of terms in the dictionary and the timescale. We examine what happens when the number of terms of the nonlinear dictionary is varied with respect to its ability to represent the invariant measure, Koopman eigenfunctions, and temporal autocorrelations. The dictionary comprises piecewise constant functions through a modified bisecting k-means algorithm and can efficiently scale to higher-dimensional systems.

physics.comp-ph↗

arXiv2024

Reduced Markovian Models of Dynamical Systems

Leveraging recent work on data-driven methods for constructing a finite state space Markov process from dynamical systems, we address two problems for obtaining further reduced statistical representations. The first problem is to extract the most salient reduced-order dynamics for a given timescale by using a modified clustering algorithm from network theory. The second problem is to provide an alternative construction for the infinitesimal generator of a Markov process that respects statistical features over a large range of timescales. We demonstrate the methodology on three low-dimensional dynamical systems with stochastic and chaotic dynamics. We then apply the method to two high-dimensional dynamical systems, the Kuramoto-Sivashinky equations and data sampled from fluid-flow experiments via Particle-Image Velocimetry. We show that the methodology presented herein provides a robust reduced-order statistical representation of the underlying system.

nlin.CD↗

arXiv2023

Transforming Butterflies into Graphs: Statistics of Chaotic and Turbulent Systems

We formulate a data-driven method for constructing finite volume discretizations of a dynamical system's underlying Continuity / Fokker-Planck equation. A method is employed that allows for flexibility in partitioning state space, generalizes to function spaces, applies to arbitrarily long sequences of time-series data, is robust to noise, and quantifies uncertainty with respect to finite sample effects. After applying the method, one is left with Markov states (cell centers) and a random matrix approximation to the generator. When used in tandem, they emulate the statistics of the underlying system. We apply the method to the Lorenz equations (a three-dimensional ordinary differential equation) and a modified Held-Suarez atmospheric simulation (a Flux-Differencing Discontinuous Galerkin discretization of the compressible Euler equations with gravity and rotation on a thin spherical shell). We show that a coarse discretization captures many essential statistical properties of the system, such as steady state moments, time autocorrelations, and residency times for subsets of state space.

physics.flu-dyn↗

arXiv2022

Statistical Nonlocality of Dynamically Coherent Structures

We introduce a class of stochastic advection problems amenable to analysis of turbulent transport. The statistics of the flow field are represented as a continuous time Markov process, a choice that captures the intuitive notion of turbulence as moving from one coherent structure to another. We obtain closed form expressions for the turbulent transport operator without invoking approximations. We recover the classical estimate of turbulent transport as a diffusivity tensor, the components of which are the integrated auto-correlation of the velocity field, in the limit that the operator becomes local in space and time.

physics.flu-dyn↗

arXiv2019

Wall-to-wall optimal transport in two dimensions

Gradient ascent methods are developed to compute incompressible flows that maximize heat transport between two isothermal no-slip parallel walls. Parameterizing the magnitude of velocity fields by a Péclet number $\text{Pe}$ proportional to their root-mean-square rate-of-strain, the schemes are applied to compute two-dimensional flows optimizing convective enhancement of diffusive heat transfer, i.e., the Nusselt number $\text{Nu}$ up to $\text{Pe} \approx 10^5$. The resulting transport exhibits a change of scaling from $\text{Nu}-1 \sim \text{Pe}^{2}$ for $\text{Pe} < 10$ in the linear regime to $\text{Nu} \sim \text{Pe}^{0.54}$ for $\text{Pe} > 10^3$. Optimal fields are observed to be approximately separable, i.e., products of functions of the wall-parallel and wall-normal coordinates. Analysis employing a separable ansatz yields a conditional upper bound $\lesssim \text{Pe}^{6/11} = \text{Pe}^{0.\overline{54}}$ as $\text{Pe} \rightarrow \infty$ similar to the computationally achieved scaling. Implications for heat transfer in buoyancy-driven Rayleigh-Bénard convection are discussed.

physics.flu-dyn↗

arXiv2016

A Probabilistic Optimum-Path Forest Classifier for Binary Classification Problems

Probabilistic-driven classification techniques extend the role of traditional approaches that output labels (usually integer numbers) only. Such techniques are more fruitful when dealing with problems where one is not interested in recognition/identification only, but also into monitoring the behavior of consumers and/or machines, for instance. Therefore, by means of probability estimates, one can take decisions to work better in a number of scenarios. In this paper, we propose a probabilistic-based Optimum Path Forest (OPF) classifier to handle with binary classification problems, and we show it can be more accurate than naive OPF in a number of datasets. In addition to being just more accurate or not, probabilistic OPF turns to be another useful tool to the scientific community.

cs.CV↗