An Adaptive Random Fourier Features approach Applied to Learning Stochastic Differential Equations

• URL: http://arxiv.org/abs/2507.15442v1
• Date: Mon, 21 Jul 2025 09:52:33 GMT
• Title: An Adaptive Random Fourier Features approach Applied to Learning Stochastic Differential Equations
• Authors: Owen Douglas, Aku Kammonen, Anamika Pandey, Raúl Tempone,
• Abstract summary: This study considers Ito diffusion processes and a likelihood-based loss function derived from the Euler-Maruyama integration introduced in citerichDiet20 and citedridi 2021stochasticdynamicalsystems.<n>Across all cases, the ARFF-based approach matches or surpasses the performance of conventional Adam-based optimization in both loss minimization and convergence speed.
• Score: 0.8947831206263182
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This work proposes a training algorithm based on adaptive random Fourier features (ARFF) with Metropolis sampling and resampling \cite{kammonen2024adaptiverandomfourierfeatures} for learning drift and diffusion components of stochastic differential equations from snapshot data. Specifically, this study considers It\^{o} diffusion processes and a likelihood-based loss function derived from the Euler-Maruyama integration introduced in \cite{Dietrich2023} and \cite{dridi2021learningstochasticdynamicalsystems}. This work evaluates the proposed method against benchmark problems presented in \cite{Dietrich2023}, including polynomial examples, underdamped Langevin dynamics, a stochastic susceptible-infected-recovered model, and a stochastic wave equation. Across all cases, the ARFF-based approach matches or surpasses the performance of conventional Adam-based optimization in both loss minimization and convergence speed. These results highlight the potential of ARFF as a compelling alternative for data-driven modeling of stochastic dynamics.

Related papers

• Diffusion Models for Solving Inverse Problems via Posterior Sampling with Piecewise Guidance [52.705112811734566]
  A novel diffusion-based framework is introduced for solving inverse problems using a piecewise guidance scheme.<n>The proposed method is problem-agnostic and readily adaptable to a variety of inverse problems.<n>The framework achieves a reduction in inference time of (25%) for inpainting with both random and center masks, and (23%) and (24%) for (4times) and (8times) super-resolution tasks.
  arXiv  Detail & Related papers  (2025-07-22T19:35:14Z)
• An evolutionary approach for discovering non-Gaussian stochastic dynamical systems based on nonlocal Kramers-Moyal formulas [19.588387760309484]
  This research endeavors to develop an evolutionary symbol sparse regression (ESSR) approach to extract non-Gaussian dynamical systems from sample path data. The efficacy and capabilities of this approach are showcased through its application to several illustrative models.
  arXiv  Detail & Related papers  (2024-09-29T03:35:01Z)
• On the Trajectory Regularity of ODE-based Diffusion Sampling [79.17334230868693]
  Diffusion-based generative models use differential equations to establish a smooth connection between a complex data distribution and a tractable prior distribution. In this paper, we identify several intriguing trajectory properties in the ODE-based sampling process of diffusion models.
  arXiv  Detail & Related papers  (2024-05-18T15:59:41Z)
• Weak Collocation Regression for Inferring Stochastic Dynamics with L\'{e}vy Noise [8.15076267771005]
  We propose a weak form of the Fokker-Planck (FP) equation for extracting dynamics with L'evy noise. Our approach can simultaneously distinguish mixed noise types, even in multi-dimensional problems.
  arXiv  Detail & Related papers  (2024-03-13T06:54:38Z)
• Response Theory via Generative Score Modeling [0.0]
  We introduce an approach for analyzing the responses of dynamical systems to external perturbations that combines score-based generative modeling with the Generalized Fluctuation-Dissipation Theorem (GFDT) The methodology enables accurate estimation of system responses, including those with non-Gaussian statistics.
  arXiv  Detail & Related papers  (2024-02-01T21:38:10Z)
• Learning Unnormalized Statistical Models via Compositional Optimization [73.30514599338407]
  Noise-contrastive estimation(NCE) has been proposed by formulating the objective as the logistic loss of the real data and the artificial noise. In this paper, we study it a direct approach for optimizing the negative log-likelihood of unnormalized models.
  arXiv  Detail & Related papers  (2023-06-13T01:18:16Z)
• A Geometric Perspective on Diffusion Models [57.27857591493788]
  We inspect the ODE-based sampling of a popular variance-exploding SDE. We establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm.
  arXiv  Detail & Related papers  (2023-05-31T15:33:16Z)
• Reservoir Computing with Error Correction: Long-term Behaviors of Stochastic Dynamical Systems [5.815325960286111]
  We propose a data-driven framework combining Reservoir Computing and Normalizing Flow to study this issue. We verify the effectiveness of the proposed framework in several experiments, including the Van der Pal, El Nino-Southern Oscillation simplified model, and Lorenz system.
  arXiv  Detail & Related papers  (2023-05-01T05:50:17Z)
• Reflected Diffusion Models [93.26107023470979]
  We present Reflected Diffusion Models, which reverse a reflected differential equation evolving on the support of the data. Our approach learns the score function through a generalized score matching loss and extends key components of standard diffusion models.
  arXiv  Detail & Related papers  (2023-04-10T17:54:38Z)
• Stochastic Interpolants: A Unifying Framework for Flows and Diffusions [16.95541777254722]
  A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo & VandenEijnden (2023), enabling the use of a broad class of continuous-time processes called stochastic interpolants' These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way.
  arXiv  Detail & Related papers  (2023-03-15T17:43:42Z)
• Score-based Continuous-time Discrete Diffusion Models [102.65769839899315]
  We extend diffusion models to discrete variables by introducing a Markov jump process where the reverse process denoises via a continuous-time Markov chain. We show that an unbiased estimator can be obtained via simple matching the conditional marginal distributions. We demonstrate the effectiveness of the proposed method on a set of synthetic and real-world music and image benchmarks.
  arXiv  Detail & Related papers  (2022-11-30T05:33:29Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.