Learning stochastic multiscale models through normalizing flows

• URL: http://arxiv.org/abs/2605.09718v1
• Date: Sun, 10 May 2026 19:31:37 GMT
• Title: Learning stochastic multiscale models through normalizing flows
• Abstract summary: We develop a principled data-driven framework for learning effective dynamics from a single observed path.<n>A central challenge is that the reduced model depends on an intractable and often unknown PDE.<n>We introduce a novel learning framework that parameterizes the invariable distribution using normalizing flows.
• Score: 0.4010598744735379
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Many systems in physics, engineering, and biology exhibit multiscale stochastic dynamics, where low-dimensional slow variables evolve under the influence of high-dimensional fast processes. In practice, observations are often limited to a single trajectory of the slow component, while the fast dynamics remain unobserved, making statistical learning challenging. Approaches based on partial differential equations (PDE), such as Fokker-Planck formulations, aim to characterize the evolution of probability densities, typically requiring dense space-time data or grid-based solvers. In contrast, we adopt a trajectory-based perspective and develop a data-driven framework for learning effective stochastic dynamics from a single observed path. We model the dynamics by coupled multiscale stochastic differential equations (SDEs) and first obtain a principled model reduction through stochastic averaging. Unlike generic model reduction techniques such as PCA, this respects the dynamical structure of the original system and explicitly incorporates the interaction between slow and fast scales. A central challenge, however, is that the reduced model depends on the invariant distribution of the fast process, which is a solution to an intractable and often unknown PDE. We introduce a novel learning framework that parameterizes the invariant distribution using normalizing flows, enabling expressive density modeling in the latent fast-variable space. The flow is trained end-to-end by optimizing a penalized likelihood objective induced by the reduced stochastic dynamics. Furthermore, we develop a Bayesian variational inference procedure for uncertainty quantification, employing a second normalizing flow to approximate the posterior distribution over model parameters. This yields a scalable approach to capturing epistemic uncertainty in multiscale systems.

Related papers

• A deep learning framework for jointly solving transient Fokker-Planck equations with arbitrary parameters and initial distributions [22.987262163354732]
  This paper introduces a deep learning-based pseudo-analytical probability solution (PAPS)<n>PAPS simultaneously resolves transient FPE solutions for arbitrary multi-modal initial distributions, system parameters, and time points.<n>Our work establishes a paradigm for scalable probabilistic modeling of multi-dimensional, parameterized systems.
  arXiv  Detail & Related papers  (2026-04-07T15:28:53Z)
• Stochastic Deep Learning: A Probabilistic Framework for Modeling Uncertainty in Structured Temporal Data [0.0]
  I propose a novel framework that integrates differential equations (SDEs) with deep generative models to improve uncertainty in machine learning applications involving structured and temporal data.<n>This approach, termed Latent Differential Inference (SLDI), embeds an It SDE in the latent space of a variational autoencoder.<n>The drift and diffusion terms of the SDE are parameterized by neural networks, enabling data-driven inference and generalizing classical time series models to handle irregular sampling and complex dynamic structure.
  arXiv  Detail & Related papers  (2026-01-08T18:53:59Z)
• Towards Fast Coarse-graining and Equation Discovery with Foundation Inference Models [6.403678133359229]
  latent dynamics in high-dimensional recordings are often characterized by a much smaller set of effective variables.<n>Most machine learning approaches tackle these tasks jointly by training autoencoders together with models that enforce dynamical consistency.<n>We propose to decouple the two problems by leveraging the recently introduced Foundation Inference Models (FIMs)<n>A proof of concept on a double-well system with semicircle diffusion, embedded into synthetic video data, illustrates the potential of this approach for fast and reusable coarse-graining pipelines.
  arXiv  Detail & Related papers  (2025-10-14T15:17:23Z)
• Self-Supervised Coarsening of Unstructured Grid with Automatic Differentiation [55.88862563823878]
  In this work, we present an original algorithm to coarsen an unstructured grid based on the concepts of differentiable physics.<n>We demonstrate performance of the algorithm on two PDEs: a linear equation which governs slightly compressible fluid flow in porous media and the wave equation.<n>Our results show that in the considered scenarios, we reduced the number of grid points up to 10 times while preserving the modeled variable dynamics in the points of interest.
  arXiv  Detail & Related papers  (2025-07-24T11:02:13Z)
• Implicit Neural Differential Model for Spatiotemporal Dynamics [5.1854032131971195]
  We introduce Im-PiNDiff, a novel implicit physics-integrated neural differentiable solver for stabletemporal dynamics.<n>Inspired by deep equilibrium models, Im-PiNDiff advances the state using implicit fixed-point layers, enabling robust long-term simulation.<n>Im-PiNDiff achieves superior predictive performance, enhanced numerical stability, and substantial reductions in memory and cost.
  arXiv  Detail & Related papers  (2025-04-03T04:07:18Z)
• Generative Latent Neural PDE Solver using Flow Matching [8.397730500554047]
  We propose a latent diffusion model for PDE simulation that embeds the PDE state in a lower-dimensional latent space.<n>Our framework uses an autoencoder to map different types of meshes onto a unified structured latent grid, capturing complex geometries.<n> Numerical experiments show that the proposed model outperforms several deterministic baselines in both accuracy and long-term stability.
  arXiv  Detail & Related papers  (2025-03-28T16:44:28Z)
• Efficient Transformed Gaussian Process State-Space Models for Non-Stationary High-Dimensional Dynamical Systems [49.819436680336786]
  We propose an efficient transformed Gaussian process state-space model (ETGPSSM) for scalable and flexible modeling of high-dimensional, non-stationary dynamical systems.<n>Specifically, our ETGPSSM integrates a single shared GP with input-dependent normalizing flows, yielding an expressive implicit process prior that captures complex, non-stationary transition dynamics.<n>Our ETGPSSM outperforms existing GPSSMs and neural network-based SSMs in terms of computational efficiency and accuracy.
  arXiv  Detail & Related papers  (2025-03-24T03:19:45Z)
• MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation [48.41289705783405]
  We propose a PDE-embedded network with multiscale time stepping (MultiPDENet)<n>In particular, we design a convolutional filter based on the structure of finite difference with a small number of parameters to optimize.<n>A Physics Block with a 4th-order Runge-Kutta integrator at the fine time scale is established that embeds the structure of PDEs to guide the prediction.
  arXiv  Detail & Related papers  (2025-01-27T12:15:51Z)
• Learning Controlled Stochastic Differential Equations [61.82896036131116]
  This work proposes a novel method for estimating both drift and diffusion coefficients of continuous, multidimensional, nonlinear controlled differential equations with non-uniform diffusion. We provide strong theoretical guarantees, including finite-sample bounds for (L2), (Linfty), and risk metrics, with learning rates adaptive to coefficients' regularity. Our method is available as an open-source Python library.
  arXiv  Detail & Related papers  (2024-11-04T11:09:58Z)
• 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)
• 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)
• Learning to Accelerate Partial Differential Equations via Latent Global Evolution [64.72624347511498]
  Latent Evolution of PDEs (LE-PDE) is a simple, fast and scalable method to accelerate the simulation and inverse optimization of PDEs. We introduce new learning objectives to effectively learn such latent dynamics to ensure long-term stability. We demonstrate up to 128x reduction in the dimensions to update, and up to 15x improvement in speed, while achieving competitive accuracy.
  arXiv  Detail & Related papers  (2022-06-15T17:31:24Z)

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.