Stochastic Deep Learning: A Probabilistic Framework for Modeling Uncertainty in Structured Temporal Data

• URL: http://arxiv.org/abs/2601.05227v1
• Date: Thu, 08 Jan 2026 18:53:59 GMT
• Title: Stochastic Deep Learning: A Probabilistic Framework for Modeling Uncertainty in Structured Temporal Data
• Authors: James Rice,
• Abstract summary: 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: I propose a novel framework that integrates stochastic differential equations (SDEs) with deep generative models to improve uncertainty quantification in machine learning applications involving structured and temporal data. This approach, termed Stochastic Latent Differential Inference (SLDI), embeds an Itô SDE in the latent space of a variational autoencoder, allowing for flexible, continuous-time modeling of uncertainty while preserving a principled mathematical foundation. 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. A central theoretical contribution is the co-parameterization of the adjoint state with a dedicated neural network, forming a coupled forward-backward system that captures not only latent evolution but also gradient dynamics. I introduce a pathwise-regularized adjoint loss and analyze variance-reduced gradient flows through the lens of stochastic calculus, offering new tools for improving training stability in deep latent SDEs. My paper unifies and extends variational inference, continuous-time generative modeling, and control-theoretic optimization, providing a rigorous foundation for future developments in stochastic probabilistic machine learning.

Related papers

• Expanding the Chaos: Neural Operator for Stochastic (Partial) Differential Equations [65.80144621950981]
  We build on Wiener chaos expansions (WCE) to design neural operator (NO) architectures for SPDEs and SDEs.<n>We show that WCE-based neural operators provide a practical and scalable way to learn SDE/SPDE solution operators.
  arXiv  Detail & Related papers  (2026-01-03T00:59:25Z)
• HGAN-SDEs: Learning Neural Stochastic Differential Equations with Hermite-Guided Adversarial Training [3.4515388499147654]
  We introduce HGAN-SDEs, a novel GAN-based framework that leverages Neural Hermite functions to construct a structured and efficient discriminator.<n>HGAN-SDEs achieve superior sample quality and learning efficiency compared to existing generative models for SDEs.
  arXiv  Detail & Related papers  (2025-12-23T11:25:22Z)
• Unlocking Out-of-Distribution Generalization in Dynamics through Physics-Guided Augmentation [46.40087254928057]
  We present SPARK, a physics-guided quantitative augmentation plugin.<n>Experiments on diverse benchmarks demonstrate that SPARK significantly outperforms state-of-the-art baselines.
  arXiv  Detail & Related papers  (2025-10-28T09:30:35Z)
• Latent Diffusion Model Based Denoising Receiver for 6G Semantic Communication: From Stochastic Differential Theory to Application [11.385703484113552]
  We propose a novel semantic communication framework empowered by generative artificial intelligence (GAI)<n>A latent diffusion model (LDM)-based semantic communication framework is proposed that combines a variational autoencoder for semantic features extraction.<n>The proposed system is a training-free framework that supports zero-shot generalization, and achieves superior performance under low-SNR and out-of-distribution conditions.
  arXiv  Detail & Related papers  (2025-06-06T03:20:32Z)
• 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)
• In-Context Learning of Stochastic Differential Equations with Foundation Inference Models [6.785438664749581]
  differential equations (SDEs) describe dynamical systems where deterministic flows, governed by a drift function, are superimposed with random fluctuations, dictated by a diffusion function.<n>We introduce FIM-SDE (Foundation Inference Model for SDEs), a pretrained recognition model that delivers accurate in-context estimation of the drift and diffusion functions of low-dimensional SDEs.<n>We demonstrate that FIM-SDE achieves robust in-context function estimation across a wide range of synthetic and real-world processes.
  arXiv  Detail & Related papers  (2025-02-26T11:04:02Z)
• Neural SDEs as a Unified Approach to Continuous-Domain Sequence Modeling [3.8980564330208662]
  We propose a novel and intuitive approach to continuous sequence modeling.<n>Our method interprets time-series data as textitdiscrete samples from an underlying continuous dynamical system.<n>We derive a maximum principled objective and a textitsimulation-free scheme for efficient training of our Neural SDE model.
  arXiv  Detail & Related papers  (2025-01-31T03:47:22Z)
• Trajectory Flow Matching with Applications to Clinical Time Series Modeling [77.58277281319253]
  Trajectory Flow Matching (TFM) trains a Neural SDE in a simulation-free manner, bypassing backpropagation through the dynamics.<n>We demonstrate improved performance on three clinical time series datasets in terms of absolute performance and uncertainty prediction.
  arXiv  Detail & Related papers  (2024-10-28T15:54:50Z)
• Latent Space Energy-based Neural ODEs [73.01344439786524]
  This paper introduces novel deep dynamical models designed to represent continuous-time sequences.<n>We train the model using maximum likelihood estimation with Markov chain Monte Carlo.<n> Experimental results on oscillating systems, videos and real-world state sequences (MuJoCo) demonstrate that our model with the learnable energy-based prior outperforms existing counterparts.
  arXiv  Detail & Related papers  (2024-09-05T18:14:22Z)
• 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)
• Learning Space-Time Continuous Neural PDEs from Partially Observed States [13.01244901400942]
  We introduce a grid-independent model learning partial differential equations (PDEs) from noisy and partial observations on irregular grids. We propose a space-time continuous latent neural PDE model with an efficient probabilistic framework and a novel design encoder for improved data efficiency and grid independence.
  arXiv  Detail & Related papers  (2023-07-09T06:53:59Z)
• Capturing dynamical correlations using implicit neural representations [85.66456606776552]
  We develop an artificial intelligence framework which combines a neural network trained to mimic simulated data from a model Hamiltonian with automatic differentiation to recover unknown parameters from experimental data. In doing so, we illustrate the ability to build and train a differentiable model only once, which then can be applied in real-time to multi-dimensional scattering data.
  arXiv  Detail & Related papers  (2023-04-08T07:55:36Z)
• ConCerNet: A Contrastive Learning Based Framework for Automated Conservation Law Discovery and Trustworthy Dynamical System Prediction [82.81767856234956]
  This paper proposes a new learning framework named ConCerNet to improve the trustworthiness of the DNN based dynamics modeling. We show that our method consistently outperforms the baseline neural networks in both coordinate error and conservation metrics.
  arXiv  Detail & Related papers  (2023-02-11T21:07:30Z)
• Closed-form Continuous-Depth Models [99.40335716948101]
  Continuous-depth neural models rely on advanced numerical differential equation solvers. We present a new family of models, termed Closed-form Continuous-depth (CfC) networks, that are simple to describe and at least one order of magnitude faster.
  arXiv  Detail & Related papers  (2021-06-25T22:08:51Z)
• Accurate and Reliable Forecasting using Stochastic Differential Equations [48.21369419647511]
  It is critical yet challenging for deep learning models to properly characterize uncertainty that is pervasive in real-world environments. This paper develops SDE-HNN to characterize the interaction between the predictive mean and variance of HNNs for accurate and reliable regression. Experiments on the challenging datasets show that our method significantly outperforms the state-of-the-art baselines in terms of both predictive performance and uncertainty quantification.
  arXiv  Detail & Related papers  (2021-03-28T04:18:11Z)

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.