Random Controlled Differential Equations

• URL: http://arxiv.org/abs/2512.23670v1
• Date: Mon, 29 Dec 2025 18:25:10 GMT
• Title: Random Controlled Differential Equations
• Authors: Francesco Piatti, Thomas Cass, William F. Turner,
• Abstract summary: We introduce a training-efficient framework for time-series learning that combines random features with controlled differential equations (CDEs)<n>Only a linear readout layer is trained, resulting in fast, scalable models with strong inductive bias.<n>We evaluate both models across a range of time-series benchmarks, demonstrating competitive or state-of-the-art performance.
• Score: 1.2107297090229683
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce a training-efficient framework for time-series learning that combines random features with controlled differential equations (CDEs). In this approach, large randomly parameterized CDEs act as continuous-time reservoirs, mapping input paths to rich representations. Only a linear readout layer is trained, resulting in fast, scalable models with strong inductive bias. Building on this foundation, we propose two variants: (i) Random Fourier CDEs (RF-CDEs): these lift the input signal using random Fourier features prior to the dynamics, providing a kernel-free approximation of RBF-enhanced sequence models; (ii) Random Rough DEs (R-RDEs): these operate directly on rough-path inputs via a log-ODE discretization, using log-signatures to capture higher-order temporal interactions while remaining stable and efficient. We prove that in the infinite-width limit, these model induces the RBF-lifted signature kernel and the rough signature kernel, respectively, offering a unified perspective on random-feature reservoirs, continuous-time deep architectures, and path-signature theory. We evaluate both models across a range of time-series benchmarks, demonstrating competitive or state-of-the-art performance. These methods provide a practical alternative to explicit signature computations, retaining their inductive bias while benefiting from the efficiency of random features.

Related papers

• Is Flow Matching Just Trajectory Replay for Sequential Data? [46.770624059457724]
  Flow matching (FM) is increasingly used for time-series generation.<n>It is not well understood whether it learns a general dynamical structure or simply performs an effective "trajectory replay"<n>We show that the implied sampler is an ODE whose dynamics constitutes a nonparametric, memory-augmented continuous-time dynamical system.
  arXiv  Detail & Related papers  (2026-02-09T06:48:45Z)
• Scalable Random Wavelet Features: Efficient Non-Stationary Kernel Approximation with Convergence Guarantees [5.758073912084366]
  We introduce Random Wavelet Features (RWF), a framework that constructs scalable, non-stationary kernel approximations by sampling from wavelet families.<n>By harnessing the inherent localization and multi-resolution structure of wavelets, RWF generates an explicit feature map that captures complex, input-dependent patterns.<n>We demonstrate empirically on a range of challenging synthetic and real-world datasets that RWF outperforms stationary random features and offers a compelling accuracy-efficiency trade-off.
  arXiv  Detail & Related papers  (2026-02-01T02:56:56Z)
• When Bayesian Tensor Completion Meets Multioutput Gaussian Processes: Functional Universality and Rank Learning [53.17227599983122]
  Functional tensor decomposition can analyze multi-dimensional data with real-valued indices.<n>We propose a rank-revealing functional low-rank tensor completion (RR-F) method.<n>We establish the universal approximation property of the model for continuous multi-dimensional signals.
  arXiv  Detail & Related papers  (2025-12-25T03:15:52Z)
• TimeFlow: Towards Stochastic-Aware and Efficient Time Series Generation via Flow Matching Modeling [2.74279932215302]
  Time series data has emerged as a critical research topic due to its broad utility in supporting downstream time series mining tasks.<n>We propose TimeFlow, a novel flow matching framework that integrates a encoder-only architecture.<n>Our model consistently outperforms strong baselines in generation quality, diversity, and efficiency.
  arXiv  Detail & Related papers  (2025-11-11T08:28:26Z)
• RDPM: Solve Diffusion Probabilistic Models via Recurrent Token Prediction [17.005198258689035]
  Diffusion Probabilistic Models (DPMs) have emerged as the de facto approach for high-fidelity image synthesis.<n>We introduce a novel generative framework, the Recurrent Diffusion Probabilistic Model (RDPM), which enhances the diffusion process through a recurrent token prediction mechanism.
  arXiv  Detail & Related papers  (2024-12-24T12:28:19Z)
• Derivative-Free Guidance in Continuous and Discrete Diffusion Models with Soft Value-Based Decoding [84.3224556294803]
  Diffusion models excel at capturing the natural design spaces of images, molecules, DNA, RNA, and protein sequences. We aim to optimize downstream reward functions while preserving the naturalness of these design spaces. Our algorithm integrates soft value functions, which looks ahead to how intermediate noisy states lead to high rewards in the future.
  arXiv  Detail & Related papers  (2024-08-15T16:47:59Z)
• 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)
• 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)
• 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.