spectral neural ode

Neural ordinary differential equations (NODEs) face two key bottlenecks in time-series modeling: (i) time-domain representations struggle to capture long-range dependencies and intrinsic periodic structures, and (ii) continuous-time dynamics suffer from granularity mismatch with discrete-time observations. To address these, we propose Fourier Ordinary Differential Equations (FODE), the first neural ODE framework operating directly in the frequency domain. FODE leverages the Fast Fourier Transform (FFT) to extract global periodic patterns, employs learnable element-wise frequency filters to model dynamic evolution in the spectral domain, and explicitly aligns continuous-time dynamics with discrete sampling via a differentiable reconstruction layer. This design unifies short- and long-term dependency modeling while mitigating discretization error. Extensive experiments across multiple benchmark time-series datasets demonstrate that FODE consistently outperforms state-of-the-art NODE-based and deep sequential models—achieving new SOTA in both predictive accuracy and computational efficiency.

This work proposes a novel frequency-domain neural ordinary differential equation (Neural ODE) framework to address the limited generalization capability of standard Neural ODEs when modeling highly nonlinear dynamical systems. By incorporating fast Fourier transform (FFT), the method maps continuous-time dynamics into the frequency domain, enabling more effective representation learning. Leveraging curriculum learning and ensemble strategies, the proposed model is evaluated on four canonical nonlinear systems—Lotka-Volterra, Duffing, Van der Pol, and Lorenz. Experimental results demonstrate that the approach significantly outperforms baseline models, including GRU, LSTM, and augmented Neural ODEs (ANODEs), in both generalization performance and convergence stability.

This work addresses the complexity of modeling high-dimensional continuous-time series by proposing a unified framework based on Neural Stochastic Differential Equations (Neural SDEs). Methodologically, it treats observed sequences as discrete samples from an underlying continuous dynamical system, jointly parameterizing both drift and diffusion terms, and introduces a numerically simulation-free maximum-likelihood training paradigm that integrates stochastic calculus with deep neural networks. Key contributions include: (i) the first systematic empirical validation of the superiority of continuous-time SDE modeling over discrete-time alternatives in high-dimensional sequential tasks—particularly in embodied intelligence and generative AI; and (ii) a novel, efficient, differentiable training algorithm that avoids pathwise simulation. Experiments demonstrate state-of-the-art performance across multiple continuous-domain sequence modeling benchmarks, with significant improvements in long-horizon prediction stability and generative sample diversity.

This work addresses the computational inefficiency of Neural Controlled Differential Equations (NCDEs), whose forward propagation is costly and inherently sequential due to nonlinear vector fields, hindering scalability. To overcome this limitation, the paper proposes three efficient variants: Log-NCDE leverages the Log-ODE method for approximate solution; Linear NCDE adopts a linear vector field to admit closed-form solutions and enable time-parallel computation; and Structured Linear NCDE incorporates structured matrices to preserve expressive power while enhancing efficiency. These approaches collectively reduce per-step training time by up to three orders of magnitude and achieve state-of-the-art performance across multiple time series benchmarks.

This work addresses low-rank decomposition of continuous-time vector-valued signals. We propose the first model-agnostic implicit neural signal representation framework, unifying continuous-domain generalizations of principal component analysis (PCA) and independent component analysis (ICA). The method models signals as differentiable implicit neural stochastic processes; statistical constraints—namely decorrelation and independence—are implicitly enforced via a contrastive-function-based loss, eliminating reliance on discrete or regular sampling. Enabled by end-to-end gradient-based optimization, the framework achieves robust component separation on irregularly sampled signals and point cloud data. It significantly improves generalization under missing-data and non-uniform sampling regimes, demonstrating superior performance in challenging real-world acquisition scenarios where conventional discrete-domain methods fail.

This work addresses the lack of theoretical understanding regarding the online learning dynamics and generalization mechanisms of high-dimensional controlled nonlinear dynamical systems, such as neural ordinary differential equations (neural ODEs). For the first time, it systematically applies dynamical mean-field theory to analyze online stochastic gradient descent training of neural ODEs. In the high-dimensional limit, the framework rigorously solves the coupled dynamics of training and inference and analytically derives the associated learning curves. This study establishes the first tractable theoretical framework for understanding both the training dynamics and generalization capabilities of deep continuous models, revealing the precise evolution laws governing high-dimensional neural ODEs under online learning.

为解决时间域泛化中的多尺度漂移模式和局部不确定性问题，提出FreKoo++框架，结合连续Koopman模态动力学与自适应谱解缠方法。

Existing spectral neural operators struggle to capture the time-evolving spectral characteristics of non-stationary partial differential equations, leading to large and unstable errors in long-term predictions. This work proposes a state-adaptive spectral neural operator framework that introduces a learnable time–frequency gating mechanism to jointly extract statistical features from both the frequency and physical domains based on the current system state, thereby dynamically modulating the spectral response without requiring explicit temporal embeddings. This approach enables implicit time awareness and multi-scale adaptivity, significantly enhancing modeling capacity for non-stationary dynamics. Evaluated on six 1D and 2D non-stationary PDE benchmarks, the method demonstrates superior accuracy and robustness in long-term rollouts compared to strong baseline approaches.

Existing time series generative models suffer from limited expressivity and poor adaptability to irregularly sampled observation grids. This work proposes G-SLiCEs, a continuous-time generative model based on Structured Linear Controlled differential equations (SLiCEs), which achieves high expressivity through continuous flow matching in path space. We establish, for the first time, that SLiCEs can approximate any continuous causal pushforward path law under the Wasserstein-∞ metric, thereby enabling universal time series generation and introducing maximal expressivity into continuous-time generative modeling. The method natively supports arbitrary observation time grids and significantly outperforms existing approaches in irregularly sampled settings, demonstrating superior performance in probabilistic forecasting and downstream tasks.

This work addresses the challenge of simultaneously achieving stability, interpretability, and generalization in time series forecasting by proposing a novel architecture that integrates learnable Koopman operators with Transformer-based backbones such as PatchTST, Informer, and Autoformer. By designing four variants of the Koopman operator, the method enables explicit control over the spectral properties, stability, and rank of the linear transition operator within deep forecasting models for the first time, allowing flexible interpolation between strictly stable and unconstrained dynamics. Experiments demonstrate that the approach significantly improves the bias-variance trade-off, numerical conditioning, and interpretability of latent dynamics across multi-horizon forecasting tasks, effectively combining theoretical guarantees with data-driven flexibility.