exponential operator splitting

To address the high-throughput, numerical stability, and computational efficiency requirements for matrix exponential evaluation in generative AI, this paper proposes a novel adaptive Taylor series evaluation algorithm. Unlike classical approaches—such as Paterson–Stockmeyer or Padé approximants combined with scaling-and-squaring—the method jointly optimizes the Taylor expansion order and scaling factor to minimize floating-point operations under a prescribed relative error tolerance (<1e−12), while integrating dynamic error control within a scaling-evaluation-squaring framework. Theoretical analysis guarantees numerical stability and achieves a Pareto improvement in both accuracy and asymptotic complexity. Empirically, on large-scale generative tasks—including diffusion models and continuous-time flow matching—the algorithm achieves an average 2.3× speedup over state-of-the-art methods. The implementation is open-sourced and validated across mainstream GPU platforms.

Existing discretization methods for continuous-time linear systems separately approximate the dynamics, input, and process noise matrices—leading to error accumulation and computational inconsistency. To address this, we propose a unified analytical discretization method based on a single matrix exponential computation. Our approach achieves the first joint closed-form discretization of all three state-space matrices (A, B, G), eliminating errors inherent in conventional zero-order-hold (ZOH) discretization that relies on multi-step numerical integration or component-wise approximations. Efficient matrix exponential evaluation is performed via the scaling-and-squaring method combined with Padé approximation. Experimental validation on canonical LTI systems confirms strict equivalence to exact ZOH discretization, with discretization error reduced by one to two orders of magnitude and computational time decreased by approximately 40%. The method thus significantly enhances accuracy, numerical consistency, and computational efficiency.

This work addresses the challenge of effectively extending kernel-based methods—originally developed for deterministic dynamical systems—to stochastic differential equations (SDEs) for approximating eigenfunctions of the Koopman operator. By leveraging the Feynman–Kac path integral representation, the study unifies three distinct kernel constructions—variational principles, Green’s function convolutions, and resolvent operators—into a coherent framework for stochastic systems with diffusion, establishing a corresponding reproducing kernel Hilbert space (RKHS) approximation scheme. Theoretically, under uniform ellipticity, these three approaches are shown to be equivalent, revealing that diffusion enhances numerical conditioning through elliptic regularization. The analysis further provides error bounds that separate RKHS approximation error from Monte Carlo sampling error. Numerical experiments on the Ornstein–Uhlenbeck process, nonlinear SDEs, and high-dimensional systems demonstrate the method’s efficacy, showing that moderate diffusion significantly improves numerical stability.

This work addresses two key challenges in Koopman operator spectral estimation within reproducing kernel Hilbert spaces (RKHS): low spectral accuracy and the frequent non-membership of eigenfunctions in the original function space. To resolve these, we propose JetEDMD—a novel method that constructs an intrinsic observable-driven rigged RKHS framework. For the first time, jet differential geometry is integrated into RKHS-based Koopman estimation: we define an extended Koopman operator on the rigged space, enabling explicit capture of nontrivial eigenfunctions and high-fidelity spectral analysis. Theoretically, we establish rigorous convergence rates and error bounds. Empirically, JetEDMD achieves significantly reduced eigenvalue estimation errors and improved long-term trajectory prediction on benchmark nonlinear dynamical systems—including van der Pol, Duffing, Hénon, and Lorenz—thereby overcoming the numerical accuracy limitations inherent to conventional EDMD.

To address spectral information loss and multiplicative structure degradation in finite-dimensional approximations of the Koopman operator—caused by heuristic selection of observable function dictionaries—this paper proposes a novel structure-guided Dynamic Mode Decomposition (DMD) framework. The core innovation lies in the first explicit incorporation of the Koopman operator’s intrinsic multiplicative structure into the DMD modeling process, achieved via constrained matrix optimization that jointly enables adaptive observable selection and structural preservation. The method enjoys theoretically guaranteed convergence and significantly improves spectral estimation accuracy and robustness to noise. Experiments on a simple pendulum, the Lorenz system, and real-world fluid flow data demonstrate superior spectral fidelity and stability compared to standard DMD.

该研究通过将矩阵指数替换为有理映射，提出了一种用于求解Fokker-Planck方程的高效且保持正性的DF-ADI方案。

本文介绍了一种通过计算交换算子的联合特征向量来解决多项式根和张量分解问题的方法，并分析了其多重结构。

本文提出了一种基于Strang算子分裂近似的框架，用于解决随机波动率模型下的数值期权定价问题，避免了条件积分方差的评估。

This work addresses spectral pollution in extended dynamic mode decomposition (EDMD) caused by non-invariant dictionaries, which distorts the eigenvalues of the Koopman operator. To resolve this issue, the authors propose a projected Koopman approximation framework that constructs compatible subspaces via forward-intersection chains, enabling accurate preservation of Koopman eigenpairs associated with nonzero eigenvalues without requiring dictionary invariance. The approach integrates coordinate-wise and $L^2(\mu)$-orthogonal projection geometries, leveraging singular value decomposition to formulate a filtered EDMD variant. Numerical experiments on the Kronecker flow, polynomial systems, and the Van der Pol oscillator demonstrate significant suppression of spectral pollution: the coordinate projector recovers the local equilibrium spectrum independently of the sampling measure, while the $L^2(\mu)$ projector accurately approximates the limit-cycle spectrum.

This study addresses the numerical instability of neural spectral architectures (NeuSA) when solving stiff differential equations by proposing a novel framework that integrates neural spectral methods with exponential time differencing (ETD). The approach leverages spectral representations and high-order exponential integrators to precisely handle linear stiff terms, while employing physics-informed neural networks to model nonlinear residuals, thereby achieving efficient decoupling of stiff–nonstiff dynamics. Experimental results demonstrate that the proposed framework attains both high stability and accuracy on stiff PDE benchmarks. Furthermore, it supports the identification of unknown physical parameters through inverse problem learning, offering a reliable new paradigm for modeling stiff systems.