Despite the fast progress in neural operator learning, long-sequence modeling still is a standing challenge whereby latent states have been introduced with techniques well derived. Diverging from existing methods that treat latent states as transient variables or decoupled representations, CoEvol-NO introduces a persistent state to establish a co-evolutionary framework, where the latent state and mesh sequence are updated jointly and bidirectionally. Inspired by classical numerical methods, we model the layer-wise state evolution as a Predictor-Corrector (PC) process. Specifically, a "Predictor'' generates a tentative target, followed by a "Corrector'' that refines the persistent state via an {error-driven update mechanism}. Furthermore, our theoretical analysis reveals that the widely used \textit{direct substitution} and \textit{residual update} paradigms are essentially {first-order approximations} of this error-driven correction under different loss assumptions. We theoretically prove that CoEvol-NO achieves strict linear time complexity. Extensive experiments on five standard benchmarks and two large-scale industrial design tasks demonstrate that CoEvol-NO consistently achieves state-of-the-art (SOTA) performance.

Modeling complex dynamics from observational data is fundamental to scientific discovery and artificial intelligence. However, existing approaches are often plagued by the entanglement of static state representations and instantaneous motion, leading to accumulated errors and off-manifold hallucinations where predicted trajectories violate intrinsic geometric constraints. To address this, we propose Geometric Flow Grounding, a unified framework that enforces dynamic evolution strictly along the tangent bundle of the learned data manifold via a differentiable Neural Tangent Projection Layer. By geometrically decoupling state representation from tangential dynamics, our method generalizes across diverse data regimes. In scientific discovery, GFG reduces numerical aliasing and improves long-horizon stability in sparse dynamical systems, while recovering interpretable gene regulatory motifs from single-cell data. For trustworthy AI, the projection residual provides a zero-shot metric for deepfake video detection by revealing inconsistencies with the implicit flow of pre-trained world models. Our results establish manifold-constrained projection as a universal operator for both discovering natural laws and verifying synthetic content. Code will be available at \url{https://github.com/yuchang97/GFG-public}

Time-Dependent Differential Equations (TDDEs) model dynamical processes across science and engineering, but time-critical applications require solvers that deliver high-fidelity trajectories under stringent latency constraints. Most existing TDDE solvers are limited by time discretization, forcing a latency-accuracy trade-off where smaller step sizes capture high-fidelity trajectories but incur prohibitive runtime, while larger steps meet real-time budgets at the cost of trajectory distortion. Dynamical System Machines (DSMs) offer a promising alternative by computing through continuous physical evolution, yet existing DSMs struggle to capture the spatiotemporal complexity of TDDEs. This work introduces DS-TS, a novel TDDE solver that is both accurate and efficient by leveraging the unique computational advantages of DSMs. DS-TS integrates three key innovations: (1) Excitatory-Inhibitory Inspired Coupling to better model complex spatial interactions; (2) State-aware Dynamic Nonlinearity to enable rich inter-node interactions and state-dependent spatiotemporal correlations; and (3) Hierarchical Temporal Integration to capture high-order temporal dependencies. Experiments demonstrate that DS-TS achieves high-fidelity solutions while delivering orders-of-magnitude improvements in speed ($\sim 10^3\times$) and energy efficiency ($\sim 10^5\times$) compared to baseline solvers.

Bayesian filtering is a well-known problem that aims to estimate plausible states of a dynamical system from observations. Among existing approaches to solve this problem, particle filters are theoretically exact for non-linear dynamics and observations, but suffer from poor scalability in high dimensions. In this work, we show that diffusion-based emulators of dynamical systems can be used to implement, without additional training, an optimal variant of particle filters that has remained largely unexplored due to implementation challenges with classical numerical solvers. Experiments on nonlinear chaotic systems, including atmospheric dynamics, demonstrate that the proposed approach successfully scales particle filtering to high-dimensional settings.