predictive dynamics modeling

Existing time-series forecasting models suffer from limited performance due to inadequate modeling of data dynamics. Method: This paper proposes PRO-DYN, a framework that integrates a learnable dynamic module as a standalone, structurally intact component at the model’s output stage—rather than dispersing it across the architecture. Grounded in dynamical systems modeling principles, we conduct systematic ablation studies across diverse backbone networks to validate two key design principles: (i) optimal placement of the dynamic module at the final layer, and (ii) its structural integrity—i.e., inseparability into subcomponents. Contribution/Results: Empirical evaluation on multiple benchmark datasets demonstrates consistent and significant improvements in forecasting accuracy. Moreover, the gains generalize across architectures, establishing PRO-DYN as an interpretable, reusable paradigm for dynamic enhancement in time-series modeling.

This work addresses the fundamental problem of selecting the training horizon for autoregressive models predicting dynamical systems—balancing insufficient long-term trend capture (short horizons) against optimization difficulty due to error accumulation (long horizons). We formally characterize this trade-off through the geometry of the loss landscape: in chaotic systems, training with long horizons induces exponential growth in loss ruggedness; in limit-cycle systems, the growth is linear. Our analysis integrates dynamical systems theory with optimization landscape theory and is validated numerically. Furthermore, we empirically demonstrate that models trained with longer horizons exhibit superior short-horizon generalization performance. Collectively, these results yield an interpretable, generalizable principle for training horizon selection in autoregressive forecasting. The derived error-growth laws and generalization properties are empirically confirmed across diverse dynamical systems, including chaotic, limit-cycle, and quasi-periodic regimes.

This work investigates the dynamic evolution of layer-wise and global accuracy during Forward-Forward (FF) algorithm training. Addressing three core questions—(i) layer-wise accuracy dynamics, (ii) the impact of depth on convergence speed, and (iii) the correlation between per-layer accuracy and overall model performance—we propose an analytical framework based on dual forward passes and layer-local loss functions, enabling fine-grained accuracy tracking and correlation modeling across layers. We systematically uncover a “deep-layer lag” phenomenon in FF networks: shallow layers attain high accuracy significantly earlier than deep layers, and their early accuracy strongly predicts final model performance (Pearson *r* > 0.95). This provides a mechanistic explanation for effective backpropagation-free learning. Our findings empirically validate the hierarchical, cooperative nature of FF training at the dynamical level, advancing theory for brain-inspired efficient learning.

Addressing the challenge of long-term dynamic forecasting under high noise and irregular sampling, this paper proposes Phy-SSM, a generalized state-space model incorporating partial physical knowledge. Methodologically, it decouples prior physical laws into known and unknown state matrices embedded within the SSM architecture, and introduces a physics-informed state regularization term to enforce latent states to satisfy underlying dynamical constraints—thereby enhancing generalizability and solution uniqueness. Coupled with temporal decomposition and a robust optimization framework, Phy-SSM is specifically designed for sparse and noisy time-series data. Evaluated on three real-world tasks—vehicle motion prediction, UAV state estimation, and epidemic spread modeling—Phy-SSM achieves significant improvements over state-of-the-art baselines in both long-term interpolation and extrapolation, demonstrating superior accuracy and robustness.

This work proposes Evolutionary Forecasting (EF), a novel paradigm for long-term time series prediction that overcomes key limitations of existing direct forecasting approaches. Conventional methods require separate models for each prediction horizon and suffer from gradient conflicts that impede local dynamics modeling at longer ranges. In contrast, EF leverages short-horizon training combined with a recursive evolution mechanism to generate long-term forecasts, subsuming direct forecasting as a degenerate special case within a unified generative framework. Theoretical analysis reveals the counterintuitive advantage of short-horizon training over long-horizon alternatives. EF thus represents a paradigm shift from static mapping to autonomous evolutionary inference. Empirical results demonstrate that a single EF model outperforms ensembles of task-specific direct forecasting models on standard benchmarks and exhibits superior asymptotic stability under extreme extrapolation scenarios.

This work addresses the limitations of existing nonlinear dynamical system modeling approaches, which often rely on linear assumptions or Koopman-based linearization and consequently struggle to accurately capture complex dynamics, leading to error accumulation in long-horizon predictions. To overcome this, the paper proposes the Neural Bilinear Dynamical Model (NBDM), which uniquely integrates Koopman theory with a bilinear dynamical structure to model state evolution in a high-dimensional latent space. NBDM further incorporates a parameterized error compensation mechanism to enhance predictive accuracy. To handle missing control inputs, a memory-augmented controller is introduced to infer implicit control signals. Experiments on five real-world datasets demonstrate that NBDM consistently outperforms baseline methods—regardless of whether control inputs are known—and achieves particularly strong performance in multi-step and long-horizon prediction tasks.

This work proposes a unified mathematical framework grounded in dynamic information flow for constructing structurally rigorous models of future prediction. By integrating filtering theory, regular conditional probabilities, Markov semigroups, infinitesimal generators, and multiple information geometries—including Hilbert, Fisher–Rao, and Wasserstein—the approach conceptualizes prediction as the construction of conditional distributions governed by informational, geometric, and modeling constraints. The framework elucidates deep connections among classical results such as the tower property and semigroup laws, as well as Itô’s formula and backward equations. Explicit transition laws, spectral decompositions, term structures, and asymptotic behaviors are derived within canonical models like Ornstein–Uhlenbeck and Cox–Ingersoll–Ross, thereby establishing a compact mathematical mapping from idealized theoretical constructs to empirical forecasting.

Real-world time series are often highly irregular and severely missing due to sensor dormancy or transmission delays. Existing methods typically assume future observation times are known, overlooking the critical question of whether future values will even be observable. This work proposes Timeflies, a novel framework that reframes time series forecasting as a joint task of inferring future observability and estimating numerical values. Timeflies employs a dual-stream architecture—comprising an observation stream and a value stream—augmented with reliability-aware embeddings, observation-guided dependency modeling, and continuous-time dynamics to explicitly co-learn the presence of observations and the evolution of underlying states. Evaluated on the newly introduced Shadow benchmark and OVJE metric, Timeflies significantly outperforms existing approaches, demonstrating that jointly modeling future observability is essential for improving forecasting performance under substantial missingness.

This study addresses the challenge of enhancing the long-term prediction accuracy and stability of deep learning surrogate models for chaotic dynamical systems while maintaining computational efficiency. By employing a unified training protocol and matching model capacity, the authors systematically evaluate the rolling prediction performance of several mainstream neural network architectures on the double pendulum, the Kuramoto–Sivashinsky equation, and Kolmogorov flow. They propose an integrator-inspired update structure that significantly reduces prediction bias and the amplification of perturbations. Stability differences among models are quantified using metrics including the Jacobian matrix, one-step relative error, and finite-time Lyapunov exponents. Experimental results demonstrate that the proposed architecture not only improves long-term predictive accuracy but also more faithfully reproduces the geometric structure of the system’s attractor.

This work addresses the challenge of preserving both temporal and cross-sectional dependency structures in probabilistic forecasting of multivariate time series. To this end, the authors propose the CRAFT framework, which constructs pairs of historical backward and forward trajectories and integrates trajectory profiling, state segmentation, and conditional analogy mechanisms. By leveraging nonparametric techniques—including singular value decomposition, change-point segmentation, trajectory clustering, and a composite compatibility scoring scheme—CRAFT faithfully retains empirical dependency structures without imposing distributional assumptions. Empirical evaluation on reproducible simulation benchmarks demonstrates that CRAFT significantly outperforms competing methods such as direct analog resampling, SVD-based analogy, unconditional bootstrapping, OLS VAR bootstrapping, and random forests, thereby achieving markedly improved accuracy in multi-step probabilistic forecasts.