state-space modeling

Existing dynamical system forecasting models suffer from significant limitations in long-horizon prediction accuracy, modeling of long-range dependencies, capture of chaotic evolution, and extrapolation under scarce-data conditions. This paper introduces the first Mamba-enhanced state-space model (SSM) for physics-informed machine learning (PIML), innovatively embedding Mamba’s structured state evolution mechanism into a neural operator framework while integrating parameter remapping and quantitative systems pharmacology priors. We design a rigorous extrapolation benchmark encompassing chaotic systems and multiscale dynamics to systematically address generalization bottlenecks under long-range dependencies, strong nonlinearity, and data scarcity. Experiments demonstrate state-of-the-art performance across diverse interpolation and extrapolation tasks with the lowest computational overhead. In real-world evaluation of anticancer drug efficacy, the method achieves highly robust predictions using only minimal clinical data—marking a critical advance in interpretable, sample-efficient PIML for complex biological dynamics.

This work addresses the fundamental question of how state space models (SSMs) can perform zero-shot in-context learning to predict the next state of arbitrary dynamical systems without parameter fine-tuning. Method: We propose HiPPO-Prophecy, a novel weight construction method for SSMs grounded in the HiPPO framework, enabling both continuous- and discrete-time modeling. Our approach theoretically establishes that continuous SSMs can asymptotically approximate the derivative of any input signal, and we derive provable next-state prediction guarantees for discrete SSMs. Through rigorous signal-theoretic analysis and asymptotic error bound derivation, we obtain an explicit upper bound on the derivative approximation error. Results: Experiments demonstrate high-accuracy zero-shot state prediction across diverse dynamical systems. This work provides the first theoretical characterization of SSMs’ capacity for modeling dynamical system evolution in zero-shot settings, significantly enhancing both their theoretical interpretability and practical applicability.

Subgrid-scale errors in chaotic dynamical systems modeling induce long-term prediction instability and statistical distortions. Method: This paper proposes an explicit bidirectional (fine-to-coarse and coarse-to-fine) multiscale data-driven modeling framework that integrates dynamical system embedding, chaotic time-series modeling, and physics-constrained learning. Contribution/Results: It introduces the first neural architecture explicitly designed to preserve both multiscale information flow and numerical stability. Applied to climate subgrid-scale parameterization, the method enables physically consistent error correction. Experiments demonstrate substantial improvements in long-term forecast stability and statistical fidelity—quantified via invariant measures and temporal correlations—while effectively mitigating the “missing physics” problem arising from unresolved small-scale processes in baseline chaotic models.

Addressing the challenge of balancing long-term prediction stability and computational efficiency in modeling long sequences, this paper proposes the Linear Oscillatory State Space (LinOSS) model. LinOSS is grounded in the forced harmonic oscillator differential equation and employs a nonnegative diagonal state matrix to ensure lightweight, intrinsic stability. It leverages implicit-explicit discretization and a fast associative parallel scan algorithm to enable efficient, scalable sequence processing. Theoretically, LinOSS is proven to possess universal function approximation capability and preserve time-reversal symmetry. In long-range forecasting tasks up to 50k steps, LinOSS achieves nearly 2× higher accuracy than Mamba and 2.5× higher than LRU, significantly outperforming existing state-of-the-art methods.

This paper addresses discrete-time interconnected systems whose subsystem dynamics and interconnection topology are partially unknown. Method: We propose a data-driven, compositional approach to construct finite-state abstractions for formal verification and distributed controller synthesis. Subsystems are modeled individually from input-output data, and—novelly—the unknown static interconnection mapping is treated as a learnable object, enabling its symbolic abstraction. Compositionality and rigorous error propagation analysis ensure that the resulting abstraction strictly satisfies an approximate simulation relation. Contribution/Results: We theoretically establish scalability and verifiability of the abstraction. Experiments demonstrate substantial mitigation of the curse of dimensionality, enabling high-precision, low-complexity controller synthesis while preserving formal guarantees.

This work addresses the lack of verifiability in learned world models when deployed in high-assurance systems by proposing a novel framework that integrates classical model order reduction (MOR) with modern world modeling. The approach combines proper orthogonal decomposition (POD) with an encoder–decoder architecture, incorporates physics-informed error bounds derived from physical priors, and employs measurement-driven action-conditioned modeling to ensure verifiable closed-loop predictions, exceptional data efficiency, and physical consistency. By systematically unifying MOR theory with contemporary world model paradigms, this study establishes a new modeling methodology that simultaneously achieves reliability and performance for safety-critical applications.

The theoretical foundations underlying single-layer Transformer modeling of time-series data remain poorly understood, particularly regarding its representational capacity and inherent limitations in capturing dynamical processes. Method: We formulate causal self-attention as a linear, history-dependent recurrence relation and analyze it through the lens of dynamical systems theory and delay embedding theory, conducting both linear and nonlinear case studies. Contribution/Results: We establish that the convexity constraint imposed by softmax attention induces systematic distortion in modeling linear oscillatory systems—a previously unrecognized limitation. Conversely, we demonstrate that Transformers can autonomously perform delay embedding and state reconstruction in partially observable nonlinear systems. Our analysis precisely characterizes the boundary conditions under which Transformers succeed or fail in time-series modeling and identifies the fundamental determinants of zero-shot forecasting performance. These findings provide critical theoretical grounding for developing trustworthy, principled time-series models.

Traditional time series forecasting is constrained by static, one-shot, model-centric paradigms that struggle to support dynamic reasoning and continual learning. This work proposes Agent-based Time Series Forecasting (ATSF), introducing an agent framework into the field for the first time and reconceptualizing prediction as a multi-round workflow encompassing perception, planning, action, reflection, and memory. ATSF enables tool invocation, feedback integration, and experiential evolution. Through three implementation pathways—workflow-based design, agent reinforcement learning, and hybrid agent architectures—ATSF establishes a novel forecasting paradigm that is interactive, evolvable, and supports iterative refinement. This study not only opens an agent-oriented research direction for time series forecasting but also systematically articulates its technical pathways, key challenges, and future opportunities.

This work addresses a fundamental limitation in current time series modeling approaches: their general lack of grounding in the underlying dynamical systems, which impedes long-term statistical forecasting, generalization to unseen regimes (e.g., critical transitions), and sample-efficient learning. The paper presents the first systematic argument for the foundational value of a dynamical systems perspective in time series modeling and introduces a novel paradigm—Dynamical System Reconstruction (DSR)—that infers latent dynamical mechanisms directly from observational data. This approach substantially enhances model interpretability, generalization capability, and computational efficiency, enabling reliable long-horizon prediction, theoretical performance bound analysis, and effective modeling under low-data regimes. The framework provides both theoretical foundations and practical pathways toward next-generation foundation models for time series.

Existing probabilistic programming languages lack native support for dynamic systems—particularly state-space models—hindering the broader adoption of Bayesian methods in this domain. This work introduces dynestyx, a library that provides first-class, unified, and user-friendly support for state-space models within a probabilistic programming framework. dynestyx enables flexible specification of priors, accommodates both discrete- and continuous-time dynamics, handles mixed-effects data, and facilitates joint Bayesian inference over latent states and model parameters with full uncertainty quantification. By doing so, this contribution substantially enhances the accessibility, flexibility, and practical utility of dynamic system modeling across statistics, signal processing, and machine learning.