arXiv:2606.11251cs.LG2026-06

用可读结构建模多变量动态系统,让关系与运动协同演化。

Mechanical Field Networks: Structured Neural Dynamics for Multivariate Systems

论文配图:Mechanical Field Networks: Structured Neural Dynamics for Multivariate Systems
图 1 · 摘自论文原文
  • 将所有变量统一到共享场态中,通过可学习的力学规律更新状态。
  • 在40维洛伦兹-96测试中,8步预测R²达0.798±0.018,局部耦合识别精度1.000。
  • 适合需要解释性动力学结构的场景,如神经信号与生态数据建模。

许多多变量动态系统仅通过轨迹可观测,其联合动态机制隐含难察。现有方法或强加可解释动力学,或学习灵活状态转移,但交互结构通常预先设定或隐含于模型中。本文提出MF-Net,一种递归动力学模型,将所有变量置于共享场态中,并通过可学习的关系律更新该状态。每个变量携带场分量,这些分量通过可学习的机械过渡共同演化。这里的‘机械’指关系驱动运动的组织方式,学习到的关系塑造状态依赖流、场响应及运动趋势,推动场态前进。结构由滚动过程本身生成:学习关系影响场的运动方式,同一内部量同时支持预测与结构读出。在已知规律交互系统、混沌基准、真实神经记录和生态时间序列上,MF-Net实现具有竞争力的短中程预测能力,并保持可解释的结构读出。在40维洛伦兹-96测试中,八步预测的R²为0.798±0.018;跨五组种子,学习关系矩阵恢复局部耦合支持,局部/非局部强度比为19.80±1.00,Precision@K为1.000±0.000。MF-Net提供一种结构可读的动力学建模框架,学习关系通过前向演化训练,在真实数据上可解释为观测条件下功能性的预测耦合。

原文摘要 · Abstract (English)

Many multivariate dynamical systems are observed only through trajectories, leaving the mechanisms governing their joint dynamics hidden. Existing approaches can impose interpretable dynamics or learn flexible state transitions, yet the resulting interaction structure is typically either specified in advance or left implicit within the learned dynamics. We introduce MF-Net, a recurrent dynamical model that represents all variables in a shared field state and updates this state through a learned relation law. Each variable carries a field component, and these components evolve jointly through a learnable mechanical transition. Here, mechanical refers to the relation-to-motion organization of the transition, where learned relations shape state-dependent flows, field responses, and motion tendencies that move the field state forward. The resulting structure is part of the rollout itself: learned relations influence how the field moves, and the same internal quantities support both forecasting and structural readout. Across known-law interaction systems, chaotic benchmarks, real neural recordings, and ecological time series, MF-Net achieves competitive short- and medium-horizon forecasting while retaining inspectable structural readout. On the 40-dimensional Lorenz--96 testbed, MF-Net achieves an eight-step $R^2$ of $0.798\pm0.018$; across five seeds, its learned relation matrix recovers the local coupling support with a local/nonlocal strength ratio of $19.80\pm1.00$ and Precision@$K$ of $1.000\pm0.000$. MF-Net provides a structure-readable dynamical modeling framework in which learned relations are trained through forward evolution and, on real data, interpreted as functional predictive couplings under appropriate observational limits.

动力系统可解释模型神经动力学时间序列建模

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