Mechanical Field Networks: Structured Neural Dynamics for Multivariate Systems
Mechanical Field Networks: Structured Neural Dynamics for Multivariate Systems
机械场网络：用于多变量系统的结构化神经动力学
Abstract: 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.
摘要： 许多多变量动力系统仅能通过轨迹进行观测，这使得支配其联合动力学的机制处于隐藏状态。现有的方法要么强加可解释的动力学，要么学习灵活的状态转换，然而由此产生的交互结构通常要么是预先指定的，要么在学习到的动力学中保持隐式。
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.
我们引入了 MF-Net，这是一种循环动力学模型，它将所有变量表示在一个共享的场状态中，并通过学习到的关系定律来更新该状态。每个变量携带一个场分量，这些分量通过可学习的机械转换共同演化。在此，“机械”指的是转换中“关系到运动”的组织方式，其中学习到的关系塑造了状态相关的流、场响应以及推动场状态向前发展的运动趋势。
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.
由此产生的结构本身就是演化过程的一部分：学习到的关系影响场的运动方式，而相同的内部量既支持预测，也支持结构读取。在已知定律的交互系统、混沌基准测试、真实神经记录和生态时间序列中，MF-Net 在实现具有竞争力的短中期预测的同时，保留了可检查的结构读取能力。
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.
在 40 维的 Lorenz—96 测试平台上，MF-Net 实现了 $0.798\pm0.018$ 的八步 $R^2$；在五个随机种子下，其学习到的关系矩阵恢复了局部耦合支持，局部/非局部强度比为 $19.80\pm1.00$，Precision@$K$ 为 $1.000\pm0.000$。MF-Net 提供了一个结构可读的动力学建模框架，其中学习到的关系通过前向演化进行训练，并在真实数据上，在适当的观测限制下被解释为功能性预测耦合。