We introduce \emph{Dynamical Physics-Modeled Neural Networks} (DynPMNNs), a continuous-time deep learning architecture in which each hidden layer is defined as the solution of an ordinary differential equation. Unlike classical feed-forward networks, this approach replaces static activation functions with time-evolving dynamical systems, providing a biologically inspired interpretation of hidden-layer behavior and enabling the integration of physically meaningful models. The framework is rigorously grounded in Reproducing Kernel Banach Spaces (RKBSs), allowing DynPMNNs to be characterized as finite-dimensional solutions of an abstract training problem and revealing structural connections with standard neural networks. We present a concrete implementation based on the FitzHugh--Nagumo model for neuronal activation, where numerical ODE solvers are embedded into the computational graph via Euler-type schemes. Both network weights and dynamical parameters are trained jointly. Through experiments on the California Housing dataset, we compare DynPMNNs with Neural ODEs (NODEs) and Closed-form Continuous-Time Networks (CfCs). Despite using fewer trainable parameters, DynPMNNs achieve competitive performance. These results position DynPMNNs as a principled bridge between dynamical systems and deep learning, with promising directions for further research in expressivity, stability, and physics-based modeling.

翻译：我们引入了**动态物理建模神经网络**（DynPMNNs），这是一种连续时间深度学习架构，其中每个隐藏层被定义为常微分方程的解。与传统前馈网络不同，该方法用时间演化动力系统替代静态激活函数，为隐藏层行为提供了受生物学启发的解释，并能够整合具有物理意义的模型。该框架严格建立在再生核巴拿赫空间（RKBSs）基础上，使得DynPMNNs可被表征为抽象训练问题的有限维解，并揭示了其与标准神经网络的结构性联系。我们基于FitzHugh–Nagumo神经元激活模型提出了具体实现方案，通过欧拉类型格式将数值ODE求解器嵌入计算图。网络权重与动力学参数被联合训练。通过在加利福尼亚住房数据集上的实验，我们将DynPMNNs与神经常微分方程（NODEs）及闭式连续时间网络（CfCs）进行了比较。尽管使用的可训练参数更少，DynPMNNs仍实现了具有竞争力的性能。这些结果确立了DynPMNNs作为动力系统与深度学习之间原则性桥梁的地位，为后续在可表达性、稳定性及基于物理建模方面的研究指明了方向。