Physics-Modeled Neural Networks

2026-05-05

AI Summary

This work proposes Dynamic Physical Modeling Neural Networks (DynPMNNs), a novel architecture that enhances model interpretability and representational capacity by integrating principles from physical dynamical systems. Specifically, the hidden layers are formulated as solutions to ordinary differential equations (ODEs), replacing conventional static activation functions with time-evolving dynamical systems—such as the biologically inspired FitzHugh–Nagumo model—and embedded within an end-to-end trainable framework grounded in Reproducing Kernel Banach Space (RKBS) theory. This study presents the first instantiation of such dynamical models within neural network hidden layers and establishes theoretical connections to classical architectures under the RKBS formalism. Empirical evaluation on the California Housing dataset demonstrates that DynPMNNs achieve performance comparable to Neural ODEs and CfCs while using significantly fewer parameters, thereby confirming their modeling efficacy and expressive efficiency.

Abstract

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.

Problem

• Physics-Modeled Neural Networks
• Dynamical Systems
• Ordinary Differential Equations
• Neural ODEs
• Reproducing Kernel Banach Spaces

Innovation

• Dynamical Physics-Modeled Neural Networks
• Ordinary Differential Equations
• Reproducing Kernel Banach Spaces
• FitzHugh–Nagumo model
• Continuous-time deep learning