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Physics-Modeled Neural Networks

topic: othertop score: 90released: 2026-05-12first surfaced: 2026-05-12arXivPDFlinked_to_results2026-05-12

Authors: Raul Felipe-Sosa, Angel Martin del Rey, Maria Flores Ceballos

arXiv · PDF

Summary

The authors introduce Dynamical Physics-Modeled Neural Networks (DynPMNNs), where each hidden layer is the solution of an ordinary differential equation (ODE) instead of a static activation function. They ground the framework in Reproducing Kernel Banach Spaces, implement it using the FitzHugh–Nagumo neuronal model with Euler-type ODE solvers embedded in the computational graph, and train both network weights and dynamical parameters jointly. On the California Housing dataset, DynPMNNs achieve competitive performance with fewer trainable parameters than Neural ODEs and Closed-form Continuous-Time Networks.

Main takeaways:

  • Each hidden layer is defined as the time-evolving solution of an ODE, replacing static activations with dynamical systems inspired by biology/physics.
  • The framework is formalized using Reproducing Kernel Banach Spaces, connecting it rigorously to standard neural network theory.
  • A concrete implementation uses the FitzHugh–Nagumo model (a classic neuronal activation model) with numerical ODE solvers embedded in the training graph.
  • On California Housing, DynPMNNs match or beat Neural ODEs and CfCs despite using fewer parameters.

Relevance

Not related to my persona/midtraining work — this is a neural architecture paper focused on ODE-based hidden layers for tabular regression, with no obvious connection to behavior installation, steering, or prompt-based conditioning.

Abstract

arXiv:2605.08176v1 Announce Type: new 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.