UC3 — Pseudo-Hamiltonian Neural Networks

Objectives

  • Understand why standard neural networks fail at long-horizon dynamical system modeling

  • Learn about port-Hamiltonian decomposition into conservation, dissipation, and external forces

  • See how physics structure enables interpretability and out-of-distribution generalization

  • Know the status of SINTEF’s phlearn package integration

Repository: wp7-UC3-pseudo-hamiltonian-neural-networks Reference Implementation: github.com/SINTEF/pseudo-hamiltonian-neural-networks Contributors: Sølve Eidnes, Kjetil Olsen Lye (SINTEF Digital)

The Problem

Standard neural networks trained to model physical systems learn to predict the next state but have no built-in notion of:

  • Energy conservation — total energy should be preserved in closed systems

  • Dissipation — energy should decay predictably due to friction and damping

  • External forcing — energy input from external sources should be separated

Without these constraints, neural networks can produce physically implausible trajectories, especially over long time horizons.

Approach: Port-Hamiltonian Decomposition

UC3 decomposes system dynamics into three physically meaningful components, each modeled by a separate sub-network:

        graph TD
    A[System State x] --> B[Conservation Network<br>Energy-preserving Hamiltonian]
    A --> C[Dissipation Network<br>Energy loss and damping]
    A --> D[External Force Network<br>State-dependent forcing]
    B --> E[dx/dt]
    C --> E
    D --> E

Component

Physics Role

Sub-Network

Conservation

Energy-preserving Hamiltonian dynamics

Skew-symmetric structure

Dissipation

Energy loss (friction, damping)

Positive semi-definite structure

External Force

State-dependent forcing terms

General neural network

This decomposition is rooted in port-Hamiltonian theory and ensures that each learned component is physically interpretable. A researcher can inspect what the model attributes to dissipation versus external forcing, for example.

Key Innovations

  • Symmetric fourth-order integration schemes improve training with sparse and noisy data

  • Decomposable architecture means each component can be inspected independently

  • Modified dynamics: learned models remain valid when external forces are changed or removed — standard neural networks cannot do this

Results

The approach outperforms standard neural networks on dynamical systems benchmarks:

Benchmark

Description

Forced/damped mass-spring

Classical mechanics with dissipation

Complex tank systems

Fluid dynamics with multiple interacting tanks

PDEs

Partial differential equations with conservation laws

Reference publications (prior work by SINTEF):

  • Eidnes et al., Journal of Computational Physics (2023)

  • Eidnes et al., Applied Mathematics and Computation (2024)

The phlearn Package

The reference implementation is maintained by SINTEF as the open-source phlearn Python package:

pip install -e phlearn/

The package provides:

  • Pre-built PHNN architectures

  • Training loops with physics-aware losses

  • Integration schemes (symplectic, symmetric fourth-order)

  • Example notebooks for standard benchmarks

Status

UC3 is led by SINTEF, building on their prior research. The WP7 repository integrates the phlearn package with a full test suite and CI/CD pipeline.

Like UC2, UC3 shows that embedding physics into the architecture produces models that generalize better than pure data-driven alternatives.

Quick Start

git clone https://github.com/NAICNO/wp7-UC3-pseudo-hamiltonian-neural-networks.git
cd pseudo-hamiltonian-neural-networks
pip install -e phlearn/
pytest tests/ -v

Keypoints

  • Standard neural networks lack notions of energy conservation, dissipation, and forcing

  • PHNNs decompose dynamics into three physically interpretable components

  • Port-Hamiltonian structure ensures each component has the correct mathematical properties

  • Learned models remain valid when external forces are modified or removed

  • Based on prior work published in Journal of Computational Physics (2023) and Applied Mathematics and Computation (2024)

  • The phlearn package provides ready-to-use PHNN architectures and training utilities