Introduction to Pseudo-Hamiltonian Neural Networks

Objectives

• Understand what Pseudo-Hamiltonian Neural Networks (PHNNs) are

• Learn why physics-preserving structure matters for dynamical systems

• Know the 3-component decomposition: conservative + dissipative + external force

• Understand the difference between PHNNs and standard neural networks

• Know the repository structure and key references

Why This Matters

The Scenario: An engineer simulating a damped mechanical system notices that a standard neural network predicts energy increasing over time – a physical impossibility for a dissipative system. The model fits short-horizon training data well but diverges on longer predictions.

The Research Question: Can we build neural networks that respect energy conservation and dissipation by design, so that long-horizon predictions remain physically valid – even when external forces change?

What This Episode Gives You: The big picture – how PHNNs decompose dynamics into three physically meaningful sub-networks, why this architecture outperforms standard approaches, and what the repository contains.

Overview

Standard neural networks trained on physical systems have no built-in notion of energy conservation or dissipation. Over long time horizons, they drift into physically impossible states – predicting perpetual motion, energy creation, or unbounded trajectories.

Pseudo-Hamiltonian Neural Networks (PHNNs) solve this by decomposing system dynamics into three physically meaningful components:

• Conservative sub-network – captures energy-preserving Hamiltonian dynamics (skew-symmetric structure)

• Dissipative sub-network – models energy loss through damping, friction, or heat transfer (positive semi-definite)

• External force sub-network – learns state-dependent forcing terms that can be modified or removed at inference time

Each component is rooted in port-Hamiltonian theory and is physically interpretable. The framework supports both ODEs (mechanical systems, circuits) and PDEs (wave equations, reaction-diffusion).

PHNN vs Standard Neural Network

Property	Standard NN	PHNN
Energy conservation	Not guaranteed	Built into architecture
Long-horizon stability	Degrades rapidly	Physically valid
Force modification	Requires retraining	Swap at inference time
Interpretability	Black box	Each sub-network has physical meaning
Training data needed	More	Less (physics provides inductive bias)

3-Component Decomposition

The core equation of a pseudo-Hamiltonian system:

dx/dt = (S(x) - R(x)) * grad_H(x) + F(x)

Where:

• H(x) – Hamiltonian (total energy), learned by the energy network

• S(x) – Skew-symmetric matrix (energy-conserving structure), learned by the S-network

• R(x) – Positive semi-definite matrix (dissipation), learned by the R-network

• F(x) – External force, learned by the F-network

Repository Structure

Component	Location
phlearn package	phlearn/ (SINTEF’s PHNN library)
ODE systems	phlearn/phlearn/phsystems/ode/
PDE systems	phlearn/phlearn/phsystems/pde/
Neural network architectures	phlearn/phlearn/phnns/
Example notebooks	example_scripts/
Demonstrator notebook	demonstrator-v1.orchestrator.ipynb
Setup scripts	setup.sh, vm-init.sh

Using AI Coding Assistants

If you are using an AI coding assistant, the repository includes an AGENT.md file with setup instructions. Tell your assistant:

“Read AGENT.md and help me run the PHNN demonstrator on my NAIC VM.”

What You Will Learn

Episode	Topic
02	Provisioning a NAIC VM
03	Setting up the environment
04	PHNN theory and port-Hamiltonian formulation
05	ODE systems: mass-spring, training, and comparison
06	Running the demonstrator notebook
07	PDE extensions: KdV, Cahn-Hilliard, BBM
08	FAQ and troubleshooting

References

• Eidnes, S., Stasik, A.J., Sterud, C., Benth, E., and Lye, K.O. (2023). Pseudo-Hamiltonian neural networks for learning partial differential equations. Journal of Computational Physics, 500, 112738.

• Eidnes, S. and Lye, K.O. (2024). Pseudo-Hamiltonian neural networks with state-dependent external forces. Applied Mathematics and Computation, 459, 128242.

• SINTEF Digital – phlearn package: https://github.com/SINTEF/pseudo-hamiltonian-neural-networks

Keypoints

• PHNNs decompose dynamics into conservative, dissipative, and external force components

• Each sub-network has built-in structural constraints guaranteeing physical validity

• Standard NNs drift into impossible states; PHNNs remain valid over long horizons

• External forces can be modified or removed at inference time without retraining

• The phlearn package by SINTEF provides the reference implementation for both ODEs and PDEs

• All code, examples, and the demonstrator notebook are included in this repository