PHNN Theory and Port-Hamiltonian Formulation

Objectives

Understand the port-Hamiltonian formulation for dynamical systems
Learn how each sub-network (S, R, F) enforces physical constraints
Understand symmetric integration and why it matters
Know how separation of concerns enables modular inference

Prerequisites

This episode covers the mathematical foundations. You do not need to understand every equation to use the demonstrator – but understanding the structure will help you interpret results.

Port-Hamiltonian Systems

A pseudo-Hamiltonian system describes the time evolution of a state vector x as:

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

Each term has a precise physical meaning:

Term	Name	Property	Physical Role
H(x)	Hamiltonian	Scalar function	Total energy of the system
S(x)	Structure matrix	Skew-symmetric	Energy-conserving coupling
R(x)	Dissipation matrix	Positive semi-definite	Energy dissipation (damping)
F(x)	External force	Unconstrained	External energy input/output

The key insight: S being skew-symmetric guarantees that x^T S x = 0, so the conservative part never creates or destroys energy. R being positive semi-definite guarantees that energy only decreases through dissipation.

Neural Network Architecture

Each term is parameterized by a separate neural network with built-in structural constraints:

        graph TD
    X[State x] --> H_NET[H-network<br/>Energy estimator]
    X --> S_NET[S-network<br/>Skew-symmetric]
    X --> R_NET[R-network<br/>Pos. semi-definite]
    X --> F_NET[F-network<br/>External force]

    H_NET --> GRAD[grad H]
    S_NET --> STRUCT["(S - R)"]
    R_NET --> STRUCT

    STRUCT --> MULT["(S - R) * grad H"]
    MULT --> SUM["+"]
    F_NET --> SUM
    SUM --> DXDT["dx/dt"]

    style S_NET fill:#d4edda
    style R_NET fill:#fff3cd
    style F_NET fill:#d1ecf1
    style H_NET fill:#f8d7da

S-network (Skew-Symmetric)

The S-network outputs a matrix that is anti-symmetrized:

S_out = A - A^T

This guarantees S = -S^T by construction, so energy is always conserved through this channel.

R-network (Positive Semi-Definite)

The R-network outputs a matrix that passes through a positive semi-definite projection:

R_out = B * B^T

This guarantees x^T R x >= 0, so energy can only decrease through dissipation – never increase.

F-network (External Force)

The F-network has no structural constraint. It learns arbitrary state-dependent forcing that can inject or remove energy from the system.

Separation of Concerns

        graph LR
    TRAINED[Trained PHNN] --> INF1["Inference: Full system<br/>(S - R) grad H + F"]
    TRAINED --> INF2["Inference: Free system<br/>(S - R) grad H"]
    TRAINED --> INF3["Inference: New force<br/>(S - R) grad H + F'"]

    style INF1 fill:#d4edda
    style INF2 fill:#fff3cd
    style INF3 fill:#d1ecf1

Because each sub-network has a physical interpretation, you can:

Remove the external force at inference time to study the free (unforced) system
Swap in a different forcing function without retraining the model
Inspect the learned Hamiltonian to verify the energy landscape is plausible
Analyze dissipation to check which states lose energy and at what rate

This is impossible with a standard black-box neural network.

Symmetric Integration

Training uses a symmetric fourth-order integration scheme rather than standard Euler or RK4. This matters because:

Lower-order integrators (Euler, RK4) introduce systematic bias in energy estimates
With sparse or noisy training data, this bias corrupts the learned Hamiltonian
Symmetric integrators preserve the geometric structure of Hamiltonian systems

The phlearn package provides midpoint and symmetric integrators via the train() function.

Energy Budget

For a well-trained PHNN, the energy budget at each time step satisfies:

dH/dt = -grad_H^T * R * grad_H + grad_H^T * F
         \___ dissipation ___/   \__ forcing __/

Without external force: dH/dt <= 0 (energy always decreases or stays constant)
With external force: energy can increase or decrease depending on F

This is the fundamental guarantee that makes PHNNs physically valid.

Keypoints

Port-Hamiltonian systems decompose dynamics into conservative, dissipative, and external force terms
The S-network is anti-symmetrized by construction, guaranteeing energy conservation
The R-network is projected to positive semi-definite, guaranteeing non-negative dissipation
Symmetric integration preserves geometric structure during training
Separation of concerns allows modifying forces at inference time without retraining
The energy budget dH/dt = -grad_H^T R grad_H + grad_H^T F is guaranteed by architecture