PDE Extensions

Objectives

  • Understand how PHNNs extend from ODEs to PDEs

  • Know the supported PDE systems: KdV, Cahn-Hilliard, BBM, KdV-Burgers

  • Learn about conservation properties for spatially extended systems

  • Run PDE example notebooks

From ODEs to PDEs

The port-Hamiltonian framework extends naturally to spatially extended systems. Instead of a finite state vector x, PDE systems operate on fields u(x, t) discretized on a spatial grid.

The structure is the same:

du/dt = (S - R) * delta_H/delta_u + F

Where S, R, and H now operate on the discretized spatial field. The phlearn package handles spatial derivatives, boundary conditions, and grid discretization internally.

Supported PDE Systems

KdV (Korteweg-de Vries)

u_t + u * u_x + u_xxx = 0

  • Models shallow water waves and soliton dynamics

  • Conserves mass and energy in the Hamiltonian formulation

  • See example_scripts/kdv_example.ipynb

Cahn-Hilliard

u_t = div(M * grad(mu)),    mu = -eps^2 * laplacian(u) + f'(u)

  • Models phase separation and spinodal decomposition

  • Conserves total mass; free energy decreases monotonically

  • See example_scripts/cahn_hilliard_example.ipynb

BBM (Benjamin-Bona-Mahony)

u_t + u_x + u * u_x - u_xxt = 0

  • Regularized alternative to KdV for long wave propagation

  • Better dispersion properties than KdV for numerical computation

  • See example_scripts/bbm_example.ipynb

KdV-Burgers

u_t + u * u_x + u_xxx - nu * u_xx = 0

  • Combines KdV dispersion with Burgers diffusion

  • Models dispersive-diffusive wave propagation

  • See example_scripts/kdv_burgers_example.ipynb

Additional Systems

The phlearn package also includes:

System

Module

Description

Heat equation

heat_system.py

Diffusion on 1D domain

Allen-Cahn

allen_cahn_system.py

Reaction-diffusion with bistable potential

Perona-Malik

perona_malik_system.py

Nonlinear diffusion (image processing)

Conservation Properties

A key advantage of the PHNN-PDE framework is that conservation laws are respected by the architecture:

Property

Standard NN

PHNN-PDE

Mass conservation

Approximate

Exact (skew-symmetric S)

Energy dissipation

Not guaranteed

Guaranteed (positive semi-definite R)

Soliton preservation

Degrades over time

Maintained

Long-time stability

Blows up

Bounded

For the KdV equation, the PHNN preserves both mass and energy to near machine precision. For Cahn-Hilliard, total mass is conserved while free energy decreases monotonically.

Running PDE Examples

cd ~/pseudo-hamiltonian-neural-networks
source venv/bin/activate
jupyter lab --no-browser --ip=127.0.0.1 --port=8888

The comprehensive PDE tutorial is example_scripts/phnn_pde_examples.ipynb. Individual system notebooks provide focused examples for each equation.

GPU Recommended

PDE training involves larger state vectors (spatial grids of 64-256 points) and benefits significantly from GPU acceleration. On CPU, PDE examples may take 30-60 minutes; on GPU (L40S), they typically complete in 10-15 minutes.

PDE System Implementation

The phlearn PDE systems are located in phlearn/phlearn/phsystems/pde/. Each system defines:

  • The Hamiltonian functional H[u]

  • The structure operator S (typically a spatial derivative operator)

  • The dissipation operator R (if applicable)

  • Boundary conditions (periodic or non-periodic)

  • Spatial discretization parameters

Keypoints

  • PHNNs extend naturally from ODEs to PDEs via discretized port-Hamiltonian operators

  • Supported systems: KdV, Cahn-Hilliard, BBM, KdV-Burgers, heat, Allen-Cahn, Perona-Malik

  • Conservation properties (mass, energy) are enforced by the network architecture

  • PDE training benefits from GPU acceleration due to larger state vectors

  • The phnn_pde_examples.ipynb notebook provides a comprehensive PDE tutorial

  • Individual notebooks exist for each PDE system in example_scripts/