koopman-graph 0.1.0

Graph Neural Networks with Koopman operator theory for spatiotemporal graph dynamics

KoopmanGraph

Graph Neural Networks with Koopman operator theory for spatiotemporal graph dynamics

Documentation | Quickstart | Examples | Contributing

---

KoopmanGraph is an open-source PyTorch library that combines Graph Neural Networks (GNNs) with Koopman operator theory to model spatiotemporal dynamics on graphs. Instead of treating node states as flat vectors, KoopmanGraph lifts features into a latent space with topology-aware encoders, advances them via a learned linear Koopman operator, and decodes predictions back to physical node features.

The result is a topology-aware alternative to vector-based Koopman methods — well suited for smart grids, traffic networks, epidemic modeling, and other networked dynamical systems.

Why KoopmanGraph?

Koopman theory encodes nonlinear dynamics into a linear domain where evolution is simple matrix multiplication and spectral analysis reveals system behavior. Existing deep Koopman packages often ignore graph structure, while GNN forecasting methods typically lack explicit linear latent dynamics.

KoopmanGraph bridges that gap:

• Topology-aware lifting — GCN and GAT encoders propagate information along edges before Koopman evolution.
• Explicit linear dynamics — A learnable finite-dimensional Koopman matrix K governs latent evolution.
• Multi-step forecasting — Roll out future graph snapshots from a single initial state.
• Built on PyTorch Geometric — Native Data objects, standard GNN layers, and familiar training APIs.

Key Features

Feature	Description
GraphKoopmanModel	End-to-end encode → Koopman advance → decode pipeline
GNNEncoder / GATEncoder	Topology-aware latent lifting with GCN or multi-head attention
KoopmanOperator	Learnable linear propagator with configurable initialization
Consistency losses	Forward and backward latent linearity constraints
GraphSnapshotSequence	Time-ordered container for PyG graph snapshots
Benchmark datasets	Synthetic, grid, IEEE 118-bus, and METR-LA traffic benchmarks
Jupyter tutorials	Nine end-to-end notebooks with real networked datasets
Tested & documented	≥80% coverage enforced in CI, Sphinx docs on Read the Docs

Architecture

Each prediction step follows three stages:

  Node features x_t          Latent state z_t           Predicted x_{t+1}
  (N × F, on graph)    →    (N × d, on graph)     →    (N × F, on graph)

       ┌──────────┐              ┌──────────┐              ┌──────────┐
  x_t  │  GNN     │  z_t         │ Koopman  │  z_{t+1}     │  GNN     │  x_{t+1}
  ───► │ Encoder  │ ───►   ───►  │    K     │ ───►   ───►  │ Decoder  │ ───►
       └──────────┘              └──────────┘              └──────────┘
         (lifting)              (linear step)              (reconstruction)

During training, the model minimizes:

1. Reconstruction — Autoencoder fidelity between input and decoded node features.
2. Forward consistency — Latent states should satisfy z_{t+1} \approx K z_t.
3. Backward consistency — Inverse linear evolution in latent space.

Installation

KoopmanGraph requires Python 3.10+, PyTorch, and PyTorch Geometric. Install those first, then install KoopmanGraph:

pip install koopman-graph

For development from source:

git clone https://github.com/tjkessler/KoopmanGraph.git
cd KoopmanGraph
pip install -e ".[dev]"

For documentation builds:

pip install -e ".[docs]"
cd docs && make html

See the installation guide for platform-specific PyTorch/PyG wheels and verification steps. Release workflow and version policy are documented in CONTRIBUTING.md.

Quickstart

Train a model on a synthetic spatiotemporal graph and predict five future snapshots:

import torch
from koopman_graph import GNNDecoder, GNNEncoder, GraphKoopmanModel
from koopman_graph.datasets import SyntheticDynamicGraphBenchmark

data_sequence = SyntheticDynamicGraphBenchmark.generate(
    num_nodes=20,
    num_timesteps=30,
    in_channels=3,
    seed=42,
    noise_std=0.01,
)

encoder = GNNEncoder(3, 64, 64)
decoder = GNNDecoder(64, 64, 3)
model = GraphKoopmanModel(
    encoder=encoder,
    decoder=decoder,
    latent_dim=64,
    time_step=0.1,
)

torch.manual_seed(0)
history = model.fit(data_sequence, epochs=20, lr=1e-3)
future_graphs = model.predict(data_sequence[0], steps=5)

print(f"Final loss: {history.loss[-1]:.6f}")
print(f"Predicted {len(future_graphs)} snapshots, shape: {future_graphs[0].x.shape}")

Expected output:

Final loss: <float>
Predicted 5 snapshots, shape: torch.Size([20, 3])

More detail: Quickstart guide · API reference

Built-in Datasets

Benchmark	Domain	Description
SyntheticDynamicGraphBenchmark	Synthetic	Laplacian diffusion on path/ring graphs
GridDynamicGraphBenchmark	Synthetic	Laplacian diffusion on a 4-connected 2D lattice
AnisotropicAdvectionGridBenchmark	Synthetic	Directional advection with asymmetric edge weights
IEEE118DynamicBenchmark	Power systems	IEEE 118-bus topology with simulated voltage/load dynamics
MetrLaTrafficBenchmark	Traffic	METR-LA sensor graph with cached speed snapshots

Examples

Jupyter tutorials in the examples/ directory cover training, evaluation, and analysis workflows:

Notebook	Topic
01_synthetic_graph.ipynb	End-to-end synthetic graph dynamics
02_ieee118_bus.ipynb	IEEE 118-bus power network
03_traffic_network.ipynb	METR-LA traffic forecasting
04_grid_attention.ipynb	GAT encoder on grid graphs
05_custom_data.ipynb	Bring your own graph sequences
06_epidemic_ring.ipynb	Epidemic spread on ring topologies
07_koopman_spectrum.ipynb	Koopman eigenvalue analysis
08_loss_stability.ipynb	Loss weighting and training stability
09_topology_ablation.ipynb	Topology ablation study

Development

Run the test suite and coverage check locally:

pytest tests/ -v --cov=koopman_graph --cov-report=term-missing --cov-fail-under=80

Lint and format:

ruff check src/ tests/
ruff format --check src/ tests/

See CONTRIBUTING.md for the full development workflow, pre-commit hooks, and pull request guidelines.

Citation

If you use KoopmanGraph in your research, please cite the repository:

@software{koopmangraph2026,
  author       = {Travis Kessler},
  title        = {KoopmanGraph: Graph Neural Networks with Koopman Operator Theory},
  year         = {2026},
  url          = {https://github.com/tjkessler/KoopmanGraph},
  version      = {0.1.0},
}

License

KoopmanGraph is released under the Apache License 2.0.