KoopmanGraph

Graph Neural Networks with Koopman operator theory for spatiotemporal graph dynamics

Documentation | Quickstart | Examples | Contributing | Support | Code of Conduct

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.
• Spectral interpretability — Eigendecomposition of the learned operator with continuous-time growth rates and spatial mode shapes.
• Built on PyTorch Geometric — Native Data objects, standard GNN layers, and familiar training APIs.

Key Features

Stability mode selection: use dense or odo when you want a soft prior (odo bounds ρ(K) via the operator 2-norm but lacks a strict ε-interior certificate; continuous odo needs eigenvalue loss on the true spectrum); choose schur, dissipative, or lyapunov when you need eigenvalues mathematically forced inside the unit disk (see 11_long_horizon_stability.ipynb vs 08_loss_stability.ipynb).

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} ≈ K z_t (consistent-autoencoder style constraint).
3. Backward consistency — Inverse linear evolution in latent space (same lineage).

These losses package established deep-Koopman training ideas for graph-structured states; they are not claimed as a new theoretical contribution.

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 history is in CHANGELOG.md; 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 · Architecture · API reference

Built-in Datasets

Examples

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

Development

Run the test suite and coverage check locally:

pytest tests/ -v --cov=koopman_graph --cov-report=term-missing --cov-fail-under=90

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. For usage questions, see Support (GitHub Discussions). Community standards are in the Code of Conduct. User-facing release notes live in CHANGELOG.md.

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},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.21404269},
  url          = {https://github.com/tjkessler/KoopmanGraph},
  version      = {0.4.0},
}

License

KoopmanGraph is released under the Apache License 2.0.