physarum-labs 0.2.0

Differentiable Linear Programming via Physarum dynamics

physarum labs

A clean MIT-licensed PyTorch implementation of Physarum-inspired differentiable solvers, unifying four lines of recent research:

1. Meng, Ravi, Singh (AAAI 2021) — Differentiable LP layer via Physarum dynamics
2. Solé & Pla-Mauri (arXiv Nov 2025) — Lagrangian variational framework
3. Schick et al. (PRX Life 2026) — Peristaltic mechanism (substrate inspiration)
4. Pietak & Levin (iScience 2025) — Regulatory Network Machine paradigm

What's here

• physarum_labs/lp.py — Clean MIT-licensed PyTorch reimplementation of the Physarum-inspired differentiable LP solver. Drop-in replacement for Sinkhorn-based optimal transport layers. No Magic Leap license — completely independent of the SuperGlue codebase.

• physarum_labs/variational.py — End-to-end VariationalNetworkMachine combining the LP solver with Solé's Lagrangian framework. Shows how to add transport dissipation, sparsity, and entropy regularization on top of the base solver, and how to learn edge affinities end-to-end.

• examples/cvxpy_comparison.py — Validates the Physarum solver against scipy.optimize.linprog on a range of problem sizes and iteration counts, and produces a convergence plot at assets/convergence.png.

Why this exists

The only public working implementation of Meng 2021 lives inside the yingxin-jia/Superglue-with-Physarum-Dynamics repository, which inherits Magic Leap's non-commercial academic use only license. Anyone wanting to commercialize Physarum Labs must reimplement — which is what this codebase does, cleanly.

Quick start

Install via PyPI

pip install physarum-labs

Install from source

git clone https://github.com/Physarum-Lab/physarum-labs
cd physarum-labs
pip install -e ".[dev]"  # includes scipy and matplotlib for benchmarks

Run the demos

python -m physarum_labs.lp                # LP solver self-test
python -m physarum_labs.variational       # maze, matching, learning loop
python examples/cvxpy_comparison.py       # validates against scipy LP solver

Run the tests

pytest tests/ -v

import torch
from physarum_labs import PhysarumLPLayer

# Cost matrix (lower = better match)
scores = torch.rand(4, 5)
solver = PhysarumLPLayer(unmatch_score=-1.0, max_iter=20)
transport_plan, loss = solver(scores)

# transport_plan shape: (5, 6) — augmented with bin for unmatched
# transport_plan[:-1, :-1] gives the actual matching scores

The algorithm

Physarum dynamics solve a doubly-stochastic optimal transport problem by iteratively updating a flux vector:

x_{k+1} = (1 - h) * x_k + h * W * A^T * p

where W = diag(x/c), c is the cost vector, A encodes row/column sum constraints, and p solves (A W A^T) p = 1.

This is mathematically equivalent to a steepest-descent on the LP problem (see Meng 2021), and to a least-action variational principle (see Solé & Pla-Mauri 2025). It's also the same family of dynamics that Pietak & Levin use to describe gene regulatory networks as analog computers.

License

MIT. Use freely, including commercially.

References

• Meng, Z., Ravi, S. N., & Singh, V. (2021). Physarum Powered Differentiable Linear Programming Layers and Applications. AAAI. arXiv:2004.14539

• Solé, R., & Pla-Mauri, J. (2025). Cognition as least action: the Physarum Lagrangian. arXiv:2511.08531

• Schick, L., et al. (2026). Decision-Making in Light-Trapped Slime Molds Involves Active Mechanical Processes. PRX Life. DOI: 10.1103/rv7g-d9kx

• Pietak, A., & Levin, M. (2025). Harnessing the analog computing power of regulatory networks with the Regulatory Network Machine. iScience. DOI: 10.1016/j.isci.2025.112536

• Bajpai, S., Lucas-DeMott, A., Murugan, N. J., Levin, M., & Kurian, P. (2025). Morphological computational capacity of Physarum polycephalum. arXiv:2510.19976