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How Petri Nets Make Predictions

From discrete tokens to continuous differential equations — and why the structure of a net tells you more than you'd expect.

Two Ways to Run a Petri Net

Every Petri net can be executed in two fundamentally different ways:

The continuous version turns any Petri net into a system of ordinary differential equations (ODEs). You've seen ODEs before if you've studied physics or population biology — they describe how quantities change over time.

Mass-Action Kinetics: The Bridge

The rule for converting a Petri net to ODEs comes from chemistry, where it's called mass-action kinetics:

For a transition with rate k and two input places with token counts x and y:

Each place's token count changes based on the transitions connected to it:

This is exactly how chemical kinetics works, which is why Petri nets model chemistry so naturally. But the same math works for any system with competing flows.

Example: Predator-Prey

The Predator-Prey demo has just two places (prey, predator) and three transitions:

• prey_reproduce (rate 1.0): prey → 2× prey
• predation (rate 0.01): prey + predator → 2× predator
• predator_death (rate 0.6): predator → (nothing)

The ODE system is:

These are the Lotka-Volterra equations — one of the most famous ODE systems in ecology. We didn't write them by hand. They emerged automatically from the Petri net structure.

See the population oscillations →

How Tic-Tac-Toe Uses This

This is where it gets surprising. The Tic-Tac-Toe demo has 34 places and 36 transitions. The ODE simulation doesn't "play" the game. Instead, it reveals the strategic value of each cell.

Here's why it works:

1. Board cells are places. Each of the 9 cells has three places: empty, X, and O.
2. Moves are transitions. Playing X in cell (0,0) consumes a token from "empty_00" and produces one in "x_00".
3. Win patterns are transitions too. A transition like "x_win_row0" requires tokens in x_00, x_01, and x_02 — all three cells in the top row.
4. ODE simulation lets everything flow. Token counts become fractional. Cells that participate in more win patterns attract more flow.

No game tree search. No minimax algorithm. The topology of the net — which places connect to which transitions — encodes the strategic structure. The ODE solver just reads it out.

Watch the ODE reveal strategy →

The Pattern

This same technique works across every ODE demo on this site:

• Coffee Shop — ODE predicts when each ingredient runs out based on order rates and restock capacity
• Knapsack — items compete for limited capacity via mass-action kinetics; the ODE converges on the optimal selection
• Enzyme Kinetics — the classic Michaelis-Menten saturation curve emerges from three transitions
• Thermostat — ODE shows temperature oscillation around the setpoint under bang-bang control
• Vet Clinic — piecewise ODE predicts patient throughput, resource bottlenecks, and revenue over a workday

In every case, you define the structure (places, transitions, arcs, rates) and the ODE solver reveals the behavior (curves, equilibria, oscillations, bottlenecks).

What Makes This Different

Most simulation tools require you to write equations by hand. With Petri nets:

• The structure is the model. Draw circles and rectangles, connect them with arrows.
• The equations are derived. Mass-action kinetics generates the ODE system automatically.
• The same model works both ways. Step through it discretely to understand the logic, then run the ODE to see long-term behavior.

Try the ODE Solver

Open any model in the Petri Net Viewer and click the play button (▶) to run the ODE simulation. You'll see token counts evolve as smooth curves. Adjust transition rates and initial token counts to explore different scenarios.

Open the Petri Net Viewer →