Hybrid Models: Rules Plus Learning
Flow-Lenia, published in December 2022 (arXiv:2212.07906), modified a single assumption in Bert Chan’s Lenia system and changed what the system could do. In standard Lenia, the update rule parameters — the growth function shape, the kernel radius, the time step — are set globally: every cell in the grid runs the same physics. Flow-Lenia made those parameters local. Each cell carries its own parameter values as part of its state vector. Parameters can diffuse between cells. The parameters themselves evolve, governed by the same update dynamics as everything else.
The result: a system where multiple “species” of self-organizing organisms — each governed by slightly different local rules — could coexist in the same grid, interact, and compete. The CA rules themselves had become evolvable. But the parameters were not hand-specified; they were learned by gradient descent to produce target behaviors. Flow-Lenia sits at the intersection of two design principles that are usually kept separate: the structural rules of cellular automata and the learned parameters of neural networks.
That intersection is becoming a research area.
The Architecture
The general pattern is straightforward. A base system provides structural rules: a CA neighborhood topology, an ABM interaction graph, or the conservation laws and symmetries of a physical system. These rules are fixed. Embedded within that structure are parameters that are learned rather than set: diffusion rates, agent preference weights, kernel shapes, coupling constants. A training procedure — gradient descent, evolutionary algorithms, or Bayesian optimization — adjusts the learned parameters to minimize a loss function measuring the gap between the system’s behavior and some target.
The family is broad. Neural ODEs (Chen et al., 2018) are differential equations where the right-hand side is parameterized by a neural network — the structural form (it’s an ODE, it respects conservation of probability) is fixed, and the dynamics are learned. Physics-informed neural networks (PINNs) take the same approach: the network must satisfy known physical constraints (incompressibility, boundary conditions, thermodynamic laws) while learning to predict within those constraints. Flow-Lenia applies the principle to cellular automata specifically.
In each case, the architecture exploits domain knowledge twice. Fixing the structural rules reduces the hypothesis space: you’re not searching over all possible functions, but over functions consistent with known constraints. The structural rules also make the learned parameters meaningful — because the framework is interpretable, a diffusion constant learned from data refers to something concrete.
What Hybrid Models Gain
The realistic range is the primary gain. A pure CA with hand-specified rules can capture the qualitative character of a phenomenon — reaction-diffusion produces Turing patterns that resemble zebrafish stripes. But matching the specific spatial frequency, the defect density, the transition dynamics requires parameter values that are hard to set manually. A hybrid system learns those values from data.
Flow-Lenia demonstrated this with complexity rather than realism: the locally-varying parameters allowed the system to sustain genuinely new phenomena — multi-species coexistence, parameter gradients, organisms that modify their own local physics as they move. None of this was achievable with globally-fixed Lenia parameters. The added degrees of freedom opened new parts of the behavior space.
In scientific applications, the gain is data-fitting power. Reaction-diffusion models of skin patterning have fixed structural form (two coupled PDEs) but unknown rate constants. Fitting those constants against observed patterns requires a hybrid approach: the structural constraints reduce the search space, and learning finds the parameters. The result is a model that both explains qualitatively (Turing mechanism) and predicts quantitatively (specific stripe spacing).
The Attribution Problem
When interesting behavior emerges from a hybrid model, it is not always clear what caused it.
In a pure CA, the mechanism is explicit. Life’s gliders emerge from the B3/S23 rule: birth with exactly three neighbors, survival with two or three. You can verify this by modifying the rule and observing what changes. The causal structure is transparent.
In a hybrid model, the structural rules and the learned parameters are both necessary. Removing either changes the system in ways that are not independent. If you ablate the learned parameters (set them to uniform values), you change the structural constraints that the rules were operating within. If you change the structural rules, the learned parameters no longer have the same meaning — the diffusion constant doesn’t diffuse the same thing. The system is coupled in a way that resists controlled ablation.
This matters most in scientific applications. The goal of building a hybrid reaction-diffusion model is usually not to reproduce observed patterns but to understand what mechanism produces them. A model that reproduces the patterns while mixing structural and learned contributions gives you predictive accuracy without mechanistic clarity. You have a compressed description of what happens, not an explanation of why.
Mordvintsev’s Neural CA (2020) makes this concrete. The system reliably regenerates its target pattern after damage. The mechanism is distributed across several hundred learned weights, each contributing marginally to the cell’s next state. You cannot identify which weights implement “detect damage” versus which implement “grow toward target” — the computation is entangled. The regeneration behavior is real, but the mechanism as represented in the model is not the same as the mechanism in biological cells, even though the analogy is suggestive. The model demonstrates that distributed local rules can produce regeneration; it does not explain how biological cells achieve it.
Where the Interpretability Problem Stands
Existing tools give partial purchase. Activation patching — systematically zeroing out learned parameters and measuring what breaks — can identify which parameters are load-bearing for a specific behavior. Sensitivity analysis measures how much the output changes under small perturbations to each parameter. In Flow-Lenia, you can ask: which cells’ parameter values are most responsible for the stability of a particular organism?
These methods work better on small hybrid models than on large ones, and they address local questions (what does this parameter do?) rather than structural questions (why does this combination of structural rules and learned parameters produce self-organization?). The latter requires a theoretical framework that connects learned parameter values to structural properties of the dynamical system — fixed points, stability boundaries, bifurcation structure.
That framework does not yet exist for general hybrid systems. Physics-informed networks can inherit theoretical guarantees from the structural constraints they respect, and some progress has been made in understanding their learned components through this lens. But for hybrid CAs and hybrid ABMs, the theory is thin.
What remains unsolved is an interpretability method that can identify which learned parameters are doing structurally interesting work — implementing a genuine mechanism — versus which are absorbing noise in the training data. That distinction is what separates a hybrid model that explains from a hybrid model that merely fits. Until it is solved, the most powerful hybrid systems will be accurate in ways we cannot fully account for.