A framework for the use of generative modelling in non-equilibrium statistical mechanics

TL;DR

The paper addresses how to model open, interacting, non-equilibrium systems using generative models of their dependency structure, formalized via the variational free energy principle ($F$). It shows that, under a Markov blanket partition, the dynamics of internal states can be recast as a gradient flow on $F(\mu,b)$, providing an as-if Bayesian inference interpretation while maintaining a distinction between the model and the physical system. Through two toy demonstrations—cellular morphogenesis and periodically firing cells—it demonstrates how minimizing $F$ guides self-organization and rhythmic activity, offering tractable explanations for complex dynamics. The work argues that this framework yields a map of possible maps for physical systems, delivering a principled lens for non-equilibrium statistical mechanics that separates scientific modelling from literal cognitive processes, while acknowledging potential reification and the interpretive nature of the inference story.

Abstract

We discuss an approach to mathematically modelling systems made of objects that are coupled together, using generative models of the dependence relationships between states (or trajectories) of the things comprising such systems. This broad class includes open or non-equilibrium systems and is especially relevant to self-organising systems. The ensuing variational free energy principle (FEP) has certain advantages over using random dynamical systems explicitly, notably, by being more tractable and offering a parsimonious explanation of why the joint system evolves in the way that it does, based on the properties of the coupling between system components. The FEP is a method whose use allows us to build a model of the dynamics of an object as if it were a process of variational inference, because variational free energy (or surprisal) is a Lyapunov function for its dynamics. In short, we argue that using generative models to represent and track relations amongst subsystems leads us to a particular statistical theory of interacting systems. Conversely, this theory enables us to construct nested models that respect the known relations amongst subsystems. We point out that the fact that a physical object conforms to the FEP does not necessarily imply that this object performs inference in the literal sense; rather, it is a useful explanatory fiction which replaces the `explicit' dynamics of the object with an `implicit' flow on free energy gradients -- a fiction that may or may not be entertained by the object itself.

Figures (7)

• Figure 1: Plots of the target extracellular gradients (\ref{['main:a']}), encoding of the target signal in the cells (\ref{['main:b']}) and softmax expectations of the identities of each cell (\ref{['main:c']}). In Figure \ref{['main:a']} colours denote different signal expressions and black dots denote no data, with the final locations of the cells starred.
• Figure 2: Here morphogenesis is demonstrated by cells migrating to align with expectations based on sensory input, modelled by a concomitant decrease of free energy in time.
• Figure 3: Dynamics of each cell, with final locations starred. The gradients here match those of Figure \ref{['migration-fig']}. It can be seen that (speaking somewhat heuristically) each cell moves to fulfil its expectations about the signals it should encounter, whilst expressing the signals associated with its current beliefs about its place in the target ensemble.
• Figure 4: Evolution of the internal state under $\mathcal{L}$ and under $F$. The dynamics of internal states in two regimes, plotted on a polar spiral showing the evolution in the phase in time. The dynamics under a gradient descent on $\mathcal{L}$ and on $F$ show excellent agreement over all timescales.
• Figure 5: Evolution of the active state under $\mathcal{L}$ and under $F$. The dynamics of the active state in two regimes were also plotted; again, excellent agreement is observed.