by Noumenal Labs
The tl;dr
In this blog post, we describe the new approach to macroscopic physics discovery that we are developing at Noumenal Labs. This approach enables us to build machine intelligences that can learn grounded world models directly from data
Macroscopic physics discovery means discovering the different macroscopic objects and object types that generate time series data, as well as the object-type-specific rules that govern their behavior — and to do so directly from data, in an unsupervised manner
Grounded world models are structured generative models that explicitly encode the structure of macroscopic objects, as well as the sparse structure of their interactions and the causal, object-type-specific rules that govern their behavior and exchange
Objects and their boundaries
The approach to macroscopic physics discovery that we discuss here is based on an object-centered approach to statistical physics, coupled to the free energy principle (FEP) — which forms the core of a generalized modeling method that has been used to mathematically describe arbitrary objects that persist in random dynamical systems. The FEP begins with a stipulative mathematical definition of a “thing” or “object”: An object that exists physically and that can be reidentified over time must, by definition, be separable from other things in its environment via a boundary, over the timescale that it exists. This obvious tautology has been described as a mathematical “theory” of “every” “thing”. This is because, if we could not separate an object from what it is not, at least for the purposes of identifying it, then it would not be sensible to label it as an object. However, it is a productive tautology, in that it leads to a minimal information theoretic formulation of system identification and an associated taxonomy of macroscopic objects.
In the FEP literature, this boundary between subsystems is formalized as a Markov blanket. In the statistical literature, generically speaking, a Markov blanket is used to identify a set of variables that impose conditional independence between a given set and all other variables. Consider a set of variables X. The Markov blanket of a given “internal” subset of variables Z ⊂ X is defined as a set B ⊂ X, such that Z is conditionally independent of all variables not in Z or B, given the blanket B. In other words, internal and external variables are independent of each other, conditioned on the blanket.
In the setting of statistics, Markov blankets are useful because they allow us to identify the set of variables that we need to track in order to perform inference on a specified set of internal variables. The smallest blanket, called a Markov boundary, can then be used to define an efficient message passing algorithm for optimal inference of the internal variables.
In the FEP literature, Markov blankets are usually used to formalize the notion of object. The Markov blanket is particularly useful in this context because the statistics of the Markov blanket can be used to fully define distinct object types. This is because the statistics of the Markov blanket or boundary variables fully characterize the interactions between the object and the world, i.e., the input-relations of that object. As a result, when we consider a world made of Markov blanketed objects, we can abstract away internal variables and only concern ourselves with representations of the boundary variables and the flow of information between the boundary variables of different objects.
A simplified version of this technical definition of an object is implicit in statistical physics. In that domain, large numbers of particles and the forces of their collisions are summarized by identifying a small number of internal variables (density, pressure, flux, etc.) that determine how the stuff inside the volume is related to what goes on at the boundary of the fluid element, i.e., how it affects surrounding fluid elements that have been given a similar treatment. Crucially, the rules that govern the interaction between two fluid elements (i.e., objects of the same type) will differ from those that govern the interaction between a fluid element and, say, a solid surface or semipermeable membrane. That is, for the complex macroscopic objects that generally concern us, relations or interactions are object-type-specific, and the substrate that mediates those interactions (fluid fluxes in this example) is specified by the underlying microscopic components (water molecules).
Similarly, in classical mechanics, the interactions between objects are formalised as being mediated by forces. Force vectors or fields provide a common, generic, uniformly applicable mathematical language to express the various ways in which objects interact with each other. Determining the composite effect of a set of forces is easy: One simply combines them by adding them up at any given location. This approach is highly effective at the microscopic level. For example, to determine how a complex object or system behaves one can monitor all forces impinging upon it at every possible location or point of contact. This level of detailed modeling is often unnecessary — and most certainly not what our brains do when modeling the behavior of macroscopic objects like bouncing balls or melting ice. Rather, we use a more generalized notion of forces that are relational and object-type-specific: An arrow pierces an apple, but bounces off an armored plate. This is a significant departure from a traditional approach to statistical physics which a priori focuses on a very restrictive, highly local, notion of what a force is or what the domain of fluxes should be. Rather, it is more about the effective macroscopic rules that govern interactions.
Why does this matter to macroscopic physics discovery?
This generalization of the notion of force is key to understanding the utility of this approach — because it provides us with a more flexible and expressive language in which to express the interactions between macroscopic objects. Crucially, these representations are learnable from data, and are thereby apt to close the gap between the physical and the simulated worlds. Thus, rather than tying this notion of forces to traditional forces vectors or the fluxes of the underlying microscopic substrate, we can simply learn from data the effective ‘forces’ that govern the behavior of the macroscopic objects.
This isn’t just a mathematical nicety: It has several applications and implications. This is especially useful in the context of simulating physical systems or virtual assets, as one would in a video game engine. Contemporary game engines simulate game worlds by using the common language of classical mechanical forces, much as we discussed above — that is, objects exert their effects on each other via simple, composable force vectors. The problem with this approach in practice is that classical mechanics can be “stiff” in the sense that small errors in force vector estimate can lead to wildly unrealistic dynamics. This is commonly observed in video game environments when you bump into a chair and it shoots off to the ceiling. This ultimately results from too much emphasis on detailed microscopic physics. By learning a smooth approximate physics from data, these idiosyncrasies can be eliminated.
Markov blankets and physics discovery
So how does this actually work? At the highest level, the aim is to sensibly partition high dimensional time series data into subsystems or objects defined by Markov blankets. At the same time we seek to characterize the interactions between the objects as being drawn from a class of generalized forces. Critically, the specific subset of forces that govern the interaction of a particular object pair are dependent upon the types of the objects.
Taken together, this suggests a highly structured unsupervised model that explicitly labels observable elements according to their current role in a given system and explicitly labels interactions by their type. The general idea behind the algorithm is that observations are labeled according to which “objects” that they are currently a part of. It functions much like image segmentation, but the focus is on dynamical systems segmentation. The algorithm is unsupervised with the overall objective of accurate prediction (surprise minimization) using the simplest or most parsimonious segmentation.
We adopt a probabilistic modeling approach that stipulates a latent embedding for each macroscopic object that indicates its core properties: its type, its location, its shape, features, domain of influence, and so on. From that embedding, we then infer a sparse graph that represents the limited number of channels via which objects can interact with each other and the types of those interactions. Crucially, this algorithm is unsupervised and learns directly from data, using a form of Bayesian attention to identify the macroscopic objects and the type dependent rules that govern how they interact.
Conclusion
To summarize, the approach to macroscopic physics discovery that Noumenal Labs is pioneering makes it possible for machine intelligences to learn the grounded world models that are necessary for them to act appropriately in the world — and to enhance our understanding and decision making. Our approach enables us to learn such grounded world models directly from time series data, in an unsupervised manner. This represents a generational leap forward in research and development of machine intelligence.