Cellular Automata

Cellular automata (CA) are canonical mathematical models of emergent self-organization. The are fully-discrete spatially-extended dynamical systems; space is discretized on a lattice with sites taking values from a finite alphabet, and time evolves in discrete steps according to a local update rule. The Elementary Cellular Automata (ECA) have one spatial dimension with a binary alphabet and nearest-neighbor local interactions, as depicted above.

The local causal states can be reconstructed from CA spacetime fields and used to produce an associated latent state field with a shared coordinate geometry. Spacetime symmetries in the latent field capture CA patterns as generalized symmetries in the observable field. These CA patterns correspond to sets of homogeneous spatial configurations that are invariant under the CA dynamic.

Because the local causal states capture the internal structure of these spacetime symmetries (i.e. their quotient groups), they can be used to identify the structural details of deviations from the symmetries. In particular, coherent structures are formally defined as spatially-localized, temporally-persistent broken symmetries.

In the example below, the observable CA values are given as white and black squares for '0' and '1', respectively. The local causal state label for each spacetime point is overlaid on top with colored letters or numbers. The explicit symmetry regions of the CA is period four in space and time with two distinct temporal phases. In space (horizontal slices), the pattern for the first phase is '...10001000100010001...'. This pattern is captured by the overlaid local causal state labels (in blue) '...ABCDABCDABCDA...'. Similarly, the other phase is '...0111011101110...' with corresponding state labels (also in blue) '...EFGHEFGHEFGHE...'. Numeric state labels in non-blue colors correspond to spatially-localized and temporally-persistent sets of states the break the above spacetime symmetry. These correspond to three coherent structures; the right-traveling gamma-plus and left-traveling gamma-minus 'particles' which collide to create the beta 'particle'. The gamma particles can be understood as phase slips between the two temporal phases of the background symmetry. This is precisely captured by the local causal states; for example a spatial configuration containing a gamma would have a local causal state pattern of the form: '...ABCD..gamma states..EFGH...'. As can be seen in the image, the states that make up the beta particle consist of states from both gammas and out-of-phase background states, indicating a phase slip still occurs 'inside' the beta particle.

Lagrangian Coherent Structures

Coherent structures in far-from-equilibrium systems can be understood as key organizing features that heavily dictate the overall macroscopic dynamics. In fluid flows they form a "hidden skeleton" that has a large influence over material transport. Providing a principled and robust mathematical accounting of these complex structures is a challenging open problem. From a machine learning perspective, coherent structure discovery is a fundamentally unsupervised problem, since ground truth does not exist for these objects. The local causal states provide a means towards defining a ground truth, based on physical principles like broken (generalized) symmetries.

Below we show an example of an unsupervised segmentation analysis of coherent vortices in a direct numerical simulation of two-dimensional turbulent flow. On the left is a snapshot of the vorticity field. In the middle is a corresponding snapshot from a latent local causal state field. Each pixel is assigned a local latent variable (a local causal state) and each unique color represents a unique local state. We can see that the reconstruction parameters chosen in this case isolate vortices ('non-white' colored states) on top of a Euclidean symmetry background (the 'white' colored state). In particular, a large "coherence time" was chosen, thus identifying vortices as the most coherent objects in the flow, with short-scale fluctuations in the background potential flow all being mapped to a single background state. There is, however, still "structure" in the background flow, and this structure can be captured, in addition to the vortices, by changing reconstruction parameters. A snapshot local causal state field from such a reconstruction is shown on the right. Unique colors again correspond to unique local causal states. Like-signed vortices are still assigned to a unique set of states, but the background potential flow is no longer assigned a single state. Colored bounding boxes are shown for comparison with state-of-the-art Lagrangian techniques, as outlined in A Critical Comparison of Lagrangian Methods for Coherent Structure Detection, Hadjighasem et. al.. Green boxes are vortices identified by the geodesic and LAVD methods (shown in Figure 9 (k) and (l), respectively). Red boxes are vortices identified by LAVD, but not geodesic. Yellow boxes are structural signatures of vortices identified as broken symmetries by the local causal states, but not identified by either geodesic or LAVD.

Videos for the full spacetime segmentation of these turbulent flows are linked below. They show side-by-side comparisons of the observable vorticity field, on the left, and the corresponding local causal state field, on the right.

Two-dimensional turbulence 1: -- Segmentation video for the "Additional Structural Detail" local causal state reconstruction shown on the right above.

Two-dimensional turbulence 2: -- Segmentation video for the "Isolated Vortices" local causal state reconstruction shown in the center above.

Two-dimensional turbulence 3: -- Segmentation video for a local causal state reconstruction similar to the previous, but trained using the squared vorticity field as it provides a stronger separation between the vortices and the background potential flow.

The local causal states only require spacetime fields for reconstruction. These can come from simulations, as with the turbulence example above, but can also come from observational data. For example, a similar spacetime segmentation analysis is performed on interpolated video data of the clouds of Jupiter, taken from the NASA Cassini spacecraft.

Local Causal State Segmentation of the Clouds of Jupiter

These results, and the high-performance computing implementation in Python required to achieve them, are detailed in the following manuscript:

Extreme Weather Events

We are currently working on using this technique for unsupervised segmentation of extreme weather events in large, high-resolution climate data sets. Alternative warming scenarios are being simulated using sophisticated climate models, but principled and automated segmentation methods are required to identify weather events on a pixel-level basis in the outputs of these simulations to answer detailed questions about how extreme weather events may be affected by climate change.

Below is a link to a local causal state segmentation analysis of the water vapor field from the CAM5.1 general circulation climate model. As can be seen, this analysis is inadequate for isolating extreme weather events, like hurricanes and atmospheric rivers, with unique sets of local causal states. Further development is required, including multivariate local causal state analysis that takes into account other physical observables.

Local Causal State Segmentation of the Water Vapor Field from the CAM5.1 Climate Model

Predictive Spacetime Autoencoders

The local causal states can be viewed as predictive spacetime autoencoders. An observable spacetime field can be encoded to a latent local causal state field, as with the examples shown above. In addition, a stochastic decoding can map back from the latent field to a reconstructed observable field. Moreover, Markov properties of the local causal states allow for a stochastic dynamic in the latent space that can evolve the local causal state field forward in time. Combined with the stochastic decoding, this provides a form of predictive forecasting for full spacetime fields, as demonstrated below with preliminary results for the circle map lattice (a discrete-time version of the Kuramoto model).

Manuscript in progress