Traditionally physical science operates under two fundamental assumptions: one, that the universe may be reduced to a set of consistent parts and fundamental laws that govern their behavior; two, that the universe and its phenomena may then be reconstructed from this set of fundamental laws. And yet many natural phenomena possess emergent properties that are difficult or even mathematically impossible to deduce even from a complete microscopic description [1, 2].The existence of irreducibly complex systems presents a challenge: to build a science of model discovery beyond the traditional reductionist-constructionist paradigm.
There are important questions: what is the meaning of structure, divorced from a domain-specific physical context? Can we mathematically define it and detect it? How does one build a consistent mathematical language to describe things as disparate as bird flocking behavior and the formation of galaxies? Why do patterns emerge in physical systems over time? Can we predict when ‘something new’ will emerge, and can we detect it once it has emerged [3]? What counts as something new?
At Inria we take a dual-pronged approach of tackling both these questions on both a theoretical level, by developing the aforementioned mathematical tools, and on a practical level, by applying these tools to real world systems in collaboration with domain scientists.
Our primary focus is causal state and structure discovery from time series. Our method takes in time series of observations and returns a set of inferred causal states of the system—sets of predictively equivalent observations—with an interpretable geometry [4, 5]. This is a stochastic parallel to deterministic attractor reconstruction in nonlinear dynamics. The inferred causal state set can be analyzed for insights into the intrinsic complexity and mechanistic drivers of the underlying system. When paired with inference of the governing equations of motion, it can also be used as the state set of an inherently interpretable, optimally predictive model [6, 7].
We are developing a software package based on existing code that implements our inference algorithm using kernel embedding of conditional probability distributions [4, 8]. The algorithm takes in arbitrary time series (including observations from multiple sources or heterogeneous data) and returns the inferred causal state set in the form of similarity matrices, which are then dimension reduced for visualization and analysis [8]. This software package is to be disseminated to the public. It is also to be used in the framework of a partnership between Inria and Inrae to improve modeling of carbon flux dynamics in ecosystems.
[1] P.W. Anderson. More is different. Science, 177(4047):393–396, August 1972.
[2] M. Gu, C. Weedbrook, A. Perales, and M. Nielsen. More really is different. Physica D, 238(9-10):835–839, May 2009.
[3] James P. Crutchfield. The calculi of emergence: Computation, dynamics, and induction. Physica D, 75:11–54, 1994.
[4] Nicolas Brodu and James P. Crutchfield. Discovering causal structure with reproducing-kernel Hilbert space \epsilon-machines. Chaos: An Interdisciplinary Journal of Nonlinear Science, 32(2):023103, 02 2022.
[5] Samuel P. Loomis and James P. Crutchfield. Topology, convergence, and reconstruction of predictive states. Physica D: Nonlinear Phenomena, 445:133621, 2023
[6] C. R. Shalizi. Causal Architecture, Complexity and Self-Organization in Time Series and Cellular Automata. PhD thesis, University of Wisconsin, Madison, Wisconsin, 2001.
[7] S.L. Brunton, J.L. Proctor, & J.N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci. U.S.A. 113 (15) 3932-3937, https://doi.org/10.1073/pnas.1517384113 (2016).
[8] Alexandra M. Jurgens, Nicolas Brodu; Inferring kernel -machines: Discovering structure in complex systems. Chaos 1 March 2025; 35 (3): 033162. https://doi.org/10.1063/5.0242981
