Machine Learning Hidden Symmetries

Emergence and Causality in Complex Systems: A Survey of Causal Emergence and Related Quantitative Studies

Emergence and causality are two fundamental concepts for understanding complex systems. They are interconnected. On one hand, emergence refers to the phenomenon where macroscopic properties cannot be solely attributed to the cause of individual properties. On the other hand, causality can exhibit emergence, meaning that new causal laws may arise as we increase the level of abstraction. Causal emergence (CE) theory aims to bridge these two concepts and even employs measures of causality to quantify emergence. This paper provides a comprehensive review of recent advancements in quantitative theories and applications of CE. It focuses on two primary challenges: quantifying CE and identifying it from data. The latter task requires the integration of machine learning and neural network techniques, establishing a significant link between causal emergence and machine learning. We highlight two problem categories: CE with machine learning and CE for machine learning, both of which emphasize the crucial role of effective information (EI) as a measure of causal emergence. The final section of this review explores potential applications and provides insights into future perspectives.

Strain topological metamaterials and revealing hidden topology in higher-order coordinates

Topological physics has revolutionized materials science, introducing topological phases of matter in diverse settings ranging from quantum to photonic and phononic systems. Herein, we present a family of topological systems, which we term "strain topological metamaterials", whose topological properties are hidden and unveiled only under higher-order (strain) coordinate transformations. We firstly show that the canonical mass dimer, a model that can describe various settings such as electrical circuits and optics, among others, belongs to this family where strain coordinates reveal a topological nontriviality for the edge states at free boundaries. Subsequently, we introduce a mechanical analog of the Majorana-supporting Kitaev chain, which supports topological edge states for both fixed and free boundaries within the proposed framework. Thus, our findings not only extend the way topological edge states are identified, but also promote the fabrication of novel topological metamaterials in various fields, with more complex, tailored boundaries.

Metalearning Generalizable Dynamics from Trajectories

We present the interpretable meta neural ordinary differential equation (iMODE) method to rapidly learn generalizable (i.e., not parameter-specific) dynamics from trajectories of multiple dynamical systems that vary in their physical parameters. The iMODE method learns metaknowledge, the functional variations of the force field of dynamical system instances without knowing the physical parameters, by adopting a bilevel optimization framework: an outer level capturing the common force field form among studied dynamical system instances and an inner level adapting to individual system instances. A priori physical knowledge can be conveniently embedded in the neural network architecture as inductive bias, such as conservative force field and Euclidean symmetry. With the learned metaknowledge, iMODE can model an unseen system within seconds, and inversely reveal knowledge on the physical parameters of a system, or as a neural gauge to "measure" the physical parameters of an unseen system with observed trajectories. iMODE can be generally applied to a dynamical system of an arbitrary type or number of physical parameters and is validated on bistable, double pendulum, Van der Pol, Slinky, and reaction-diffusion systems.

Machine learning of independent conservation laws through neural deflation

We introduce a methodology for seeking conservation laws within a Hamiltonian dynamical system, which we term "neural deflation." Inspired by deflation methods for steady states of dynamical systems, we propose to iteratively train a number of neural networks to minimize a regularized loss function accounting for the necessity of conserved quantities to be in involution and enforcing functional independence thereof consistently in the infinite-sample limit. The method is applied to a series of integrable and nonintegrable lattice differential-difference equations. In the former, the predicted number of conservation laws extensively grows with the number of degrees of freedom, while for the latter, it generically stops at a threshold related to the number of conserved quantities in the system. This data-driven tool could prove valuable in assessing a model's conserved quantities and its potential integrability.

Scientific discovery in the age of artificial intelligence

Artificial intelligence (AI) is being increasingly integrated into scientific discovery to augment and accelerate research, helping scientists to generate hypotheses, design experiments, collect and interpret large datasets, and gain insights that might not have been possible using traditional scientific methods alone. Here we examine breakthroughs over the past decade that include self-supervised learning, which allows models to be trained on vast amounts of unlabelled data, and geometric deep learning, which leverages knowledge about the structure of scientific data to enhance model accuracy and efficiency. Generative AI methods can create designs, such as small-molecule drugs and proteins, by analysing diverse data modalities, including images and sequences. We discuss how these methods can help scientists throughout the scientific process and the central issues that remain despite such advances. Both developers and users of AI toolsneed a better understanding of when such approaches need improvement, and challenges posed by poor data quality and stewardship remain. These issues cut across scientific disciplines and require developing foundational algorithmic approaches that can contribute to scientific understanding or acquire it autonomously, making them critical areas of focus for AI innovation.

Emergence of common concepts, symmetries and conformity in agent groups-an information-theoretic model

The paper studies principles behind structured, especially symmetric, representations through enforced inter-agent conformity. For this, we consider agents in a simple environment who extract individual representations of this environment through an information maximization principle. The representations obtained by different agents differ in general to some extent from each other. This gives rise to ambiguities in how the environment is represented by the different agents. Using a variant of the information bottleneck principle, we extract a 'common conceptualization' of the world for this group of agents. It turns out that the common conceptualization appears to capture much higher regularities or symmetries of the environment than the individual representations. We further formalize the notion of identifying symmetries in the environment both with respect to 'extrinsic' (birds-eye) operations on the environment as well as with respect to 'intrinsic' operations, i.e. subjective operations corresponding to the reconfiguration of the agent's embodiment. Remarkably, using the latter formalism, one can re-wire an agent to conform to the highly symmetric common conceptualization to a much higher degree than an unrefined agent; and that, without having to re-optimize the agent from scratch. In other words, one can 're-educate' an agent to conform to the de-individualized 'concept' of the agent group with comparatively little effort.

How close are integrable and nonintegrable models: A parametric case study based on the Salerno model

In the present work we revisit the Salerno model as a prototypical system that interpolates between a well-known integrable system (the Ablowitz-Ladik lattice) and an experimentally tractable, nonintegrable one (the discrete nonlinear Schrödinger model). The question we ask is, for "generic" initial data, how close are the integrable to the nonintegrable models? Our more precise formulation of this question is, How well is the constancy of formerly conserved quantities preserved in the nonintegrable case? Upon examining this, we find that even slight deviations from integrability can be sensitively felt by measuring these formerly conserved quantities in the case of the Salerno model. However, given that the knowledge of these quantities requires a deep physical and mathematical analysis of the system, we seek a more "generic" diagnostic towards a manifestation of integrability breaking. We argue, based on our Salerno model computations, that the full spectrum of Lyapunov exponents could be a sensitive diagnostic to that effect.

Machine learning conservation laws from differential equations

We present a machine learning algorithm that discovers conservation laws from differential equations, both numerically (parametrized as neural networks) and symbolically, ensuring their functional independence (a nonlinear generalization of linear independence). Our independence module can be viewed as a nonlinear generalization of singular value decomposition. Our method can readily handle inductive biases for conservation laws. We validate it with examples including the three-body problem, the KdV equation, and nonlinear Schrödinger equation.

Learn one size to infer all: Exploiting translational symmetries in delay-dynamical and spatiotemporal systems using scalable neural networks

We design scalable neural networks adapted to translational symmetries in dynamical systems, capable of inferring untrained high-dimensional dynamics for different system sizes. We train these networks to predict the dynamics of delay-dynamical and spatiotemporal systems for a single size. Then, we drive the networks by their own predictions. We demonstrate that by scaling the size of the trained network, we can predict the complex dynamics for larger or smaller system sizes. Thus, the network learns from a single example and by exploiting symmetry properties infers entire bifurcation diagrams.

Learning spatiotemporal chaos using next-generation reservoir computing

Forecasting the behavior of high-dimensional dynamical systems using machine learning requires efficient methods to learn the underlying physical model. We demonstrate spatiotemporal chaos prediction using a machine learning architecture that, when combined with a next-generation reservoir computer, displays state-of-the-art performance with a computational time ¹⁰- ¹⁰ times faster for training process and training data set ∼ ¹⁰ times smaller than other machine learning algorithms. We also take advantage of the translational symmetry of the model to further reduce the computational cost and training data, each by a factor of ∼10.