Scientific data, from cellular snapshots in biology to celestial distributions in cosmology, often consists of static patterns from underlying dynamical systems. These snapshots, while lacking temporal ordering, implicitly encode the processes that preserve them. This work investigates how strongly such a distribution constrains its underlying dynamics and how to recover them. We introduce the Equilibrium flow method, a framework that learns continuous dynamics that preserve a given pattern distribution. Our method successfully identifies plausible dynamics for 2-D systems and recovers the signature chaotic behavior of the Lorenz attractor. For high-dimensional Turing patterns from the Gray-Scott model, we develop an efficient, training-free variant that achieves high fidelity to the ground truth, validated both quantitatively and qualitatively. Our analysis reveals the solution space is constrained not only by the data but also by the learning model's inductive biases. This capability extends beyond recovering known systems, enabling a new paradigm of inverse design for Artificial Life. By specifying a target pattern distribution, we can discover the local interaction rules that preserve it, leading to the spontaneous emergence of complex behaviors, such as life-like flocking, attraction, and repulsion patterns, from simple, user-defined snapshots.

Imagine grasshoppers distributed evenly across a dry field. As the temperature rises, the grasshoppers start to sweat. The field catches fire in a few spots and starts to spread. What kind of burn pattern will appear in the field? If the grasshoppers sweat more, or less, or if the fire spreads faster, or slower--how will that alter the burn pattern?

Microstructure evolution in matter is often modeled numerically using field or level-set solvers, mirroring the dual representation of spatiotemporal complexity in terms of pixel or voxel data, and geometrical forms in vector graphics. Motivated by this analog, as well as the structural and event-driven nature of artificial and spiking neural networks, respectively, we evaluate their performance in learning and predicting fatigue crack growth and Turing pattern development. Predictions are made based on digital libraries constructed from computer simulations, which can be replaced by experimental data to lift the mathematical overconstraints of physics. Our assessment suggests that the leaky integrate-and-fire neuron model offers superior predictive accuracy with fewer parameters and less memory usage, alleviating the accuracy-cost tradeoff in contrast to the common practices in computer vision tasks. Examination of network architectures shows that these benefits arise from its reduced weight range and sparser connections. The study highlights the capability of event-driven models in tackling problems with evolutionary bulk-phase and interface behaviors using the digital library approach.

Swarm robots, inspired by the emergence of animal herds, are robots that assemble a large number of modules and self-organize themselves to form specific morphologies and exhibit specific functions. These modular robots perform relatively simple actions and controls, and create macroscopic morphologies and functions through the interaction of a large number of modular robots. This research focuses on such self-organizing robots or swarm robots. The proposed algorithm is a model that applies the Turing pattern, one of the self-organization models, to make a group of modules accumulate and stay within a certain region. The proposed method utilizes the area within the spots of the Turing pattern as the aggregation region of the modules. Furthermore, it considers the value corresponding to the concentration distribution within the spotted pattern of the Turing pattern model (referred to as the potential value in this research), identifies the center of the region (spotted pattern), and makes it the center of the module group. By controlling the modules in the direction of the higher potential value, it succeeds in maintaining the shape of the module group as a whole while moving. The algorithm was validated using a two-dimensional simulation model. The unit module robot was assumed to have the following properties: 1) limited self-drive, 2) no module identifier, 3) information exchange only with adjacent modules, 4) no coordinate system, and 5) only simple arithmetic and memory functions. Using these modules, the devised algorithm was able to achieve not only the creation of static forms but also the realization of the following movements: 1) modules accumulate and grow, 2) modules move to the light source, 3) exit the gap while maintaining its shape, and 4) self-replication.

Reaction-diffusion (Turing) systems are fundamental to the formation of spatial patterns in nature and engineering. These systems are governed by a set of non-linear partial differential equations containing parameters that determine the rate of constituent diffusion and reaction. Critically, these parameters, such as diffusion coefficient, heavily influence the mode and type of the final pattern, and quantitative characterization and knowledge of these parameters can aid in bio-mimetic design or understanding of real-world systems. However, the use of numerical methods to infer these parameters can be difficult and computationally expensive. Typically, adjoint solvers may be used, but they are frequently unstable for very non-linear systems. Alternatively, massive amounts of iterative forward simulations are used to find the best match, but this is extremely effortful. Recently, physics-informed neural networks have been proposed as a means for data-driven discovery of partial differential equations, and have seen success in various applications. Thus, we investigate the use of physics-informed neural networks as a tool to infer key parameters in reaction-diffusion systems in the steady-state for scientific discovery or design. Our proof-of-concept results show that the method is able to infer parameters for different pattern modes and types with errors of less than 10\%. In addition, the stochastic nature of this method can be exploited to provide multiple parameter alternatives to the desired pattern, highlighting the versatility of this method for bio-mimetic design. This work thus demonstrates the utility of physics-informed neural networks for inverse parameter inference of reaction-diffusion systems to enhance scientific discovery and design.

The high-dimesionality, non-linearity and emergent properties of complex systems pose a challenge to identifying general laws in the same manner that has been so successful in simpler physical systems. In Anderson's seminal work on why "more is different" he pointed to how emergent, macroscale patterns break symmetries of the underlying microscale laws. Yet, less recognized is that these large-scale, emergent patterns must also retain some symmetries of the microscale rules. Here we introduce a new, relational macrostate theory (RMT) that defines macrostates in terms of symmetries between two mutually predictive observations, and develop a machine learning architecture, MacroNet, that identifies macrostates. Using this framework, we show how macrostates can be identifed across systems ranging in complexity from the simplicity of the simple harmonic oscillator to the much more complex spatial patterning characteristic of Turing instabilities. Furthermore, we show how our framework can be used for the inverse design of microstates consistent with a given macroscopic property -- in Turing patterns this allows us to design underlying rule with a given specification of spatial patterning, and to identify which rule parameters most control these patterns. By demonstrating a general theory for how macroscopic properties emerge from conservation of symmetries in the mapping between observations, we provide a machine learning framework that allows a unified approach to identifying macrostates in systems from the simple to complex, and allows the design of new examples consistent with a given macroscopic property.

Although intelligent and adept at image recognition and classification, deep neural networks can still be vulnerable to adversarial perturbations, i.e., small but queer details in an image that causes errors in neural network output. Some of these are universal. They tend to interfere with the neural network when placed on any input. A research paper presented at the 35th AAAI Conference on Artificial Intelligence by researchers at Skoltech demonstrated that patterns that cause neural networks to make mistakes in image recognition are, in fact, similar to Turing patterns found all around the Neural network world. This result can help design defenses for pattern recognition systems that are currently susceptible to attacks.

State-of-the-art deep classifiers are intriguingly vulnerable to universal adversarial perturbations: single disturbances of small magnitude that lead to misclassification of most inputs. This phenomena may potentially result in a serious security problem. Despite the extensive research in this area, there is a lack of theoretical understanding of the structure of these perturbations. In image domain, there is a certain visual similarity between patterns, that represent these perturbations, and classical Turing patterns, which appear as a solution of non-linear partial differential equations and are underlying concept of many processes in nature. In this paper, we provide a theoretical bridge between these two different theories, by mapping a simplified algorithm for crafting universal perturbations to (inhomogeneous) cellular automata, the latter is known to generate Turing patterns. Furthermore, we propose to use Turing patterns, generated by cellular automata, as universal perturbations, and experimentally show that they significantly degrade the performance of deep learning models. We found this method to be a fast and efficient way to create a data-agnostic quasi-imperceptible perturbation in the black-box scenario.