We give an explicit construction of the generating set of a colored operad that implements theta theory in the mathematical model of Minimalism in generative linguistics, in the form of a coloring algorithm for syntactic objects. We show that the coproduct operation on workspaces allows for a recursive implementation of the theta criterion. We also show that this filtering by coloring rules on structures freely formed by Merge is equivalent to a process of structure formation by a colored version of Merge: the form of the generators of the colored operad then implies the dichotomy is semantics between External and Internal Merge, where Internal Merge only moves to non-theta positions.

Whether deep neural networks can exhibit emergent behaviour is not only relevant for understanding how deep learning works, it is also pivotal for estimating potential security risks of increasingly capable artificial intelligence systems. Here, we show that training deep neural networks gives rise to emergent weight morphologies independent of the training data. Specifically, in analogy to condensed matter physics, we derive a theory that predict that the homogeneous state of deep neural networks is unstable in a way that leads to the emergence of periodic channel structures. We verified these structures by performing numerical experiments on a variety of data sets. Our work demonstrates emergence in the training of deep neural networks, which impacts the achievable performance of deep neural networks.

This work presents a novel means for understanding learning dynamics and scaling relations in neural networks. We show that certain measures on the spectrum of the empirical neural tangent kernel, specifically entropy and trace, yield insight into the representations learned by a neural network and how these can be improved through architecture scaling. These results are demonstrated first on test cases before being shown on more complex networks, including transformers, auto-encoders, graph neural networks, and reinforcement learning studies. In testing on a wide range of architectures, we highlight the universal nature of training dynamics and further discuss how it can be used to understand the mechanisms behind learning in neural networks. We identify two such dominant mechanisms present throughout machine learning training. The first, information compression, is seen through a reduction in the entropy of the NTK spectrum during training, and occurs predominantly in small neural networks. The second, coined structure formation, is seen through an increasing entropy and thus, the creation of structure in the neural network representations beyond the prior established by the network at initialization. Due to the ubiquity of the latter in deep neural network architectures and its flexibility in the creation of feature-rich representations, we argue that this form of evolution of the network's entropy be considered the onset of a deep learning regime.

Autonomous shape and structure formation is an important problem in the domain of large-scale multi-agent systems. In this paper, we propose a 3D structure representation method and a distributed structure formation strategy where settled agents guide free moving agents to a prescribed location to settle in the structure. Agents at the structure formation frontier looking for neighbors to settle act as beacons, generating a surface gradient throughout the formed structure propagated by settled agents. Free-moving agents follow the surface gradient along the formed structure surface to the formation frontier, where they eventually reach the closest beacon and settle to continue the structure formation following a local bidding process. Agent behavior is governed by a finite state machine implementation, along with potential field-based motion control laws. We also discuss appropriate rules for recovering from stagnation points. Simulation experiments are presented to show planar and 3D structure formations with continuous and discontinuous boundary/surfaces, which validate the proposed strategy, followed by a scalability analysis.

Despite recent progress in understanding the double descend phenomena [1, 2, 3, 4] we still do not have a complete theory of generalisation in over-parameterised models. Two recent empirical observations, neural collapse [5] and grokking (generalisation beyond over-fitting) [6], can help us understand the training and generalisation properties of over-parameterised models. Neural collapse occurs in the terminal phase of training, i.e. the phase with zero train error. It refers to the collapse of the N dimensional, last-layer features (input to the last/classification layer) [5] to a (C 1)- dimensional equiangular tight frame (ETF) structure, where C is the number of classes. The feature vectors converge towards the vertices of the ETF structure such that features for each class are close to one vertex.

We train a neural network model to predict the full phase space evolution of cosmological N-body simulations. Its success implies that the neural network model is accurately approximating the Green's function expansion that relates the initial conditions of the simulations to its outcome at later times in the deeply nonlinear regime. We test the accuracy of this approximation by assessing its performance on well understood simple cases that have either known exact solutions or well understood expansions. These scenarios include spherical configurations, isolated plane waves, and two interacting plane waves: initial conditions that are very different from the Gaussian random fields used for training. We find our model generalizes well to these well understood scenarios, demonstrating that the networks have inferred general physical principles and learned the nonlinear mode couplings from the complex, random Gaussian training data. These tests also provide a useful diagnostic for finding the model's strengths and weaknesses, and identifying strategies for model improvement. We also test the model on initial conditions that contain only transverse modes, a family of modes that differ not only in their phases but also in their evolution from the longitudinal growing modes used in the training set. When the network encounters these initial conditions that are orthogonal to the training set, the model fails completely. In addition to these simple configurations, we evaluate the model's predictions for the density, displacement, and momentum power spectra with standard initial conditions for N-body simulations. We compare these summary statistics against N-body results and an approximate, fast simulation method called COLA. Our model achieves percent level accuracy at nonlinear scales of $k\sim 1\ \mathrm{Mpc}^{-1}\, h$, representing a significant improvement over COLA.

Using machine learning methods, researchers at TU Graz can predict the structure formation of functionalized molecules at the interfaces of hybrid materials. Now they have also succeeded in looking behind the driving forces of this structure formation. The production of nanomaterials involves self-assembly processes of functionalized (organic) molecules on inorganic surfaces. This combination of organic and inorganic components is essential for applications in organic electronics and other areas of nanotechnology. Until now, certain desired surface properties were often achieved on a trial-and-error basis. Molecules were chemically modified until the best result for the desired surface property was found.

Matter evolved under influence of gravity from minuscule density fluctuations. Non-perturbative structure formed hierarchically over all scales, and developed non-Gaussian features in the Universe, known as the Cosmic Web. To fully understand the structure formation of the Universe is one of the holy grails of modern astrophysics. Astrophysicists survey large volumes of the Universe and employ a large ensemble of computer simulations to compare with the observed data in order to extract the full information of our own Universe. However, to evolve trillions of galaxies over billions of years even with the simplest physics is a daunting task. We build a deep neural network, the Deep Density Displacement Model (hereafter D$^3$M), to predict the non-linear structure formation of the Universe from simple linear perturbation theory. Our extensive analysis, demonstrates that D$^3$M outperforms the second order perturbation theory (hereafter 2LPT), the commonly used fast approximate simulation method, in point-wise comparison, 2-point correlation, and 3-point correlation. We also show that D$^3$M is able to accurately extrapolate far beyond its training data, and predict structure formation for significantly different cosmological parameters. Our study proves, for the first time, that deep learning is a practical and accurate alternative to approximate simulations of the gravitational structure formation of the Universe.