Related papers: p-Adic Statistical Field Theory and Convolutional …
The field of neuroscience and the development of artificial neural networks (ANNs) have mutually influenced each other, drawing from and contributing to many concepts initially developed in statistical mechanics. Notably, Hopfield networks…
Neural Network Field Theories (NN-FTs) represent a novel construction of arbitrary field theories, including those of conformal fields, through the specification of the network architecture and prior distribution for the network parameters.…
Deep Neural Networks (DNNs) excel at many tasks, often rivaling or surpassing human performance. Yet their internal processes remain elusive, frequently described as "black boxes." While performance can be refined experimentally, achieving…
Kohn-Sham density functional theory (KS-DFT) has found widespread application in accurate electronic structure calculations. However, it can be computationally demanding especially for large-scale simulations, motivating recent efforts…
Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1D are explored. Many of our field theories are highly-interacting without free quadratic analogs.…
We construct in a rigorous mathematical way interacting quantum field theories on a p-adic spacetime. The main result is the construction of a measure on a function space which allows a rigorous definition of the partition function. The…
The precise equivalence between discretized Euclidean field theories and a certain class of probabilistic graphical models, namely the mathematical framework of Markov random fields, opens up the opportunity to investigate machine learning…
Neural network field theory (NNFT) represents fields as neural networks and samples field configurations by drawing network parameters from a probability distribution. We identify a previously unexplored architectural freedom in NNFT,…
Regression with non-Euclidean responses -- e.g., probability distributions, networks, symmetric positive-definite matrices, and compositions -- has become increasingly important in modern applications. In this paper, we propose deep…
In this paper we propose a generalization of deep neural networks called deep function machines (DFMs). DFMs act on vector spaces of arbitrary (possibly infinite) dimension and we show that a family of DFMs are invariant to the dimension of…
A common lens to theoretically study neural net architectures is to analyze the functions they can approximate. However, constructions from approximation theory may be unrealistic and therefore less meaningful. For example, a common…
Ab initio calculations are fundamentally bottlenecked for large systems by the steep computational scaling of solving self-consistent field (SCF) equations. While machine learning offers potential accelerations, existing methods often…
Density Functional Theory (DFT) is widely used for atomistic simulations. However, its reach stays limited due to several limitations such as lack of accurate exchange-correlation functional, requirement of costly O(N 3) diagonalization…
This work begins by establishing a mathematical formalization between different geometrical interpretations of Neural Networks, providing a first contribution. From this starting point, a new interpretation is explored, using the idea of…
Every known artificial deep neural network (DNN) corresponds to an object in a canonical Grothendieck's topos; its learning dynamic corresponds to a flow of morphisms in this topos. Invariance structures in the layers (like CNNs or LSTMs)…
Non-integrability in the sense of dynamical systems, also known as dynamical chaos, is a strongly nonlinear qualitative phenomenon. Its most promising theoretical descriptions are likely to emerge from non-perturbative approaches, with…
A new way of computing scattering amplitudes in a certain very important QFT (N=4 SYM) has recently been developed, in which an algebraic structure called the positive Grassmannian plays a very important role. The mathematics of the…
In order to construct examples for interacting quantum field theory models, the methods of euclidean field theory turned out to be powerful tools since they make use of the techniques of classical statistical mechanics. Starting from an…
We present a deep neural network representation of the AdS/CFT correspondence, and demonstrate the emergence of the bulk metric function via the learning process for given data sets of response in boundary quantum field theories. The…
Quantum field theory (QFT) for interacting many-electron systems is fundamental to condensed matter physics, yet achieving accurate solutions confronts computational challenges in managing the combinatorial complexity of Feynman diagrams,…