Numerical simulation is becoming increasingly used to support theoretical effort into understanding the turbulence problem. We develop theoretical ideas related to the transfer and dissipation of energy, which clarify long-standing issues with the energy balance in isotropic turbulence. These ideas are supported by results from large scale numerical simulations. Due to the large number of degrees of freedom required to capture all the interacting scales of motion, the increase in computational power available has only recently allowed flows of interest to be realised. A parallel pseudo-spectral code for the direct numerical simulation (DNS) of isotropic turbulence has been developed. Some discussion is given on the challenges and choices involved. The DNS code has been extensively benchmarked by reproducing well established results from literature. The DNS code has been used to conduct a series of runs for freely-decaying turbulence. Decay was performed from a Gaussian random field as well as an evolved velocity field obtained from forced simulation. Since the initial condition does not describe developed turbulence, we are required to determine when the field can be considered to be evolved and measurements are characteristic of decaying turbulence. We explore the use of power-law decay of the total energy and compare with the use of dynamic quantities such as the peak dissipation rate, maximum transport power and velocity derivative skewness. We then show how this choice of evolved time affects the measurement of statistics. In doing so, it is found that the Taylor dissipation surrogate, u^3 / L, is a better surrogate for the maximum inertial flux than dissipation. Stationary turbulence has also been investigated, where we ensure that the energy input rate remains constant for all runs and variation is only introduced by modifying the fluid viscosity (and lattice size). We present results for Reynolds numbers up to Rλ = 335 on a 1024^3 lattice. Using different methods of vortex identification, the persistence of intermittent structure in an ensemble average is considered and shown to be reduced as the ensemble size increases. The longitudinal structure functions are computed for smaller lattices directly from an ensemble of realisations of the real-space velocity field. From these, we consider the generalised structure functions and investigate their scaling exponents using direct analysis and extended self-similarity (ESS), finding results consistent with the literature. An exploitation of the pseudo-spectral technique is used to calculate second- and third-order structure functions from the energy and transfer spectra, with a comparison presented to the real-space calculation. An alternative to ESS is discussed, with the second-order exponent found to approach 2/3. The dissipation anomaly is then considered for both forced and free-decay. Using different choices of the evolved time for a decaying simulation, we show how the behaviour of the dimensionless dissipation coefficient is affected. The Karman-Howarth equation (KHE) is studied and a derivation of a work term presented using a transformation of the Lin equation. The balance of energy represented by the KHE is then investigated using the pseudo-spectral method mentioned above. The consequences of this new input term for the structure functions are discussed. Based on the KHE, we develop a model for the behaviour of the dimensionless dissipation coefficient that predicts Cɛ= Cɛ(∞)+CL/RL. DNS data is used to fit the model. We find Cɛ(∞) = 0.47 and CL = 19.1 for forced turbulence, with excellent agreement to the data. Theoretical methods based on the renormalization group and statistical closures are still being developed to study turbulence. The dynamic RG procedure used by Forster, Nelson and Stephen (FNS) is considered in some detail and a disagreement in the literature over the method and results is resolved here. An additional constraint on the loop momentum is shown to cause a correction to the viscosity increment such that all methods of evaluation lead to the original result found by FNS. The application of statistical closure and renormalized perturbation theory is discussed and a new two-time model probability density functional presented. This has been shown to be self-consistent to second order and to reproduce the two-time covariance equation of the local energy transfer (LET) theory. Future direction of this work is discussed.

The question of the possibility of intelligent machines is fundamentally intertwined with the machines’ ability to reason. Or not. The developments of the recent years point in a completely different direction : What we need is simple, generic but scalable algorithms that can keep learning on their own. This thesis is an attempt to find theoretical explanations to the findings of recent years where empirical evidence has been presented in support of phase transitions in neural networks, power law behavior of various entities, and even evidence of algorithmic universality, all of which are beautifully explained in the context of statistical physics, quantum field theory and statistical field theory but not necessarily in the context of deep learning where no complete theoretical framework is available. Inspired by these developments, and as it turns out, with the overly ambitious goal of providing a solid theoretical explanation of the empirically observed power laws in neu- ral networks, we set out to substantiate the claims that renormalization group theory may be the sought-after theory of deep learning which may explain the above, as well as what we call algorithmic universality. / La question de la possibilité de machines intelligentes est intimement liée à la capacité de ces machines à raisonner. Ou pas. Les développements des dernières années indiquent une direction complètement différente : ce dont nous avons besoin sont des algorithmes simples, génériques mais évolutifs qui peuvent continuer à apprendre de leur propre chef. Cette thèse est une tentative de trouver des explications théoriques aux constatations des dernières années où des preuves empiriques ont été présentées en faveur de transitions de phase dans les réseaux de neurones, du comportement en loi de puissance de diverses entités, et même de l'universialité algorithmique, tout cela étant parfaitement expliqué dans le contexte de la physique statistique, de la théorie quantique des champs et de la théorie statistique des champs, mais pas nécessairement dans le contexte de l'apprentissage profond où aucun cadre théorique complet n'est disponible. Inspiré par ces développements, et comme il s'avère, avec le but ambitieux de fournir une explication théorique solide des lois de puissance empiriquement observées dans les réseaux de neurones, nous avons entrepris de étayer les affirmations selon lesquelles la théorie du groupe de renormalisation pourrait être la théorie recherchée de l'apprentissage profond qui pourrait expliquer cela, ainsi que ce que nous appelons l'universialité algorithmique.