Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks

Geist, Moritz; Petersen, Philipp; Raslan, Mones; Schneider, Reinhold; Kutyniok, Gitta

doi:10.1007/s10915-021-01532-w

Cited by 53 publications

(20 citation statements)

References 56 publications

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“…With this idea in mind, AI methods, in particular in their incarnation through Neural Network approximations, have also been proposed to solve partial differential equations. These were first introduced in [92] and have recently received renewed attention, see for example [93][94][95][96].…”

Section: Neural Network Methods For Solving Pdesmentioning

confidence: 99%

Hyperbolic compactification of M-theory and de Sitter quantum gravity

2022

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We present a mechanism for accelerated expansion of the universe in the generic case of negative-curvature compactifications of M-theory, with minimal ingredients. M-theory on a hyperbolic manifold with small closed geodesics supporting Casimir energy -- along with a single classical source (7-form flux) -- contains an immediate 3-term structure for volume stabilization at positive potential energy. Hyperbolic manifolds are well-studied mathematically, with an important rigidity property at fixed volume. They and their Dehn fillings to more general Einstein spaces exhibit explicit discrete parameters that yield small closed geodesics supporting Casimir energy. The off-shell effective potential derived by M. Douglas incorporates the warped product structure via the constraints of general relativity, screening negative energy. Analyzing the fields sourced by the localized Casimir energy and the available discrete choices of manifolds and fluxes, we find a regime where the net curvature, Casimir energy, and flux compete at large radius and stabilize the volume. Further metric and form field deformations are highly constrained by hyperbolic rigidity and warping effects, leading to calculations giving strong indications of a positive Hessian, and residual tadpoles are small. We test this via explicit back reacted solutions and perturbations in patches including the Dehn filling regions, initiate a neural network study of further aspects of the internal fields, and derive a Maldacena-Nunez style no-go theorem for Anti-de Sitter extrema for a range of parameters. A simple generalization incorporating 4-form flux produces axion monodromy inflation. As a relatively simple de Sitter uplift of the large-N M2-brane theory, the construction applies to de Sitter holography as well as to cosmological modeling, and introduces new connections between mathematics and the physics of string/M theory compactifications.

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Section: Neural Network Methods For Solving Pdesmentioning

confidence: 99%

Hyperbolic compactification of M-theory and de Sitter quantum gravity

2022

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“…PDEs are ubiquitous in science; developing efficient numerical solvers is of paramount importance. Many methods have been developed to attack this problem, including recent exploration of Machine Learning methods (A partial list includes [23][24][25][26][27][28][29][30][31] ). These methods promise more versatility and the ability to avoid the curse of dimensionality, the exponential increase in complexity with the number of dimensions.…”

Section: Pde Solving Examplesmentioning

confidence: 99%

Born-Infeld (BI) for AI: Energy-Conserving Descent (ECD) for Optimization

Luca¹,

Silverstein²

2022

Preprint

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We introduce a novel framework for optimization based on energy-conserving Hamiltonian dynamics in a strongly mixing (chaotic) regime and establish its key properties analytically and numerically. The prototype is a discretization of Born-Infeld dynamics, with a squared relativistic speed limit depending on the objective function. This class of frictionless, energy-conserving optimizers proceeds unobstructed until slowing naturally near the minimal loss, which dominates the phase space volume of the system. Building from studies of chaotic systems such as dynamical billiards, we formulate a specific algorithm with good performance on machine learning and PDE-solving tasks, including generalization. It cannot stop at a high local minimum and cannot overshoot the global minimum, yielding an advantage in non-convex loss functions, and proceeds faster than GD+momentum in shallow valleys.

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“…Namely, in function approximation, under certain conditions, single-hidden-layer neural networks which called shallow neural networks can approximate well continuous functions on bounded domains. Networks with many hidden-layers called deep neural networks which revolutionized the field of approximation theory e.g., [1,3,4,5,6,12,11,16,19,20,21] and when solving partial differential equations using deep learning techniques [2,7,8,13,18,22]. In the literature, there exist several results about approximation properties of deep neural networks, where authors use different activation functions in order to unraveling the extreme efficiency of deep neural networks.…”

Section: Introductionmentioning

confidence: 99%