Hamiltonian neural networks for solving equations of motion

Mattheakis, Marios; Sondak, David; Protopapas, Pavlos

doi:10.1103/physreve.105.065305

Cited by 54 publications

(29 citation statements)

References 37 publications

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“…where q and p represent, respectively, position and momenta of discrete particles. A similar approach is followed in [70]. The same loss function in Eq.…”

Section: Neural Network For Conservative Systemsmentioning

confidence: 99%

Thermodynamics of learning physical phenomena

Chinesta¹

2022

Preprint

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Thermodynamics could be seen as an expression of physics at a high epistemic level. As such, its potential as an inductive bias to help machine learning procedures attain accurate and credible predictions has been recently realized in many fields. We review how thermodynamics provides helpful insights in the learning process. At the same time, we study the influence of aspects such as the scale at which a given phenomenon is to be described, the choice of relevant variables for this description or the different techniques available for the learning process.

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“…where q and p represent, respectively, position and momenta of discrete particles. A similar approach is followed in [70]. The same loss function in Eq.…”

Section: Neural Network For Conservative Systemsmentioning

confidence: 99%

Thermodynamics of learning physical phenomena

Chinesta¹

2022

Preprint

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“…Supervised PINNs can be trained on data to learn nonlinear differential operators [14], discover differential equations [21,20], and solve inverse problems [1,4,18]. Unsupervised PINNs can be trained without using any labeled data to discover analytical and differentiable solutions of ordinary [12,16] or partial differential equations (PDEs) [22,11], and eigenvalue problems [7,19,13,8].…”

Section: Introductionmentioning

confidence: 99%

First principles physics-informed neural network for quantum wavefunctions and eigenvalue surfaces

Mattheakis¹,

Schleder²,

Larson³

et al. 2022

Preprint

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Physics-informed neural networks have been widely applied to learn general parametric solutions of differential equations. Here, we propose a neural network to discover parametric eigenvalue and eigenfunction surfaces of quantum systems. We apply our method to solve the hydrogen molecular ion. This is an ab initio deep learning method that solves the Schrödinger equation with the Coulomb potential yielding realistic wavefunctions that include a cusp at the ion positions. The neural solutions are continuous and differentiable functions of the interatomic distance and their derivatives are analytically calculated by applying automatic differentiation. Such a parametric and analytical form of the solutions is useful for further calculations such as the determination of force fields.

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“…The NN consists of multiple hidden layers with trigonometric sin(•) function used as the activation function for the hidden neurons. This choice of activation has been shown to improve PINNs' performance in solving nonlinear dynamical systems [76] and high-dimensional partial differential equations [71]. The outputs of the NN are the solutions N x (t) ∈ R n and N u (t) ∈ R m .…”

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confidence: 99%

“…The outputs of the NN are the solutions N x (t) ∈ R n and N u (t) ∈ R m . We construct a neural state and control vectors that identically satisfies the initial conditions by parametrizing x(t) = x(0) + f (t)N x (t) and u(t) = u(0) + f (t)N u (t), where f (t) = 1 − e −t is a parametric function satisfying f (0) = 0 [76]. The network parameters, weights and biases, are randomly initialized and then, they are optimized by minimizing a physics-informed loss function defined by…”

mentioning

confidence: 99%

“…( 2), we employ Adam optimizer [77]. Moreover, the points t i are randomly perturbed during training iteration-this method has been shown to improve the training and the neural predictions [70,76]. Two-level system with amplitude damping and one control field.-We consider a two-level system as a proof-ofprinciple example to illustrate the power of PINNs for QC.…”

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confidence: 99%

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