Signed particles and neural networks, towards efficient simulations of quantum systems.

*(English)*Zbl 1456.81018Summary: Recently, two approaches were suggested which combine signed particles and neural networks to speed up the time-dependent simulation of quantum systems. Both specialize on the efficient computation of a multi-dimensional function defined over the phase space known as the Wigner kernel. While the first approach completely defines the network analytically, the second is based on an architecture with generalization capabilities. Although relatively simple, these networks can reduce the amount of memory needed and provide a computational speedup, but they are both affected by the use of expensive activation functions (sinusoidals). In this work, we go beyond these previous strategies and suggest a more general network consisting of a set of different hidden layers which are based on less expensive activation functions (e.g. rectified linear units), and which now predict one column of the discretized kernel at a time. This approach comes with generalization capabilities and allows the network to accurately learn a transform from the space of potentials to the space of kernels. As it is shown in our final validation test, this new approach performs very well during the simulation of quantum systems. In fact, while keeping a good accuracy, it further reduces the amount of memory required along with its computational burden.

##### MSC:

81P05 | General and philosophical questions in quantum theory |

81-10 | Mathematical modeling or simulation for problems pertaining to quantum theory |

68T05 | Learning and adaptive systems in artificial intelligence |

68T07 | Artificial neural networks and deep learning |

81S30 | Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics |

65Y10 | Numerical algorithms for specific classes of architectures |

65Y20 | Complexity and performance of numerical algorithms |

65Z05 | Applications to the sciences |

##### Keywords:

quantum mechanics; machine learning; signed particle formulation; neural networks; simulation of quantum systems
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\textit{J. M. Sellier} et al., J. Comput. Phys. 387, 154--162 (2019; Zbl 1456.81018)

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