A Tutorial on the Use of Physics-Informed Neural Networks to Compute the Spectrum of Quantum Systems

• URL: http://arxiv.org/abs/2407.20669v2
• Date: Wed, 11 Sep 2024 11:08:12 GMT
• Title: A Tutorial on the Use of Physics-Informed Neural Networks to Compute the Spectrum of Quantum Systems
• Authors: Lorenzo Brevi, Antonio Mandarino, Enrico Prati,
• Abstract summary: We show how to construct a physics-informed Neural Network (PINN) able to solve the Schr"odinger equation for a given potential. PINNs are a deep learning method that exploits Automatic Differentiation to solve Integro-Differential Equations in a mesh-free way.
• Score: 0.9374652839580183
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum many-body systems are of great interest for many research areas, including physics, biology and chemistry. However, their simulation is extremely challenging, due to the exponential growth of the Hilbert space with the system size, making it exceedingly difficult to parameterize the wave functions of large systems by using exact methods. Neural networks and machine learning in general are a way to face this challenge. For instance, methods like Tensor networks and Neural Quantum States are being investigated as promising tools to obtain the wave function of a quantum mechanical system. In this tutorial, we focus on a particularly promising class of deep learning algorithms. We explain how to construct a Physics-Informed Neural Network (PINN) able to solve the Schr\"odinger equation for a given potential, by finding its eigenvalues and eigenfunctions. This technique is unsupervised, and utilizes a novel computational method in a manner that is barely explored. PINNs are a deep learning method that exploits Automatic Differentiation to solve Integro-Differential Equations in a mesh-free way. We show how to find both the ground and the excited states. The method discovers the states progressively by starting from the ground state. We explain how to introduce inductive biases in the loss to exploit further knowledge of the physical system. Such additional constraints allow for a faster and more accurate convergence. This technique can then be enhanced by a smart choice of collocation points in order to take advantage of the mesh-free nature of the PINN. The methods are made explicit by applying them to the infinite potential well and the particle in a ring, a challenging problem to be learned by an Artificial Intelligence agent due to the presence of complex-valued eigenfunctions and degenerate states.

Related papers

• Fourier Neural Operators for Learning Dynamics in Quantum Spin Systems [77.88054335119074]
  We use FNOs to model the evolution of random quantum spin systems. We apply FNOs to a compact set of Hamiltonian observables instead of the entire $2n$ quantum wavefunction.
  arXiv  Detail & Related papers  (2024-09-05T07:18:09Z)
• Addressing the Non-perturbative Regime of the Quantum Anharmonic Oscillator by Physics-Informed Neural Networks [0.9374652839580183]
  In quantum realm, such approach paves the way to a novel approach to solve the Schroedinger equation for non-integrable systems. We investigate systems with real and imaginary frequency, laying the foundation for novel numerical methods to tackle problems emerging in quantum field theory.
  arXiv  Detail & Related papers  (2024-05-22T08:34:52Z)
• Deep Quantum Graph Dreaming: Deciphering Neural Network Insights into Quantum Experiments [0.5242869847419834]
  We use a technique called $inception$ or $deep$ $dreaming$ to explore what neural networks learn about quantum optics experiments. Our story begins by training deep neural networks on the properties of quantum systems. We find that the network can shift the initial distribution of properties of the quantum system, and we can conceptualize the learned strategies of the neural network.
  arXiv  Detail & Related papers  (2023-09-13T16:13:54Z)
• On the Interpretability of Quantum Neural Networks [0.0]
  Interpretability of artificial intelligence (AI) methods, particularly deep neural networks, is of great interest. Here, we explore the interpretability of quantum neural networks using local model-agnostic interpretability measures commonly utilized for classical neural networks. A feature of our explanations is the delineation of the region in which data samples have been given a random label, likely subjects of inherently random quantum measurements.
  arXiv  Detail & Related papers  (2023-08-22T00:43:14Z)
• An Analysis of Physics-Informed Neural Networks [0.0]
  We present a new approach to approximating the solution to physical systems - physics-informed neural networks. The concept of artificial neural networks is introduced, the objective function is defined, and optimisation strategies are discussed. The partial differential equation is then included as a constraint in the loss function for the problem, giving the network access to knowledge of the dynamics of the physical system it is modelling.
  arXiv  Detail & Related papers  (2023-03-06T04:45:53Z)
• Towards Neural Variational Monte Carlo That Scales Linearly with System Size [67.09349921751341]
  Quantum many-body problems are central to demystifying some exotic quantum phenomena, e.g., high-temperature superconductors. The combination of neural networks (NN) for representing quantum states, and the Variational Monte Carlo (VMC) algorithm, has been shown to be a promising method for solving such problems. We propose a NN architecture called Vector-Quantized Neural Quantum States (VQ-NQS) that utilizes vector-quantization techniques to leverage redundancies in the local-energy calculations of the VMC algorithm.
  arXiv  Detail & Related papers  (2022-12-21T19:00:04Z)
• A quantum algorithm for training wide and deep classical neural networks [72.2614468437919]
  We show that conditions amenable to classical trainability via gradient descent coincide with those necessary for efficiently solving quantum linear systems. We numerically demonstrate that the MNIST image dataset satisfies such conditions. We provide empirical evidence for $O(log n)$ training of a convolutional neural network with pooling.
  arXiv  Detail & Related papers  (2021-07-19T23:41:03Z)
• Conditional physics informed neural networks [85.48030573849712]
  We introduce conditional PINNs (physics informed neural networks) for estimating the solution of classes of eigenvalue problems. We show that a single deep neural network can learn the solution of partial differential equations for an entire class of problems.
  arXiv  Detail & Related papers  (2021-04-06T18:29:14Z)
• The Hintons in your Neural Network: a Quantum Field Theory View of Deep Learning [84.33745072274942]
  We show how to represent linear and non-linear layers as unitary quantum gates, and interpret the fundamental excitations of the quantum model as particles. On top of opening a new perspective and techniques for studying neural networks, the quantum formulation is well suited for optical quantum computing.
  arXiv  Detail & Related papers  (2021-03-08T17:24:29Z)
• Approximating Ground State Energies and Wave Functions of Physical Systems with Neural Networks [11.790752430770636]
  We address the problem of solving the time independent Schr"odinger equation for the ground state solution of physical systems. We propose using end-to-end deep learning approach in a variational optimization scheme for approximating the ground state energies and wave functions.
  arXiv  Detail & Related papers  (2020-11-21T01:30:52Z)
• Machine learning transfer efficiencies for noisy quantum walks [62.997667081978825]
  We show that the process of finding requirements on both a graph type and a quantum system coherence can be automated. The automation is done by using a convolutional neural network of a particular type that learns to understand with which network and under which coherence requirements quantum advantage is possible. Our results are of importance for demonstration of advantage in quantum experiments and pave the way towards automating scientific research and discoveries.
  arXiv  Detail & Related papers  (2020-01-15T18:36:53Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.