An application of neural networks for solving quantum mechanical problems has been
suggested in [1,2]. Many improvements, including an adaptation of deep neural network techniques
[3], have been proposed since. Development of a new computational technology which could lift
the curse of dimensionality, however, has not yet been completed, although some steps in this
direction have already been made [4,5].
We propose a new approach to training neural networks for approximation of quantum
Hamiltonian invariant subspaces corresponding to bound states. The approach is based on training
an artificial neural network to solve the Schr ̈odinger equation in imaginary time with initial
conditions that put the solution into an invariant subspace.
The advantage of the proposed approach is a simpler objective function which leads to better
performance.
Theoretical results are illustrated with numerical examples.

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2. I.E. Lagaris, A. Likas, and D.I. Fotiadis, Artificial neural networks in quantum mechanics // Comp.
Phys. Comm. 1997, V.104, P.1-14
3. Sirignano, J., Spiliopoulos, K., DGM: A deep learning algorithm for solving partial differential
equations// arXiv preprint arXiv:1708.07469
4. Hong Li, Qilong Zhai, Jeff Z. Y. Chen, Neural-network-based multistate solver for a static
Schr ̈odinger equation // Phys.Rev. A 2021,V. 103, P. 032405
5. V.A. Roudnev, M.M. Stepanova, Deep learning approach to high dimensional problems of quantum
mechanics // Proceedings of Science 2022, V.429, P. 13