This paper presents a novel approach to solving the quantum many-body problem using artificial neural networks (NNs). The authors introduce a variational representation of quantum states based on NNs with variable numbers of hidden neurons. This approach allows for the efficient description of both ground-state and time-dependent properties of complex interacting quantum systems. The NN-based wave function, called Neural-Network Quantum States (NQS), is trained using reinforcement learning to minimize the energy expectation value, enabling accurate simulations of quantum systems. The wave function is interpreted as a computational black box that maps many-body configurations to complex numbers representing the quantum state. The authors demonstrate that NQS can achieve high accuracy in describing equilibrium and dynamical properties of spin models in one and two dimensions. For example, in the transverse-field Ising (TFI) and antiferromagnetic Heisenberg (AFH) models, NQS outperforms traditional methods like matrix product states (MPS) and achieves results comparable to exact calculations. The NQS approach is particularly effective in handling the sign problem in quantum Monte Carlo (QMC) simulations and provides a more compact representation of quantum states compared to MPS. The authors show that increasing the number of hidden units in the NN improves the accuracy of the wave function representation, with results reaching MPS-grade accuracy in one dimension and significantly improving variational states in two dimensions. The method is also extended to time-dependent problems, where the NN weights are trained to reproduce quantum dynamics according to the Dirac-Frenkel time-dependent variational principle. This allows for the simulation of unitary time evolution, including quantum quenches in spin models. The results demonstrate that NQS can accurately describe the time evolution of quantum systems, even in the presence of complex interactions. The paper highlights the potential of NQS to solve the quantum many-body problem in regimes inaccessible to traditional numerical methods. The approach leverages the power of machine learning to capture the essential features of quantum states, offering a promising new tool for studying complex quantum systems. The authors conclude that NQS can be a powerful and flexible method for solving the quantum many-body problem, with applications ranging from fundamental questions in quantum physics to practical simulations of interacting fermions in two dimensions.This paper presents a novel approach to solving the quantum many-body problem using artificial neural networks (NNs). The authors introduce a variational representation of quantum states based on NNs with variable numbers of hidden neurons. This approach allows for the efficient description of both ground-state and time-dependent properties of complex interacting quantum systems. The NN-based wave function, called Neural-Network Quantum States (NQS), is trained using reinforcement learning to minimize the energy expectation value, enabling accurate simulations of quantum systems. The wave function is interpreted as a computational black box that maps many-body configurations to complex numbers representing the quantum state. The authors demonstrate that NQS can achieve high accuracy in describing equilibrium and dynamical properties of spin models in one and two dimensions. For example, in the transverse-field Ising (TFI) and antiferromagnetic Heisenberg (AFH) models, NQS outperforms traditional methods like matrix product states (MPS) and achieves results comparable to exact calculations. The NQS approach is particularly effective in handling the sign problem in quantum Monte Carlo (QMC) simulations and provides a more compact representation of quantum states compared to MPS. The authors show that increasing the number of hidden units in the NN improves the accuracy of the wave function representation, with results reaching MPS-grade accuracy in one dimension and significantly improving variational states in two dimensions. The method is also extended to time-dependent problems, where the NN weights are trained to reproduce quantum dynamics according to the Dirac-Frenkel time-dependent variational principle. This allows for the simulation of unitary time evolution, including quantum quenches in spin models. The results demonstrate that NQS can accurately describe the time evolution of quantum systems, even in the presence of complex interactions. The paper highlights the potential of NQS to solve the quantum many-body problem in regimes inaccessible to traditional numerical methods. The approach leverages the power of machine learning to capture the essential features of quantum states, offering a promising new tool for studying complex quantum systems. The authors conclude that NQS can be a powerful and flexible method for solving the quantum many-body problem, with applications ranging from fundamental questions in quantum physics to practical simulations of interacting fermions in two dimensions.