Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach

•An algorithm based on deep neural networks is proposed to solve high-dimensional eigenvalue problems.•The fixed-point reformulation and Feynman-Kac formula provide a natural loss function to optimize the neural networks.•The algorithm shares a similar spirit with Diffusion Monte Carlo but overcomes the shortcoming in providing the gradient.•Numerical results demonstrate the accuracy of the method in terms of both eigenvalue and eigenfunction (with gradients).•The developed algorithm is equally applicable to high-dimensional nonlinear operators.