The solution of boundary-value problems using deep learning approaches, such as the physics- informed neural networks [1] or the deep operator networks [2], have been extensively investigated in recent years. However, achieving high accuracy in the approximations obtained from these methods often remains a significant challenge. A multi-level neural network approach was proposed in [3] that allows one to iteratively reduce the errors, sometimes within machine precision, when approximating a solution using PINNs. In this work, we extend the multi-level approach to approximate linear operators using the Green operator networks (GreenONets) as described in [4]. Starting with an initial approximation of the operator, we correct the solution by considering a different Green operator network involving higher frequencies. The method enables one to iteratively reduce the high-frequency contributions present in the residuals. Numerical examples will be presented to demonstrate the efficiency of the proposed multi-level approach. References [1] M. Raissi, P. Perdikaris, and G.E. Karniadakis. Physics-informed neural networks: A deep learning frame- work for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378: 686-707, 2024. [2] L. Lu, P. Jin, G. Pang, Z. Zhang, and G.E. Karniadakis. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature machine intelligence 3.3: 218-229, 2021. [3] Z. Aldirany, R. Cottereau, M. Laforest, and S. Prudhomme. Multi-level neural networks for accurate solutions of boundary-value problems. Computer Methods in Applied Mechanics and Engineering 419: 116666, 2024. [4] Z. Aldirany, R. Cottereau, M. Laforest, and S. Prudhomme. Operator approximation of the wave equation based on deep learning of Green’s function. Computer & Mathematics with Applications, 159: 21–30, 2024.