This paper presents a framework for manipulating deformable linear objects (DLOs) with online model parameter estimation. The framework combines a neural network (NN) model trained to mimic the dynamics of DLOs with a gradient-based approach for estimating both the optimal manipulation actions and model parameters. The NN model is conditioned on analytical model parameters, allowing it to adapt to different real-world DLOs. During the online phase, the NN model is used to estimate the best manipulation action to steer the DLO from its initial to a final target configuration, while simultaneously adapting the model parameters to better match the observed deformations. The framework is tested across various DLOs, surfaces, and target shapes, demonstrating its effectiveness compared to existing methods. The NN-based model is more computationally efficient and scalable than the analytical model, making it suitable for real-time applications. The framework also enables online adaptation of model parameters, eliminating the need for task-specific data generation or complex online adaptation controllers. The results show that the proposed method achieves accurate shape control with minimal error, outperforming other approaches in terms of efficiency and adaptability. The method is applicable to a wide range of DLOs and surfaces, making it a versatile solution for robotic manipulation tasks.This paper presents a framework for manipulating deformable linear objects (DLOs) with online model parameter estimation. The framework combines a neural network (NN) model trained to mimic the dynamics of DLOs with a gradient-based approach for estimating both the optimal manipulation actions and model parameters. The NN model is conditioned on analytical model parameters, allowing it to adapt to different real-world DLOs. During the online phase, the NN model is used to estimate the best manipulation action to steer the DLO from its initial to a final target configuration, while simultaneously adapting the model parameters to better match the observed deformations. The framework is tested across various DLOs, surfaces, and target shapes, demonstrating its effectiveness compared to existing methods. The NN-based model is more computationally efficient and scalable than the analytical model, making it suitable for real-time applications. The framework also enables online adaptation of model parameters, eliminating the need for task-specific data generation or complex online adaptation controllers. The results show that the proposed method achieves accurate shape control with minimal error, outperforming other approaches in terms of efficiency and adaptability. The method is applicable to a wide range of DLOs and surfaces, making it a versatile solution for robotic manipulation tasks.