A joint optimization network for force identification on nonlinear structures

Accurate force identification is fundamental to the structural health monitoring and integrity assessment of complex engineering systems. However, nonlinear structural dynamics invalidate the superposition principle and render the inverse problem ill-posed, making dynamic force identification highly challenging. Existing model-driven and algorithm unrolling (AU) methods rely on explicit analytical forward models to compute Jacobian matrices, which are often unavailable for practical nonlinear structures. To overcome this limitation, this paper proposes a joint optimization network (JO-Net) for nonlinear structural force identification. JO-Net replaces the analytical Jacobian with an automatically differentiable surrogate model embedded in a deep unrolling framework. The surrogate network learns nonlinear forward dynamics and provides sensitivity information via automatic differentiation, while the proximal gradient descent algorithm is unrolled into a layer-wise network with a convolutional encoder–decoder serving as a learnable proximal operator. A key feature of JO-Net is the simultaneous optimization of the surrogate model and the inverse AU network, enabling adaptive refinement of structural mappings and mitigation of modeling errors. Numerical and experimental validations on nonlinear structures demonstrate that JO-Net significantly outperforms purely data-driven and model-based Kalman filter methods. By combining the surrogate model gradient with a data-driven gradient mapping matrix, the proposed method achieves a better balance between robustness and accuracy. Furthermore, JO-Net demonstrates superior robustness against measurement uncertainties and maintains reliable identification performance for unseen force–amplitude ranges within the tested nonlinear structures.