Abstract

Structural parameter identification in high-order partial differential equations (PDEs), such as the Euler-Bernoulli beam equation, remains challenging due to the computation of high-order derivative operators, particularly when structural parameters vary spatially. Current physics-informed machine learning approaches, including Physics-Informed Neural Networks (PINNs) and Universal Differential Equations (UDEs), typically require expensive automatic differentiation (AD) or adjoint calculations. These methods often fail when measurement data is noisy or boundary conditions (BCs) are unknown. This paper proposes Neural-VSI, a variational framework that parameterizes unknown fields with neural networks and reformulates the inverse problem using local variational forms with pre-computed integration weight matrices. Our method bypasses the heavy computational overhead of AD, allowing for the estimation of hard-to-measure structural parameter fields without utilizing boundary condition information. Experiments on the vibration of beams with two unknown distributed parameters demonstrate that this approach achieves a speedup of over 200$\times$ compared to strong-form baselines while maintaining robust identification accuracy.