Learning Structural Eigenmodes with Modal Operator Network (ModalONet)

Abstract

Neural operators such as the deep operator network (DeepONet) and the Laplace neural operator (LNO) are effective surrogates, but they are almost exclusively trained to reproduce the forward response of a system rather than its intrinsic structure, such as the modal properties of a structural system. We introduce ModalONet, which puts operator learning to a different use, such as recovering the modal basis, namely: mode shapes, natural frequencies, and damping ratios of a dynamical system directly from its response field, with no eigensolver and no labeled modes. Our key observation is that the DeepONet branch-trunk factorization is itself a learnable form of modal superposition: the trunk supplies continuous, mesh-free mode shapes, while an LNO branch supplies the modal coordinates in pole-residue form, so that each learned pole yields a natural frequency and a damping ratio. Training uses the response field alone, under a composite loss of reconstruction, orthonormality, and temporal projection consistency. The degenerate (equal-frequency) modes are resolved by a separable trunk and a shared frequency parameter, with post-hoc log-envelope regression improves the damping ratio estimates. Across four structural systems, namely: simply supported and cantilever Euler-Bernoulli beams and rectangular and (degenerate) square Kirchhoff plates, the ModalONet recovers the analytical modal basis with modal assurance criterion (MAC) values of at least 0.998 for every mode shape, natural frequency errors within 5%, and damping ratio errors within 7%, demonstrating the potential of neural operators as accurate and interpretable tools for modal identification.