LiftGCN: Efficient Energy-Preserving Graph Learning via Joukowski Spectral Lifting for Finite Element Stress Prediction

Details

Finite element stress fields have a property that makes standard GNNs poorly suited for them: stress concentrations near holes, notches, and loading points create sharp spatial gradients and high-frequency graph components. Conventional message-passing GNNs act as low-pass filters — they progressively smooth out exactly the information that matters most for structural integrity analysis. LiftGCN addresses this via Joukowski spectral lifting: the real spectrum of a normalized graph operator is mapped onto the unit circle through the Joukowski transform, realized as a simple second-order recurrence rather than matrix exponentials or eigendecomposition. This gives unit-modulus characteristic roots, meaning spectral magnitudes are preserved rather than attenuated with depth — an energy-preserving structure under a positive-definite metric. Each layer requires only one sparse neighborhood aggregation (O(ed) complexity), with lightweight nonlinear residuals for expressivity. The spectral stability prevents the high-frequency loss that makes deep GNNs unreliable for stress fields. Experiments on finite element stress prediction show LiftGCN achieves competitive overall accuracy while specifically improving reconstruction of stress concentrations and high-gradient structures with lower computational cost than alternatives. Code is publicly available. This is a niche but clean domain application — relevant to ML engineers building surrogate models for structural simulation.