SONAR: Sensor-Conditioned Neural Operators with Adaptive Spectral Rotation for Full-Field Prediction on Meshes

Existing physical-field prediction paradigms include autoregressive simulators and in-situ sensors reconstruction, which rely on complete prior field data or dense in-domain sensors and thus cannot be deployed under practical equipment operating conditions. Real engineering systems only provide sparse low-dimensional time-series observations. These observations mismatch target physical fields in both units and spatial locations, making it hard to infer high-dimensional field evolution from limited sensor measurements. This raises a critical challenge for high-precision physical field prediction of edge-side equipment: revealing how sparse heterogeneous observations propagate in high-dimensional field spaces and regulate global field distributions. Therefore, we define a more rigorous edge-side physics-AI task: causal next-step full-field prediction on unstructured meshes from short windows of heterogeneous observed state sequences. Duhamel–Volterra and observability analysis show that the predictable component of the field is fully spanned by causal convolutions of the observed states, unobservable field energy sets an error floor, and the optimal predictor admits a conditional spectral decomposition. We instantiate this framework in SONAR: observation history drives the rotation of mesh spectral basis fields generated by deep message passing and architecturally Gram-whitened; sensor-derived coefficients synthesize the global field, and a curvature-guided active-set residual repairs localized high-frequency errors. Experiments on elastic-beam, heat-plate, and mechanical-leg benchmarks with ablation, uncertainty, and robustness analyses show that SONAR achieves the best accuracy among compared methods under a matched information budget.