Physics-guided thermoelastic state-space learning for spindle thermal error in machine tools

Thermal error of motorized spindles is a primary source of accuracy degradation in machine tools. Existing prediction methods often exhibit limited reliability under varying operating conditions and sparse data, as physical mechanisms and structural information are insufficiently exploited, leading poor robustness and weak prediction accuracy. To address the above challenges, a physics-informed thermal error modeling and compensation framework is proposed for spindles in machine tools. An implicitly coupled thermoelastic state-space network is formulated to model temperature evolution and deformation accumulation. Sliding windows are used to extract local temperature patterns, while sensor layout and functional regions are encoded to represent structural information. Long-term thermoelastic states are propagated to capture cumulative thermal deformation. To enforce physical consistency during training, a differentiable finite-element residual regularization term is introduced. A weak-form heat-conduction constraint is embedded through learnable stiffness and mass operators and is optimized jointly with the data-driven loss. Experiments were conducted on spindle thermal datasets. The experimental results indicate that improved prediction accuracy and enhanced cross-condition generalization are obtained compared with several representative deep learning baselines. A digital twin system is further implemented to enable real-time sensing, prediction, and closed-loop compensation. Milling experiments on annular workpieces show a reduction in surface flatness error of approximately 65 %, demonstrating the practical effectiveness of the developed framework for thermal error control in precision machining.