Decoupling thermo-mechanical signals in ionic hydrogels via deep operator networks

In responsive hydrogels, concurrent thermal and mechanical stimuli induce inseparable electrical signals due to the superposition of the ionic thermoelectric and piezoionic effects, a fundamental challenge in soft ionotronics. To address this, we propose the T-DeepONet model, which integrates the superior temporal modeling capability of the transformer with the spatial encoding of the deep operator network (DeepONet) to learn the complex thermo-mechanical operator. The model is trained on a comprehensive synthetic dataset generated from experimentally validated finite element simulations, enabling T-DeepONet to map the coupled voltage fields to independent temperature and pressure distributions. By integrating transformer-based temporal modeling with DeepONet's spatial encoding, T-DeepONet resolves the distinct spatiotemporal signatures of thermal diffusion and mechanical transients, achieving 98.2% R2 accuracy across synchronous and asynchronous loading scenarios with ∼100 ms inference latency. This work establishes a general framework for real-time, field-level disentanglement in multiphysics soft matter systems, opening avenues for high-fidelity tactile perception in soft robotics and bridging advances in nonequilibrium ion transport with operator learning.