A Lightweight Data-Driven Scheme for the Inverse Problem of Morphable Surface by Electromagnetic Actuation

Abstract Four-dimensional (4D) electronics, with the ability to deform over time in response to controlled actuation, have attracted rapidly growing interest for use in soft robots, biomedical devices, and wearable systems. A scheme to determine the amplitude and distribution of the actuation for the structure to deform into target shapes is essential for 4D electronics to operate precisely. Solution to this inverse problem is not straightforward in many cases, especially when nonlinearity governs the deformation. This work proposes a lightweight data-driven framework for the inverse problem in an electromagnetically actuated soft surface that takes the form of a Cu-EGaIn conductive network. The deformation of the soft surface is controlled by adjusting the applied voltage to regulate the current distribution and consequently the force distribution. Morphology measurement via the binocular vision method, combined with cubic interpolation, establishes the relationship between displacement of feature points and applied voltage. A machine learning architecture is then constructed with a balance between accuracy and training data volume, allowing for the inversion from surface morphology to predicted voltage. The framework avoids the construction of complex electromechanical coupling models and large deformation–current library databases. Using a relatively small experimental dataset of fewer than 300 samples, the prediction accuracies for the applied voltage to drive the morphable surface to target shapes are 82.91% for the activated interface position and 84.91% for the applied voltage value. Meanwhile, the proposed scheme was also preliminarily validated for cross-device reconstruction of liquid metal flexible films fabricated using the current manufacturing process. In addition, the data-driven scheme was further validated for inverse reconstruction beyond the training distribution through lightweight and rapid transfer learning. This work demonstrates the effectiveness of machine learning for determining excitation from surface morphology in electromagnetic actuation, serving as a foundation for 4D electronics.