Mathematics of Digital Twins and Transfer Learning for PDE Models

We define a digital twin (DT) of a physical system governed by partial differential equations (PDEs) as a model for real-time simulations and control of the system behavior under changing conditions. We construct DTs using the Karhunen-Lo\`{e}ve Neural Network (KL-NN) surrogate model and transfer learning (TL). The surrogate model allows fast inference and differentiability with respect to control parameters for control and optimization. TL is used to retrain the model for new conditions with minimal additional data. We employ the moment equations to analyze TL and identify parameters that can be transferred to new conditions. The proposed analysis also guides the control variable selection in DT to facilitate efficient TL. For linear PDE problems, the non-transferable parameters in the KL-NN surrogate model can be exactly estimated from a single solution of the PDE corresponding to the mean values of the control variables under new target conditions. Retraining an ML model with a single solution sample is known as one-shot learning, and our analysis shows that the one-shot TL is exact for linear PDEs. For nonlinear PDE problems, transferring of any parameters introduces errors. For a nonlinear diffusion PDE model, we find that for a relatively small range of control variables, some surrogate model parameters can be transferred without introducing a significant error, some can be approximately estimated from the mean-field equation, and the rest can be found using a linear residual least square problem or an ordinary linear least square problem if a small labeled dataset for new conditions is available. The former approach results in a one-shot TL while the latter approach is an example of a few-shot TL. Both methods are approximate for the nonlinear PDEs.