A physics-informed multi-fidelity approach for the estimation of differential equations parameters in low-data or large-noise regimes

In this paper, we propose a multi-fidelity approach for parameter estimation problems based on Physics-Informed Neural Networks (PINNs). The proposed methods apply to models expressed by linear or nonlinear di¤erential equations, whose parameters need to be estimated starting from (possibly partial and noisy) measurements of the model’s solution. To overcome the limitations of PINNs in case only few and/or significantly noisy data are available, we propose to train a first Artificial Neural Network (ANN), that provides a low-fidelity solution, using a dataset constructed with numerical simulations of the parametrized mathematical model. Then, we express the solution as the sum of the low-fidelity solution and a correction term, provided by a second ANN, which is trained by simultaneously minimizing the distance of the solution from the data and the residual of the di¤erential equation, evaluated at a suitable set of collocation points. For the setup of the multi-fidelity formulation, we propose two alternative approaches. In the first one, we train the low-fidelity solution with data obtained for a fixed value of the parameters, representing an initial guess for the parameters to be estimated (possibly deriving from prior knowledge). In the second approach, the low-fidelity solution encodes the parametric dependence, being trained on data generated by sampling the parameter space and including the parameters themselves among its independent variables. Numerical tests performed for two benchmark problems, a steady-state and a time-dependent advection-di¤usion-reaction partial di¤erential equations, show that the first approach allows the convergence of the PINN training to accelerate significantly. The second approach, in addition to improving the convergence speed, increases the accuracy of the parameters estimate. This is emphasized in cases characterized by a small number of available measurements (possibly a¤ected by noise). Finally, we show an application of the proposed multi-fidelity approach in the context of cardiac electrophysiology. Specifically, we employ a multi-fidelity PINN to estimate an unknown parameter of a nonlinear model describing the ionic dynamics of cardiac cells, starting from noisy measurements of the transmembrane potential.