FUSE: Fast Unified Simulation and Estimation for PDEs

Partial Differential Equations (PDEs) describe the propagation of system conditions for a very wide range of physical systems. Parametric PDEs not only consider different system conditions, but the underlying solution operator is also characterized by a set of discrete parameters. Traditional numerical methods based on different discretization schemes such as Finite Differences, Finite Volumes and Finite Elements have been developed along with fast and parallelizable implementations to tackle complex problems, such as atmospheric modeling and cardiovascular biomechanics. From parametric PDEs, these methods define maps from the underlying set of discrete parameters, which describe the dynamics and the boundary/initial conditions, to physical quantities such as velocity or pressure that are continuous in the spatio-temporal domain. Despite their successful application, there still exist well-known drawbacks of traditional solvers. To describe a particular physical phenomenon, PDE parameters and solvers need to be calibrated on precise conditions that are not known a priori and cannot easily be measured in realistic applications. Therefore, iterative and thus expensive calibration procedures are considered in the cases where the parameters and conditions are inferred from data [1]. Even after the solvers are calibrated, an ensemble of solutions need to be generated to account for uncertainties in the model parameters or assess the sensitivity of the solution to different parameters which are computationally prohibitive downstream tasks [2]. For these reasons, a large variety of deep learning algorithms have recently been proposed for scientific applications, broadly categorized into surrogate and inverse modeling algorithms, to either reduce the computational time of complex simulations or infer missing discrete information from data to calibrate a simulator to precise conditions.