FUSE: Fast Unified Simulation and Estimation for PDEs

The joint prediction of continuous fields and statistical estimation of the underlying discrete parameters is a common problem for many physical systems governed by PDEs. Hitherto, it has been separately addressed by employing operator learning surrogates for field prediction while using simulation-based inference (and its variants) for statistical parameter determination. Here, we argue that solving both problems within the same framework can lead to consistent gains in accuracy and robustness. To this end, we propose a novel and flexible formulation of the operator learning problem that allows jointly predicting infinite-dimensional quantities and inferring distributions of finite-dimensional parameters and thus amortizing the cost of both the inverse and the surrogate models to a joint training step. We present the capabilities of the proposed methodology for predicting continuous and discrete biomarkers in full-body haemodynamics simulations under different levels of missing information, and in a test case for atmospheric large-eddy simulation of a two-dimensional dry cold bubble, predicting continuous time-series measurements from inferred system conditions. For both cases, we present comparisons against different baselines to demonstrate significantly increased accuracy in both the inverse and surrogate tasks.