FUSE: Fast Unified Simulation and Estimation for PDEs

TL;DR

This work proposes a novel and flexible formulation of the operator learning problem that allows jointly predicting continuous quantities and inferring distributions of discrete parameters, and thus amortizing the cost of both the inverse and the surrogate models to a joint pre-training step.

Abstract

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 continuous quantities and inferring distributions of discrete parameters, and thus amortizing the cost of both the inverse and the surrogate models to a joint pre-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. We also consider a test case for atmospheric large-eddy simulation of a two-dimensional dry cold bubble, where we infer both continuous time-series and information about the systems conditions. We present comparisons against different baselines to showcase significantly increased accuracy in both the inverse and the surrogate tasks.

Figures (19)

• Figure 1: FUSE models a posterior distribution over finite-dimensional parameters $\xi$ given infinite-dimensional functions $u$ with $d_u$ components (channels). It learns other continuous functions $s$ with $d_s$ channels from parameters $\xi$. Band-limited Fourier transforms and a lifting operator act as a bridge between finite and infinite dimensions for the forward problem. Likewise, as inference models such as FMPE or NPE require fixed-size inputs, the operator layers are conjoined with a band-limited Fourier transform to learn a fixed-size representation of the input function.
• Figure 2: ACB, propagated uncertainty: Comparison of the benchmark models on the horizontal velocities at location 1, showing the unified ensemble predictions for the sample where FUSE has the greatest $L^1$ error, i.e. a worst-case scenario. *GAROM predicts the velocity from the true parameters, while all other models predict pressure from the posterior distribution over parameters given continuous inputs.
• Figure 3: PWP, Comparison of models ability to predict the Sup. Middle Cerebral pressure and its uncertainty for the sample where FUSE has the greatest $L^1$ error, i.e. a worst-case scenario. *see Figure \ref{['fig:acb worst case prediction with uncertainty']}.
• Figure 4: ACB, sensitivity analysis and generalization: Validation of the FUSE model against numeric simulations on peak horizontal velocities $u$ at location 1 ($x=15$ km, $z=50$ m). Samples above the dashed line correspond to amplitudes larger than seen during training. The parameter resolution is $100\times 100$ for FUSE, and $(10+10)\times 10$ for the numerical samples. Figure continued in the appendix, Figure \ref{['fig:acb ood']}.
• Figure A.1: PWP: Box plots of the errors of the forward and inverse components of FUSE, as reported in Table \ref{['tab:results inv for both']}, for different levels of available input information ("test cases"). For the inverse problem, parameter samples $\xi_i$ are sampled from $\rho^{\phi}(\xi^i | u)$. For the forward problem, the output function $s$ is predicted based on the true parameter values $\xi^* \sim \rho(\xi | u)$. For the unified problem evaluating both the inverse and forward model parts, the mean $\Bar{s}$ of the ensemble prediction $s_i$ from inferred parameters $\xi_i \sim \rho^{\phi}(\xi^i | u)$ is compared to the true output time series $s$. As information is removed from the input in the different cases, it becomes more difficult to estimate $s$.