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.

翻译：连续场的联合预测与底层离散参数的统计估计是许多由偏微分方程控制的物理系统的常见问题。迄今为止，该问题一直通过采用算子学习代理进行场预测，同时使用基于仿真的推断（及其变体）进行统计参数确定来分别解决。本文认为，在同一框架内解决这两个问题可以带来精度和鲁棒性的一致提升。为此，我们提出了一种新颖且灵活的算子学习问题表述，能够联合预测连续量并推断离散参数的分布，从而将逆模型和代理模型的成本分摊到联合预训练步骤中。我们展示了所提方法在不同信息缺失水平下，针对全身血流动力学仿真中连续和离散生物标志物预测的能力。我们还考虑了一个二维干冷气泡大气大涡模拟的测试案例，在该案例中我们同时推断连续时间序列和系统状态信息。通过与不同基线的比较，我们展示了在逆任务和代理任务中均显著提升的准确性。