Recent work in scientific machine learning (SciML) has focused on incorporating partial differential equation (PDE) information into the learning process. Much of this work has focused on relatively "easy" PDE operators (e.g., elliptic and parabolic), with less emphasis on relatively "hard" PDE operators (e.g., hyperbolic). Within numerical PDEs, the latter problem class requires control of a type of volume element or conservation constraint, which is known to be challenging. Delivering on the promise of SciML requires seamlessly incorporating both types of problems into the learning process. To address this issue, we propose ProbConserv, a framework for incorporating conservation constraints into a generic SciML architecture. To do so, ProbConserv combines the integral form of a conservation law with a Bayesian update. We provide a detailed analysis of ProbConserv on learning with the Generalized Porous Medium Equation (GPME), a widely-applicable parameterized family of PDEs that illustrates the qualitative properties of both easier and harder PDEs. ProbConserv is effective for easy GPME variants, performing well with state-of-the-art competitors; and for harder GPME variants it outperforms other approaches that do not guarantee volume conservation. ProbConserv seamlessly enforces physical conservation constraints, maintains probabilistic uncertainty quantification (UQ), and deals well with shocks and heteroscedasticities. In each case, it achieves superior predictive performance on downstream tasks.

翻译：近期科学机器学习（SciML）领域的研究侧重于将偏微分方程（PDE）信息融入学习过程。大量工作主要关注相对"简单"的PDE算子（如椭圆型与抛物型），而对相对"困难"的PDE算子（如双曲型）关注较少。在数值PDE领域，后一类问题需要对某种体积单元或守恒约束进行控制，这被公认具有挑战性。实现SciML的承诺需要将这两类问题无缝融入学习过程。为解决此问题，我们提出ProbConserv框架，该框架可将守恒约束整合到通用SciML架构中。具体而言，ProbConserv通过结合守恒定律的积分形式与贝叶斯更新实现这一目标。我们以广义多孔介质方程（GPME）——这一广泛适用的参数化PDE族（能同时体现简单与困难PDE的定性特性）——为案例，对ProbConserv进行详细分析。对于简单的GPME变体，ProbConserv与最先进竞争者表现相当；而对于困难的GPME变体，它优于其他无法保证体积守恒的方法。ProbConserv可无缝实施物理守恒约束，保持概率不确定性量化（UQ），并有效处理激波与异方差性。在各类下游任务中，它均展现出卓越的预测性能。