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：科学机器学习（SciML）的最新工作聚焦于将偏微分方程（PDE）信息纳入学习过程。多数研究侧重于相对"简单"的PDE算子（如椭圆型和抛物型），而对相对"困难"的PDE算子（如双曲型）关注较少。在数值PDE领域，后一类问题需要控制某种体积元或守恒约束，而这已知具有挑战性。要兑现SciML的承诺，必须将这两类问题无缝融入学习过程。为解决这一问题，我们提出ProbConserv——一个将守恒约束融入通用SciML架构的框架。具体而言，ProbConserv将守恒律的积分形式与贝叶斯更新相结合。我们通过广义多孔介质方程（GPME）——一个广泛适用且能体现简单与困难PDE定性特性的参数化PDE族——对ProbConserv进行了详细分析。该框架在简单GPME变体上表现有效，与最先进方法性能相当；而在困难GPME变体上，它优于其他无法保证体积守恒的方法。ProbConserv能无缝施加物理守恒约束，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。