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）领域的研究重点在于将偏微分方程（PDE）信息融入学习过程中。多数工作聚焦于相对“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，后一类问题需要对一种体积元或守恒约束进行控制，这被公认为具有挑战性。实现SciML的愿景需要将两类问题无缝融入学习过程。为解决这一问题，我们提出ProbConserv框架，用于将守恒约束整合到通用SciML架构中。具体而言，ProbConserv将守恒定律的积分形式与贝叶斯更新相结合。我们对ProbConserv在广义多孔介质方程（GPME）学习中的表现进行了详细分析——GPME是一族应用广泛的参数化PDE，能同时体现简单与困难PDE的定性特征。ProbConserv在处理简单GPME变体时效果显著，与最先进竞争方法表现相当；而在处理困难GPME变体时，其性能优于其他不保证体积守恒的方法。ProbConserv能无缝执行物理守恒约束，维持概率不确定性量化（UQ），并有效处理激波和异方差性问题。在各类下游任务中，该方法均实现了卓越的预测性能。