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：近期科学机器学习领域的研究聚焦于将偏微分方程信息融入学习过程。大部分工作侧重于相对“简单”的偏微分方程算子（如椭圆型和抛物型），而对相对“困难”的偏微分方程算子（如双曲型）关注较少。在数值偏微分方程中，后一类问题需要对体积元或守恒约束进行控制，这被公认为具有挑战性。实现科学机器学习的承诺需要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框架，该框架可将守恒约束纳入通用科学机器学习架构。具体方法是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）对ProbConserv进行详细分析——该方程是一类广泛适用的参数化偏微分方程族，同时展现了简单与困难偏微分方程的定性特征。对于简单的GPME变体，ProbConserv能与最先进的竞争方法表现相当；对于困难的GPME变体，它则优于其他无法保证体积守恒的方法。ProbConserv能够无缝施加物理守恒约束、保持概率不确定性量化，并有效处理激波和异方差性问题。在各类下游任务中，该方法均取得了更优的预测性能。