Neural operators (NOs) have emerged as effective tools for modeling complex physical systems in scientific machine learning. In NOs, a central characteristic is to learn the governing physical laws directly from data. In contrast to other machine learning applications, partial knowledge is often known a priori about the physical system at hand whereby quantities such as mass, energy and momentum are exactly conserved. Currently, NOs have to learn these conservation laws from data and can only approximately satisfy them due to finite training data and random noise. In this work, we introduce conservation law-encoded neural operators (clawNOs), a suite of NOs that endow inference with automatic satisfaction of such conservation laws. ClawNOs are built with a divergence-free prediction of the solution field, with which the continuity equation is automatically guaranteed. As a consequence, clawNOs are compliant with the most fundamental and ubiquitous conservation laws essential for correct physical consistency. As demonstrations, we consider a wide variety of scientific applications ranging from constitutive modeling of material deformation, incompressible fluid dynamics, to atmospheric simulation. ClawNOs significantly outperform the state-of-the-art NOs in learning efficacy, especially in small-data regimes.

翻译：神经算子已成为科学机器学习中建模复杂物理系统的有效工具。核心理念在于直接从数据中学习支配物理定律。与其它机器学习应用不同，关于物理系统的部分先验知识（如质量、能量和动量的精确守恒）通常是已知的。当前，神经算子必须从数据中学习这些守恒律，且由于训练数据有限和随机噪声的存在，只能近似满足这些定律。本研究引入了守恒律编码神经算子（clawNOs）——一类赋予推理过程自动满足此类守恒律能力的神经算子。clawNOs通过构建解场的无散度预测，自动保证连续性方程的成立。因此，clawNOs符合最基础且最普遍的守恒律，这些定律对于正确的物理一致性至关重要。为验证效果，我们针对从材料变形本构建模、不可压缩流体动力学到大气模拟等多种科学应用进行了广泛测试。clawNOs在学习效能上显著优于现有最先进的神经算子，尤其在数据稀缺情境下表现更为突出。