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（NOs）已成为科学机器学习中建模复杂物理系统的有效工具。其核心特征在于直接从数据中学习支配性物理定律。与其他机器学习应用不同，通常可先验获知所研究物理系统的部分知识，例如质量、能量和动量等物理量需精确守恒。当前，神经算子需从数据中学习这些守恒定律，且由于有限训练数据和随机噪声，仅能近似满足这些定律。本文提出守恒定律编码神经算子（clawNOs），这是一类通过推理自动满足此类守恒定律的神经算子。clawNOs构建于解场的无散度预测之上，从而自动保证连续性方程成立。因此，clawNOs严格符合对确保物理一致性至关重要的最基础且最普遍的守恒定律。为验证其性能，我们考虑了从材料变形本构建模、不可压缩流体力学到大气模拟等广泛科学应用。clawNOs在学习效能上显著优于当前最先进的神经算子，尤其在数据稀缺场景下表现突出。