Role of physics in physics-informed machine learning

Physical systems are characterized by inherent symmetries, one of which is encapsulated in the units of their parameters and system states. These symmetries enable a lossless order-reduction, e.g., via dimensional analysis based on the Buckingham theorem. Despite the latter’s benefits, machine learning (ML) strategies for the discovery of constitutive laws seldom subject experimental and/or numerical data to dimensional analysis. We demonstrate the potential of dimensional analysis to sig-nificantly enhance the interpretability and generalizability of ML-discovered secondary laws. Our numerical experiments with creeping fluid flow past solid ellipsoids show how dimensional analysis enables both deep neural networks and sparse regression to reproduce old results, e.g., Stokes law for a sphere, and generate new ones, e.g., an expression for an ellipsoid misaligned with the flow direction. Our results suggest the need to incorporate other physics-based symmetries and invariances into ML-based techniques for equation discovery.

物理系统具有内在的对称性，其中之一体现在其参数和系统状态的单位中。这些对称性能够实现无损的降阶，例如通过基于白金汉定理的量纲分析。尽管后者有诸多益处，但用于发现本构定律的机器学习（ML）策略很少对实验和/或数值数据进行量纲分析。我们展示了量纲分析在显著提高由机器学习发现的二级定律的可解释性和普适性方面的潜力。我们对粘性流体绕过固体椭球体流动的数值实验表明，量纲分析如何使深度神经网络和稀疏回归既能重现旧的结果，例如球体的斯托克斯定律，又能产生新的结果，例如与流动方向不一致的椭球体的表达式。我们的结果表明，有必要将其他基于物理的对称性和不变性纳入基于机器学习的方程发现技术中。