With the recent success of representation learning methods, which includes deep learning as a special case, there has been considerable interest in developing techniques that incorporate known physical constraints into the learned representation. As one example, in many applications that involve a signal propagating through physical media (e.g., optics, acoustics, fluid dynamics, etc), it is known that the dynamics of the signal must satisfy constraints imposed by the wave equation. Here we propose a matrix factorization technique that decomposes such signals into a sum of components, where each component is regularized to ensure that it {nearly} satisfies wave equation constraints. Although our proposed formulation is non-convex, we prove that our model can be efficiently solved to global optimality. Through this line of work we establish theoretical connections between wave-informed learning and filtering theory in signal processing. We further demonstrate the application of this work on modal analysis problems commonly arising in structural diagnostics and prognostics.

翻译：随着表示学习方法（深度学习为其特例）的近期成功，将已知物理约束融入所学表征的技术引起了广泛关注。例如，在涉及信号通过物理介质（如光学、声学、流体动力学等）传播的众多应用中，信号的动态特性必须满足波动方程所施加的约束。本文提出一种矩阵分解技术，将此类信号分解为若干分量之和，其中每个分量均通过正则化处理以确保其{近似}满足波动方程约束。尽管所提公式是非凸的，但我们证明该模型可高效求解全局最优解。通过此项研究，我们建立了波动信息学习与信号处理中滤波理论之间的理论联系。此外，我们进一步展示了该方法在结构诊断与预测中常见的模态分析问题上的应用。