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Wave Physics-informed Matrix Factorizations

Harsha Vardhan Tetali, Joel B. Harley, Benjamin D. Haeffele

TL;DR

This work introduces Wave-Informed Matrix Factorization (WIMF), a representation learning framework that couples data fitting with a wave-equation based regularizer to extract physically interpretable modal components. By formulating the problem as a non-convex matrix factorization with a Helmholtz-inspired constraint and leveraging structured-regularization theory, the authors prove that the model can be solved to global optimality and provide a polynomial-time algorithm. A key insight is that the polar subproblem reduces to a tractable line search over a single wavenumber and yields an interpretation as a bank of first-order Butterworth filters, linking physics, optimization, and signal processing. The method is validated on four vibration datasets (homogeneous, inhomogeneous, traveling waves, and multi-segment), where WIMF consistently outperforms non-physics baselines, especially under high noise and spatially varying wave behavior, highlighting its practical impact for modal analysis and structural diagnostics.

Abstract

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.

Wave Physics-informed Matrix Factorizations

TL;DR

This work introduces Wave-Informed Matrix Factorization (WIMF), a representation learning framework that couples data fitting with a wave-equation based regularizer to extract physically interpretable modal components. By formulating the problem as a non-convex matrix factorization with a Helmholtz-inspired constraint and leveraging structured-regularization theory, the authors prove that the model can be solved to global optimality and provide a polynomial-time algorithm. A key insight is that the polar subproblem reduces to a tractable line search over a single wavenumber and yields an interpretation as a bank of first-order Butterworth filters, linking physics, optimization, and signal processing. The method is validated on four vibration datasets (homogeneous, inhomogeneous, traveling waves, and multi-segment), where WIMF consistently outperforms non-physics baselines, especially under high noise and spatially varying wave behavior, highlighting its practical impact for modal analysis and structural diagnostics.

Abstract

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.
Paper Structure (32 sections, 6 theorems, 72 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 32 sections, 6 theorems, 72 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Wave-Informed Matrix Factorization
  5. Problem Formulation
  6. Model Optimization
  7. Solving the Polar Problem
  8. Wave-Informed Factorization Algorithm
  9. Gradient Descent Update
  10. Solve Polar Problem
  11. Stopping Condition and Growing the Factorization
  12. Signal Processing Interpretation of the Regularizer
  13. Experimental Evaluation
  14. Homogeneous Vibration Data with Fixed Boundaries
  15. Inhomogeneous Vibration Data With Spatial Decay
  16. ...and 17 more sections

Key Result

Theorem 1

For eq:main_obj, the polar problem in eq:polar_def is where $\mathbf{A}(k)$ is as defined in eq:Ak. Further, if we define $k^*$ as Then the optimal values of $\mathbf{d},\mathbf{x}, k$ are given as $\mathbf{d}^* = \mathbf{A}(k^*)^{-1/2} \widebar \mathbf{d}$, $\mathbf{x}^* = \widebar \mathbf{x}$, and $k^*$. Where $\widebar \mathbf{d}$ and $\widebar \mathbf{x}$ are the left and right singular vecto

Figures (5)

  • Figure 1: Butterworth Filter on the spectral-like domain
  • Figure 2: (a) Homogeneously vibrating medium marked with different colors at three different time instances for a short spatial interval at the start. (b) Non-Homogeneously (exponentially decaying over space) vibration data marked with different colors at three different time instances for a short spatial interval at the start. (c) Traveling wave data marked with different colors at three different time instances for a short spatial interval at the start. (d) Data of two mediums joined end-to-end and vibrating together marked with different colors at different time instances for a short spatio-temporal interval around the joint.
  • Figure 3: Spectra (with respect to Eigen-basis of $\mathbf{L}$) of decaying exponentials $e^{-\beta_1 \ell} \sin (\pi \ell)$, $e^{-\beta_2 \ell} \sin (2\pi \ell)$ and $e^{-\beta_3} \sin(3\pi \ell)$ (indicated using blue, red and yellow colors respectively) for (a) $\ell \in [0,1]$, since here L=1 (b) $\ell \in [0,10]$, since here L=10
  • Figure 4: (a) Homogeneous Vibration
  • Figure 6: (a) Two columns from the basis obtained from PCA and the absolute value in the transformed domain (transformed to the eigenbasis of $\mathbf{L}$) (b) Two columns from the basis obtained from wave-informed matrix factorization and the absolute value in the transformed domain

Theorems & Definitions (11)

  • Theorem 1
  • Proposition 1
  • proof
  • Theorem \ref{thm:polar}
  • proof
  • Lemma 1
  • proof
  • Theorem \ref{thm:polar}
  • proof
  • Corollary 1: Adapted from Cor 13 in malherbe2017global
  • ...and 1 more