A Dynamical Low-Rank Approximation Framework for Continuous-Time Low-Rank Factor Dynamics in Oscillatory PDE Systems

In many large-scale scientific computing problems, the solution update evolves on a low-rank manifold and is factored as ΔW = BA, where B ∈ R^(m×r) and A ∈ R^(r×n) with r ≪ m, n. This paper identifies the mathematical limitations of Spectral (SVD)-based initialization of the low-rank factors on oscillatory PDE operators. Using the Dynamical Low-Rank Approximation (DLRA) framework of Koch and Lubich, we derive continuous-time evolution equations for the factor time-derivatives Ḃ and Ȧ that constrain the optimization trajectory to the tangent space of the low-rank manifold. The resulting ODE system is integrated using standard numerical schemes such as Forward Euler or Runge-Kutta methods. This continuous-time formulation mathematically and empirically shields the low-rank manifold from high-frequency truncation noise that collapses standard unconstrained optimizers. We establish sharp spectral bounds via Bessel function analysis proving that rank-N approximations cannot resolve wavenumber-k oscillatory fields when N < k, explaining the catastrophic failure of spectral initialization on indefinite operators. The geometric filter is validated on the Helmholtz equation and a 16-million-parameter Kolmogorov flow simulation on a 4096×4096 grid, achieving 0.0001% relative L2 error at rank-16 in 134 seconds versus 252 seconds for the explicit DNS baseline.