Accelerating SAV-based optimization via randomized low-rank Hessian approximation

We propose a new optimization method, the Nyström-enhanced relaxed scalar auxiliary variable method (N-RSAV), which incorporates curvature information into the RSAV framework to accelerate convergence while preserving an unconditional modified energy dissipation law. Existing RSAV-based methods rely solely on first-order information and often suffer from slow convergence, particularly for ill-conditioned problems such as those arising in physics-informed neural networks (PINNs). To address this limitation, we design the linear operator in the RSAV scheme using approximate Hessian information obtained from a randomized low-rank Nyström approximation. To preserve the dissipation structure, we enforce positive semidefiniteness through eigenvalue truncation. Furthermore, we introduce an adaptive strategy that reuses the approximate Hessian based on the deviation between the original and modified energies, significantly reducing computational cost. We also provide a convergence analysis of the RSAV scheme with a general positive semidefinite operator under the Polyak-Lojasiewicz (PL) condition and establish corresponding convergence guarantees for N-RSAV under the PL condition and an additional convexity assumption. Numerical experiments on ill-conditioned problems with effectively low-rank structure, including convex quadratic problems and training of PINNs, demonstrate that the proposed methods achieve substantially faster convergence than conventional RSAV-based approaches.

Structure-Preserving Physics-Informed Neural Networks With Energy or Lyapunov Structure

Recently, there has been growing interest in using physics-informed neural networks (PINNs) to solve differential equations. However, the preservation of structure, such as energy and stability, in a suitable manner has yet to be established. This limitation could be a potential reason why the learning process for PINNs is not always efficient and the numerical results may suggest nonphysical behavior. Besides, there is little research on their applications on downstream tasks. To address these issues, we propose structure-preserving PINNs to improve their performance and broaden their applications for downstream tasks. Firstly, by leveraging prior knowledge about the physical system, a structure-preserving loss function is designed to assist the PINN in learning the underlying structure. Secondly, a framework that utilizes structure-preserving PINN for robust image recognition is proposed. Here, preserving the Lyapunov structure of the underlying system ensures the stability of the system. Experimental results demonstrate that the proposed method improves the numerical accuracy of PINNs for partial differential equations. Furthermore, the robustness of the model against adversarial perturbations in image data is enhanced.

A modified model for topic detection from a corpus and a new metric evaluating the understandability of topics

This paper presents a modified neural model for topic detection from a corpus and proposes a new metric to evaluate the detected topics. The new model builds upon the embedded topic model incorporating some modifications such as document clustering. Numerical experiments suggest that the new model performs favourably regardless of the document's length. The new metric, which can be computed more efficiently than widely-used metrics such as topic coherence, provides variable information regarding the understandability of the detected topics.

Composing a surrogate observation operator for sequential data assimilation

In data assimilation, state estimation is not straightforward when the observation operator is unknown. This study proposes a method for composing a surrogate operator for a true operator. The surrogate model is improved iteratively to decrease the difference between the observations and the results of the surrogate model, and a neural network is adopted in the process. A twin experiment suggests that the proposed method outperforms approaches that use a specific operator that is given tentatively throughout the data assimilation process.