Establishing stability certificates for closed-loop systems under reinforcement learning (RL) policies is essential to move beyond empirical performance and offer guarantees of system behavior. Classical Lyapunov methods require a strict stepwise decrease in the Lyapunov function but such certificates are difficult to construct for learned policies. The RL value function is a natural candidate but it is not well understood how it can be adapted for this purpose. To gain intuition, we first study the linear quadratic regulator (LQR) problem and make two key observations. First, a Lyapunov function can be obtained from the value function of an LQR policy by augmenting it with a residual term related to the system dynamics and stage cost.

Missing data in multivariate time series are common issues that can affect the analysis and downstream applications.Although multivariate time series data generally consist of the trend, seasonal and residual terms, existing works mainly focus on optimizing the modeling for the first two items. However, we find that the residual term is more crucial for getting accurate fillings, since it is more related to the diverse changes of data and the biggest component of imputation errors.Therefore, in this study, we introduce frequency-domain information and design Frequency-aware Generative Models for Multivariate Time Series Imputation (FGTI). Specifically, FGTI employs a high-frequency filter to boost the residual term imputation, supplemented by a dominant-frequency filter for the trend and seasonal imputation. Cross-domain representation learning module then fuses frequency-domain insights with deep representations.Experiments over various datasets with real-world missing values show that FGTI achieves superiority in both data imputation and downstream applications.

Traditional diffusion-based image restoration methods utilize degraded images as conditional input to effectively guide the reverse generation process, without modifying the original denoising diffusion process.

Missing data in multivariate time series are common issues that can affect the analysis and downstream applications.Although multivariate time series data generally consist of the trend, seasonal and residual terms, existing works mainly focus on optimizing the modeling for the first two items. However, we find that the residual term is more crucial for getting accurate fillings, since it is more related to the diverse changes of data and the biggest component of imputation errors.Therefore, in this study, we introduce frequency-domain information and design Frequency-aware Generative Models for Multivariate Time Series Imputation (FGTI). Specifically, FGTI employs a high-frequency filter to boost the residual term imputation, supplemented by a dominant-frequency filter for the trend and seasonal imputation. Cross-domain representation learning module then fuses frequency-domain insights with deep representations.Experiments over various datasets with real-world missing values show that FGTI achieves superiority in both data imputation and downstream applications.

Multimodal Large Language Models (LLMs) are pivotal in revolutionizing customer support and operations by integrating multiple modalities such as text, images, and audio. Federated Prompt Learning (FPL) is a recently proposed approach that combines pre-trained multimodal LLMs such as vision-language models with federated learning to create personalized, privacy-preserving AI systems. However, balancing the competing goals of personalization, generalization, and privacy remains a significant challenge. Over-personalization can lead to overfitting, reducing generalizability, while stringent privacy measures, such as differential privacy, can hinder both personalization and generalization. In this paper, we propose a Differentially Private Federated Prompt Learning (DP-FPL) approach to tackle this challenge by leveraging a low-rank factorization scheme to capture generalization while maintaining a residual term that preserves expressiveness for personalization. To ensure privacy, we introduce a novel method where we apply local differential privacy to the two low-rank components of the local prompt, and global differential privacy to the global prompt. Our approach mitigates the impact of privacy noise on the model performance while balancing the tradeoff between personalization and generalization. Extensive experiments demonstrate the effectiveness of our approach over other benchmarks.

Nowadays, denoising diffusion probabilistic models have been adapted for many image segmentation tasks. However, existing end-to-end models have already demonstrated remarkable capabilities. Rather than using denoising diffusion probabilistic models alone, integrating the abilities of both denoising diffusion probabilistic models and existing end-to-end models can better improve the performance of image segmentation. Based on this, we implicitly introduce residual term into the diffusion process and propose ResEnsemble-DDPM, which seamlessly integrates the diffusion model and the end-to-end model through ensemble learning. The output distributions of these two models are strictly symmetric with respect to the ground truth distribution, allowing us to integrate the two models by reducing the residual term. Experimental results demonstrate that our ResEnsemble-DDPM can further improve the capabilities of existing models. Furthermore, its ensemble learning strategy can be generalized to other downstream tasks in image generation and get strong competitiveness.

For multi-scale problems, the conventional physics-informed neural networks (PINNs) face some challenges in obtaining available predictions. In this paper, based on PINNs, we propose a practical deep learning framework for multi-scale problems by reconstructing the loss function and associating it with special neural network architectures. New PINN methods derived from the improved PINN framework differ from the conventional PINN method mainly in two aspects. First, the new methods use a novel loss function by modifying the standard loss function through a (grouping) regularization strategy. The regularization strategy implements a different power operation on each loss term so that all loss terms composing the loss function are of approximately the same order of magnitude, which makes all loss terms be optimized synchronously during the optimization process. Second, for the multi-frequency or high-frequency problems, in addition to using the modified loss function, new methods upgrade the neural network architecture from the common fully-connected neural network to special network architectures such as the Fourier feature architecture, and the integrated architecture developed by us. The combination of the above two techniques leads to a significant improvement in the computational accuracy of multi-scale problems. Several challenging numerical examples demonstrate the effectiveness of the proposed methods. The proposed methods not only significantly outperform the conventional PINN method in terms of computational efficiency and computational accuracy, but also compare favorably with the state-of-the-art methods in the recent literature. The improved PINN framework facilitates better application of PINNs to multi-scale problems.