Fu‐Chen Chen

Fuchen Chen, Lijian Jiang, Guanghui Zheng

In this paper, we present a dimension reduction method to reduce the dimension of parameter space and state space and efficiently solve inverse problems. To this end, proper orthogonal decomposition (POD) and radial basis function (RBF) are combined to represent the solution of forward model with a form of variable separation. This POD-RBF method can be used to efficiently evaluate the model's output. A gradient regularization method is presented to solve the inverse problem with fast convergence. A generalized cross validation method is suggested to select the regularization parameter and differential step size for the gradient computation. Because the regularization method needs many model's evaluations. This is desirable for POD-RBF method. Thus, the POD-RBF method is integrated with the gradient regularization method to provide an efficient approach to solve inverse problems. We focus on the coefficient inversion of diffusion equations using the proposed approach. Based on different types of measurement data and different basis functions for coefficients, we present a few numerical examples for the coefficient inversion. The numerical results show that accurate reconstruction for the coefficient can be achieved efficiently.

Yun-Yun Tsai, Fu-Chen Chen, Albert Y. C. Chen et al.

Machine learning models struggle with generalization when encountering out-of-distribution (OOD) samples with unexpected distribution shifts. For vision tasks, recent studies have shown that test-time adaptation employing diffusion models can achieve state-of-the-art accuracy improvements on OOD samples by generating new samples that align with the model's domain without the need to modify the model's weights. Unfortunately, those studies have primarily focused on pixel-level corruptions, thereby lacking the generalization to adapt to a broader range of OOD types. We introduce Generalized Diffusion Adaptation (GDA), a novel diffusion-based test-time adaptation method robust against diverse OOD types. Specifically, GDA iteratively guides the diffusion by applying a marginal entropy loss derived from the model, in conjunction with style and content preservation losses during the reverse sampling process. In other words, GDA considers the model's output behavior with the semantic information of the samples as a whole, which can reduce ambiguity in downstream tasks during the generation process. Evaluation across various popular model architectures and OOD benchmarks shows that GDA consistently outperforms prior work on diffusion-driven adaptation. Notably, it achieves the highest classification accuracy improvements, ranging from 4.4\% to 5.02\% on ImageNet-C and 2.5\% to 7.4\% on Rendition, Sketch, and Stylized benchmarks. This performance highlights GDA's generalization to a broader range of OOD benchmarks.