Physics-informed neural networks (PINNs) have earned high expectations in solving partial differential equations (PDEs), but their optimization usually faces thorny challenges due to the unique derivative-dependent loss function. By analyzing the loss distribution, previous research observed the propagation failure phenomenon of PINNs, intuitively described as the correct supervision for model outputs cannot ``propagate'' from initial states or boundaries to the interior domain. Going beyond intuitive understanding, this paper provides the first formal and in-depth study of propagation failure and its root cause. Based on a detailed comparison with classical finite element methods, we ascribe the failure to the conventional single-point-processing architecture of PINNs and further prove that propagation failure is essentially caused by the lower gradient correlation of PINN models on nearby collocation points. Compared to superficial loss maps, this new perspective provides a more precise quantitative criterion to identify where and why PINN fails. The theoretical finding also inspires us to present a new PINN architecture, named ProPINN, which can effectively unite the gradient of region points for better propagation. ProPINN can reliably resolve PINN failure modes and significantly surpass advanced Transformer-based models with 46% relative promotion.

Unsupervised deep learning is a promising method in brain MRI registration to reduce the reliance on anatomical labels, while still achieving anatomically accurate transformations. For the Learn2Reg2024 LUMIR challenge, we propose fine-tuning of the pre-trained TransMorph model to improve the convergence stability as well as the deformation smoothness. The former is achieved through the FAdam optimizer, and consistency in structural changes is incorporated through the addition of gradient correlation in the similarity measure, improving anatomical alignment. The results show slight improvements in the Dice and Hd-Dist95 scores, and a notable reduction in the NDV compared to the baseline TransMorph model. These are also confirmed by inspecting the boundaries of the tissue. Our proposed method highlights the effectiveness of including Gradient Correlation to achieve smoother and structurally consistent deformations for interpatient brain MRI registration.

Recently, substantial progress has been made on the problem of high-dimensional sparse linear models [22]. In particular, Lasso has been shown to be remarkably successful, and is statistically well-behaved and generates interpretable solutions. However, in the presence of non-linearity (i.e., the relation between the covariates and response is nonlinear), boosted decision trees, deep learning models, and kernel methods are regarded as the most effective models that deliver substantial performance boost over linear models; however, their interpretability is limited. As a result, there is a significant gap between the statistical performance and the interpretability, and it is often desirable to have computationally efficient algorithms that learn interpretable models without sacrificing statistical guarantees. This raises a natural question that we aim to tackle: Is there any algorithm which has similar statistical performance to complex models, while still retaining much of the interpretability of Lasso? In this paper, we answer the above question affirmatively and propose a novel way of learning the feature non-linearities with provable statistical and computational guarantees.