Beyond Derivative Pathology of PINNs: Variable Splitting Strategy with Convergence Analysis

Physics-informed neural networks (PINNs) have recently emerged as effective methods for solving partial differential equations (PDEs) in various problems. Substantial research focuses on the failure modes of PINNs due to their frequent inaccuracies in predictions. However, most are based on the premise that minimizing the loss function to zero causes the network to converge to a solution of the governing PDE. In this study, we prove that PINNs encounter a fundamental issue that the premise is invalid. We also reveal that this issue stems from the inability to regulate the behavior of the derivatives of the predicted solution. Inspired by the \textit{derivative pathology} of PINNs, we propose a \textit{variable splitting} strategy that addresses this issue by parameterizing the gradient of the solution as an auxiliary variable. We demonstrate that using the auxiliary variable eludes derivative pathology by enabling direct monitoring and regulation of the gradient of the predicted solution. Moreover, we prove that the proposed method guarantees convergence to a generalized solution for second-order linear PDEs, indicating its applicability to various problems.

Why Rectified Power Unit Networks Fail and How to Improve It: An Effective Theory Perspective

The Rectified Power Unit (RePU) activation functions, unlike the Rectified Linear Unit (ReLU), have the advantage of being a differentiable function when constructing neural networks. However, it can be experimentally observed when deep layers are stacked, neural networks constructed with RePU encounter critical issues. These issues include the values exploding or vanishing and failure of training. And these happen regardless of the hyperparameter initialization. From the perspective of effective theory, we aim to identify the causes of this phenomenon and propose a new activation function that retains the advantages of RePU while overcoming its drawbacks.

FLEUR: An Explainable Reference-Free Evaluation Metric for Image Captioning Using a Large Multimodal Model

Most existing image captioning evaluation metrics focus on assigning a single numerical score to a caption by comparing it with reference captions. However, these methods do not provide an explanation for the assigned score. Moreover, reference captions are expensive to acquire. In this paper, we propose FLEUR, an explainable reference-free metric to introduce explainability into image captioning evaluation metrics. By leveraging a large multimodal model, FLEUR can evaluate the caption against the image without the need for reference captions, and provide the explanation for the assigned score. We introduce score smoothing to align as closely as possible with human judgment and to be robust to user-defined grading criteria. FLEUR achieves high correlations with human judgment across various image captioning evaluation benchmarks and reaches state-of-the-art results on Flickr8k-CF, COMPOSITE, and Pascal-50S within the domain of reference-free evaluation metrics. Our source code and results are publicly available at: https://github.com/Yebin46/FLEUR.

Neural Operators Learn the Local Physics of Magnetohydrodynamics

Magnetohydrodynamics (MHD) plays a pivotal role in describing the dynamics of plasma and conductive fluids, essential for understanding phenomena such as the structure and evolution of stars and galaxies, and in nuclear fusion for plasma motion through ideal MHD equations. Solving these hyperbolic PDEs requires sophisticated numerical methods, presenting computational challenges due to complex structures and high costs. Recent advances introduce neural operators like the Fourier Neural Operator (FNO) as surrogate models for traditional numerical analyses. This study explores a modified Flux Fourier neural operator model to approximate the numerical flux of ideal MHD, offering a novel approach that outperforms existing neural operator models by enabling continuous inference, generalization outside sampled distributions, and faster computation compared to classical numerical schemes.

Revealing and Utilizing In-group Favoritism for Graph-based Collaborative Filtering

When it comes to a personalized item recommendation system, It is essential to extract users' preferences and purchasing patterns. Assuming that users in the real world form a cluster and there is common favoritism in each cluster, in this work, we introduce Co-Clustering Wrapper (CCW). We compute co-clusters of users and items with co-clustering algorithms and add CF subnetworks for each cluster to extract the in-group favoritism. Combining the features from the networks, we obtain rich and unified information about users. We experimented real world datasets considering two aspects: Finding the number of groups divided according to in-group preference, and measuring the quantity of improvement of the performance.

Scalable Wasserstein Gradient Flow for Generative Modeling through Unbalanced Optimal Transport

Wasserstein Gradient Flow (WGF) describes the gradient dynamics of probability density within the Wasserstein space. WGF provides a promising approach for conducting optimization over the probability distributions. Numerically approximating the continuous WGF requires the time discretization method. The most well-known method for this is the JKO scheme. In this regard, previous WGF models employ the JKO scheme and parametrize transport map for each JKO step. However, this approach results in quadratic training complexity $O(K^2)$ with the number of JKO step $K$. This severely limits the scalability of WGF models. In this paper, we introduce a scalable WGF-based generative model, called Semi-dual JKO (S-JKO). Our model is based on the semi-dual form of the JKO step, derived from the equivalence between the JKO step and the Unbalanced Optimal Transport. Our approach reduces the training complexity to $O(K)$. We demonstrate that our model significantly outperforms existing WGF-based generative models, achieving FID scores of 2.62 on CIFAR-10 and 6.19 on CelebA-HQ-256, which are comparable to state-of-the-art image generative models.

Spatial-and-Frequency-aware Restoration method for Images based on Diffusion Models

Diffusion models have recently emerged as a promising framework for Image Restoration (IR), owing to their ability to produce high-quality reconstructions and their compatibility with established methods. Existing methods for solving noisy inverse problems in IR, considers the pixel-wise data-fidelity. In this paper, we propose SaFaRI, a spatial-and-frequency-aware diffusion model for IR with Gaussian noise. Our model encourages images to preserve data-fidelity in both the spatial and frequency domains, resulting in enhanced reconstruction quality. We comprehensively evaluate the performance of our model on a variety of noisy inverse problems, including inpainting, denoising, and super-resolution. Our thorough evaluation demonstrates that SaFaRI achieves state-of-the-art performance on both the ImageNet datasets and FFHQ datasets, outperforming existing zero-shot IR methods in terms of LPIPS and FID metrics.

Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Laws

Traditionally, classical numerical schemes have been employed to solve partial differential equations (PDEs) using computational methods. Recently, neural network-based methods have emerged. Despite these advancements, neural network-based methods, such as physics-informed neural networks (PINNs) and neural operators, exhibit deficiencies in robustness and generalization. To address these issues, numerous studies have integrated classical numerical frameworks with machine learning techniques, incorporating neural networks into parts of traditional numerical methods. In this study, we focus on hyperbolic conservation laws by replacing traditional numerical fluxes with neural operators. To this end, we developed loss functions inspired by established numerical schemes related to conservation laws and approximated numerical fluxes using Fourier neural operators (FNOs). Our experiments demonstrated that our approach combines the strengths of both traditional numerical schemes and FNOs, outperforming standard FNO methods in several respects. For instance, we demonstrate that our method is robust, has resolution invariance, and is feasible as a data-driven method. In particular, our method can make continuous predictions over time and exhibits superior generalization capabilities with out-of-distribution (OOD) samples, which are challenges that existing neural operator methods encounter.

$p$-Poisson surface reconstruction in curl-free flow from point clouds

The aim of this paper is the reconstruction of a smooth surface from an unorganized point cloud sampled by a closed surface, with the preservation of geometric shapes, without any further information other than the point cloud. Implicit neural representations (INRs) have recently emerged as a promising approach to surface reconstruction. However, the reconstruction quality of existing methods relies on ground truth implicit function values or surface normal vectors. In this paper, we show that proper supervision of partial differential equations and fundamental properties of differential vector fields are sufficient to robustly reconstruct high-quality surfaces. We cast the $p$-Poisson equation to learn a signed distance function (SDF) and the reconstructed surface is implicitly represented by the zero-level set of the SDF. For efficient training, we develop a variable splitting structure by introducing a gradient of the SDF as an auxiliary variable and impose the $p$-Poisson equation directly on the auxiliary variable as a hard constraint. Based on the curl-free property of the gradient field, we impose a curl-free constraint on the auxiliary variable, which leads to a more faithful reconstruction. Experiments on standard benchmark datasets show that the proposed INR provides a superior and robust reconstruction. The code is available at \url{https://github.com/Yebbi/PINC}.

Towards long-tailed, multi-label disease classification from chest X-ray: Overview of the CXR-LT challenge

Many real-world image recognition problems, such as diagnostic medical imaging exams, are "long-tailed" $\unicode{x2013}$ there are a few common findings followed by many more relatively rare conditions. In chest radiography, diagnosis is both a long-tailed and multi-label problem, as patients often present with multiple findings simultaneously. While researchers have begun to study the problem of long-tailed learning in medical image recognition, few have studied the interaction of label imbalance and label co-occurrence posed by long-tailed, multi-label disease classification. To engage with the research community on this emerging topic, we conducted an open challenge, CXR-LT, on long-tailed, multi-label thorax disease classification from chest X-rays (CXRs). We publicly release a large-scale benchmark dataset of over 350,000 CXRs, each labeled with at least one of 26 clinical findings following a long-tailed distribution. We synthesize common themes of top-performing solutions, providing practical recommendations for long-tailed, multi-label medical image classification. Finally, we use these insights to propose a path forward involving vision-language foundation models for few- and zero-shot disease classification.