Low-pass graph filters are fundamental for signal processing on graphs and other non-Euclidean domains. However, the computation of such filters for parametric graph families can be prohibitively expensive as computation of the corresponding low-frequency subspaces, requires the repeated solution of an eigenvalue problem. We suggest a novel algorithm of low-pass graph filter interpolation based on Riemannian interpolation in normal coordinates on the Grassmann manifold. We derive an error bound estimate for the subspace interpolation and suggest two possible applications for induced parametric graph families. First, we argue that the temporal evolution of the node features may be translated to the evolving graph topology via a similarity correction to adjust the homophily degree of the network. Second, we suggest a dot product graph family induced by a given static graph which allows to infer improved message passing scheme for node classification facilitated by the filter interpolation.

In real-world regression tasks, datasets frequently exhibit imbalanced distributions, characterized by a scarcity of data in high-complexity regions and an abundance in low-complexity areas. This imbalance presents significant challenges for existing classification methods with clear class boundaries, while highlighting a scarcity of approaches specifically designed for imbalanced regression problems. To better address these issues, we introduce a novel concept of Imbalanced Regression, which takes into account both the complexity of the problem and the density of data points, extending beyond traditional definitions that focus only on data density. Furthermore, we propose Error Distribution Smoothing (EDS) as a solution to tackle imbalanced regression, effectively selecting a representative subset from the dataset to reduce redundancy while maintaining balance and representativeness. Through several experiments, EDS has shown its effectiveness, and the related code and dataset can be accessed at https://anonymous.4open.science/r/Error-Distribution-Smoothing-762F.

Reinforcement Learning (RL) has made notable success in decision-making fields like autonomous driving and robotic manipulation. Yet, its reliance on real-time feedback poses challenges in costly or hazardous settings. Furthermore, RL's training approach, centered on "on-policy" sampling, doesn't fully capitalize on data. Hence, Offline RL has emerged as a compelling alternative, particularly in conducting additional experiments is impractical, and abundant datasets are available. However, the challenge of distributional shift (extrapolation), indicating the disparity between data distributions and learning policies, also poses a risk in offline RL, potentially leading to significant safety breaches due to estimation errors (interpolation). This concern is particularly pronounced in safety-critical domains, where real-world problems are prevalent. To address both extrapolation and interpolation errors, numerous studies have introduced additional constraints to confine policy behavior, steering it towards more cautious decision-making. While many studies have addressed extrapolation errors, fewer have focused on providing effective solutions for tackling interpolation errors. For example, some works tackle this issue by incorporating potential cost-maximizing optimization by perturbing the original dataset. However, this, involving a bi-level optimization structure, may introduce significant instability or complicate problem-solving in high-dimensional tasks. This motivates us to pinpoint areas where hazards may be more prevalent than initially estimated based on the sparsity of available data by providing significant insight into constrained offline RL. In this paper, we present conservative metrics based on data sparsity that demonstrate the high generalizability to any methods and efficacy compared to using bi-level cost-ub-maximization.

Predicting the evolution of spatiotemporal physical systems from sparse and scattered observational data poses a significant challenge in various scientific domains. Traditional methods rely on dense grid-structured data, limiting their applicability in scenarios with sparse observations. To address this challenge, we introduce GrINd (Grid Interpolation Network for Scattered Observations), a novel network architecture that leverages the high-performance of grid-based models by mapping scattered observations onto a high-resolution grid using a Fourier Interpolation Layer. In the high-resolution space, a NeuralPDE-class model predicts the system's state at future timepoints using differentiable ODE solvers and fully convolutional neural networks parametrizing the system's dynamics. We empirically evaluate GrINd on the DynaBench benchmark dataset, comprising six different physical systems observed at scattered locations, demonstrating its state-of-the-art performance compared to existing models. GrINd offers a promising approach for forecasting physical systems from sparse, scattered observational data, extending the applicability of deep learning methods to real-world scenarios with limited data availability.

Gaussian process regression is a classical kernel method for function estimation and data interpolation. In large data applications, computational costs can be reduced using low-rank or sparse approximations of the kernel. This paper investigates the effect of such kernel approximations on the interpolation error. We introduce a unified framework to analyze Gaussian process regression under important classes of computational misspecification: Karhunen-Lo\`eve expansions that result in low-rank kernel approximations, multiscale wavelet expansions that induce sparsity in the covariance matrix, and finite element representations that induce sparsity in the precision matrix. Our theory also accounts for epistemic misspecification in the choice of kernel parameters.

Recently, neural networks have been extensively employed to solve partial differential equations (PDEs) in physical system modeling. While major studies focus on learning system evolution on predefined static mesh discretizations, some methods utilize reinforcement learning or supervised learning techniques to create adaptive and dynamic meshes, due to the dynamic nature of these systems. However, these approaches face two primary challenges: (1) the need for expensive optimal mesh data, and (2) the change of the solution space's degree of freedom and topology during mesh refinement. To address these challenges, this paper proposes a neural PDE solver with a neural mesh adapter. To begin with, we introduce a novel data-free neural mesh adaptor, called Data-free Mesh Mover (DMM), with two main innovations. Firstly, it is an operator that maps the solution to adaptive meshes and is trained using the Monge-Ampère equation without optimal mesh data. Secondly, it dynamically changes the mesh by moving existing nodes rather than adding or deleting nodes and edges. Theoretical analysis shows that meshes generated by DMM have the lowest interpolation error bound. Based on DMM, to efficiently and accurately model dynamic systems, we develop a moving mesh based neural PDE solver (MM-PDE) that embeds the moving mesh with a twobranch architecture and a learnable interpolation framework to preserve information within the data. Empirical experiments demonstrate that our method generates suitable meshes and considerably enhances accuracy when modeling widely considered PDE systems. The simulation of physical phenomena is a popular research topic in many disciplines, ranging from weather forecasting (Schalkwijk et al., 2015), structural mechanics (Panthi et al., 2007) to turbulence modeling (Garnier et al., 2021). Meanwhile, due to the rapid development of deep learning techniques, there are emerging neural network based approaches designed for simulating physical systems (Li et al., 2020; Raissi et al., 2019; Brandstetter et al., 2022).

Interpolation methodologies have been widely used within the domain of indoor positioning systems. However, existing indoor positioning interpolation algorithms exhibit several inherent limitations, including reliance on complex mathematical models, limited flexibility, and relatively low precision. To enhance the accuracy and efficiency of indoor positioning interpolation techniques, this paper proposes a simple yet powerful geometric-aware interpolation algorithm for indoor positioning tasks. The key to our algorithm is to exploit the geometric attributes of the local topological manifold using manifold learning principles. Therefore, instead of constructing complicated mathematical models, the proposed algorithm facilitates the more precise and efficient estimation of points grounded in the local topological manifold. Moreover, our proposed method can be effortlessly integrated into any indoor positioning system, thereby bolstering its adaptability. Through a systematic array of experiments and comprehensive performance analyses conducted on both simulated and real-world datasets, we demonstrate that the proposed algorithm consistently outperforms the most commonly used and representative interpolation approaches regarding interpolation accuracy and efficiency. Furthermore, the experimental results also underscore the substantial practical utility of our method and its potential applicability in real-time indoor positioning scenarios.

Modeling spatiotemporal dynamical systems is a fundamental challenge in machine learning. Transformer models have been very successful in NLP and computer vision where they provide interpretable representations of data. However, a limitation of transformers in modeling continuous dynamical systems is that they are fundamentally discrete time and space models and thus have no guarantees regarding continuous sampling. To address this challenge, we present the Continuous Spatiotemporal Transformer (CST), a new transformer architecture that is designed for the modeling of continuous systems. This new framework guarantees a continuous and smooth output via optimization in Sobolev space. We benchmark CST against traditional transformers as well as other spatiotemporal dynamics modeling methods and achieve superior performance in a number of tasks on synthetic and real systems, including learning brain dynamics from calcium imaging data.

The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at https://github.com/thu-ml/NUNO.