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A primary challenge of physics-informed machine learning (PIML) is its generalization beyond the training domain, especially when dealing with complex physical problems represented by partial differential equations (PDEs). This paper aims to enhance the generalization capabilities of PIML, facilitating practical, real-world applications where accurate predictions in unexplored regions are crucial. We leverage the inherent causality and temporal sequential characteristics of PDE solutions to fuse PIML models with recurrent neural architectures based on systems of ordinary differential equations, referred to as neural oscillators. Through effectively capturing long-time dependencies and mitigating the exploding and vanishing gradient problem, neural oscillators foster improved generalization in PIML tasks. Extensive experimentation involving time-dependent nonlinear PDEs and biharmonic beam equations demonstrates the efficacy of the proposed approach. Incorporating neural oscillators outperforms existing state-of-the-art methods on benchmark problems across various metrics. Consequently, the proposed method improves the generalization capabilities of PIML, providing accurate solutions for extrapolation and prediction beyond the training data.

Seismic imaging is the numerical process of creating a volumetric representation of the subsurface geological structures from elastic waves recorded at the surface of the Earth. As such, it is widely utilized in the energy and construction sectors for applications ranging from oil and gas prospection, to geothermal production and carbon capture and storage monitoring, to geotechnical assessment of infrastructures. Extracting quantitative information from seismic recordings, such as an acoustic impedance model, is however a highly ill-posed inverse problem, due to the band-limited and noisy nature of the data. This paper introduces IntraSeismic, a novel hybrid seismic inversion method that seamlessly combines coordinate-based learning with the physics of the post-stack modeling operator. Key features of IntraSeismic are i) unparalleled performance in 2D and 3D post-stack seismic inversion, ii) rapid convergence rates, iii) ability to seamlessly include hard constraints (i.e., well data) and perform uncertainty quantification, and iv) potential data compression and fast randomized access to portions of the inverted model. Synthetic and field data applications of IntraSeismic are presented to validate the effectiveness of the proposed method.

Deep learning methods have gained considerable interest in the numerical solution of various partial differential equations (PDEs). One particular focus is physics-informed neural networks (PINN), which integrate physical principles into neural networks. This transforms the process of solving PDEs into optimization problems for neural networks. To address a collection of advection-diffusion equations (ADE) in a range of difficult circumstances, this paper proposes a novel network structure. This architecture integrates the solver, a multi-scale deep neural networks (MscaleDNN) utilized in the PINN method, with a hard constraint technique known as HCPINN. This method introduces a revised formulation of the desired solution for ADE by utilizing a loss function that incorporates the residuals of the governing equation and penalizes any deviations from the specified boundary and initial constraints. By surpassing the boundary constraints automatically, this method improves the accuracy and efficiency of the PINN technique. To address the ``spectral bias'' phenomenon in neural networks, a subnetwork structure of MscaleDNN and a Fourier-induced activation function are incorporated into the HCPINN, resulting in a hybrid approach called SFHCPINN. The effectiveness of SFHCPINN is demonstrated through various numerical experiments involving ADE in different dimensions. The numerical results indicate that SFHCPINN outperforms both standard PINN and its subnetwork version with Fourier feature embedding. It achieves remarkable accuracy and efficiency while effectively handling complex boundary conditions and high-frequency scenarios in ADE.

High-order tensor methods for solving both convex and nonconvex optimization problems have generated significant research interest, leading to algorithms with optimal global rates of convergence and local rates that are faster than Newton's method. On each iteration, these methods require the unconstrained local minimization of a (potentially nonconvex) multivariate polynomial of degree higher than two, constructed using third-order (or higher) derivative information, and regularized by an appropriate power of regularization. Developing efficient techniques for solving such subproblems is an ongoing topic of research, and this paper addresses the case of the third-order tensor subproblem. We propose the CQR algorithmic framework, for minimizing a nonconvex Cubic multivariate polynomial with Quartic Regularisation, by minimizing a sequence of local quadratic models that incorporate simple cubic and quartic terms. The role of the cubic term is to crudely approximate local tensor information, while the quartic one controls model regularization and progress. We provide necessary and sufficient optimality conditions that fully characterise the global minimizers of these cubic-quartic models. We then turn these conditions into secular equations that can be solved using nonlinear eigenvalue techniques. We show, using our optimality characterisations, that a CQR algorithmic variant has the optimal-order evaluation complexity of $\mathcal{O}(\epsilon^{-3/2})$ when applied to minimizing our quartically-regularised cubic subproblem, which can be further improved in special cases. We propose practical CQR variants that use local tensor information to construct the local cubic-quartic models. We test these variants numerically and observe them to be competitive with ARC and other subproblem solvers on typical instances and even superior on ill-conditioned subproblems with special structure.

The aim of this paper is to give a systematic mathematical interpretation of the diffusion problem on which Graph Neural Networks (GNNs) models are based. The starting point of our approach is a dissipative functional leading to dynamical equations which allows us to study the symmetries of the model. We discuss the conserved charges and provide a charge-preserving numerical method for solving the dynamical equations. In any dynamical system and also in GRAph Neural Diffusion (GRAND), knowing the charge values and their conservation along the evolution flow could provide a way to understand how GNNs and other networks work with their learning capabilities.

We study hypothesis testing under communication constraints, where each sample is quantized before being revealed to a statistician. Without communication constraints, it is well known that the sample complexity of simple binary hypothesis testing is characterized by the Hellinger distance between the distributions. We show that the sample complexity of simple binary hypothesis testing under communication constraints is at most a logarithmic factor larger than in the unconstrained setting and this bound is tight. We develop a polynomial-time algorithm that achieves the aforementioned sample complexity. Our framework extends to robust hypothesis testing, where the distributions are corrupted in the total variation distance. Our proofs rely on a new reverse data processing inequality and a reverse Markov inequality, which may be of independent interest. For simple $M$-ary hypothesis testing, the sample complexity in the absence of communication constraints has a logarithmic dependence on $M$. We show that communication constraints can cause an exponential blow-up leading to $\Omega(M)$ sample complexity even for adaptive algorithms.

Biomechanical and orthopaedic studies frequently encounter complex datasets that encompass both circular and linear variables. In most cases the circular and linear variables are (i) considered in isolation with dependency between variables neglected and (ii) the cyclicity of the circular variables disregarded resulting in erroneous decision making. Given the inherent characteristics of circular variables, it is imperative to adopt methods that integrate directional statistics to achieve precise modelling. This paper is motivated by the modelling of biomechanical data, i.e., the fracture displacements, that is used as a measure in external fixator comparisons. We focus on a data set, based on an Ilizarov ring fixator, comprising of six variables. A modelling framework applicable to the 6D joint distribution of circular-linear data based on vine copulas is proposed. The pair-copula decomposition concept of vine copulas represents the dependence structure as a combination of circular-linear, circular-circular and linear-linear pairs modelled by their respective copulas. This framework allows us to assess the dependencies in the joint distribution as well as account for the cyclicity of the circular variables. Thus, a new approach for accurate modelling of mechanical behaviour for Ilizarov ring fixators and other data of this nature is imparted.

Physics-informed neural networks (PINNs) are a powerful approach for solving problems involving differential equations, yet they often struggle to solve problems with high frequency and/or multi-scale solutions. Finite basis physics-informed neural networks (FBPINNs) improve the performance of PINNs in this regime by combining them with an overlapping domain decomposition approach. In this work, FBPINNs are extended by adding multiple levels of domain decompositions to their solution ansatz, inspired by classical multilevel Schwarz domain decomposition methods (DDMs). Analogous to typical tests for classical DDMs, we assess how the accuracy of PINNs, FBPINNs and multilevel FBPINNs scale with respect to computational effort and solution complexity by carrying out strong and weak scaling tests. Our numerical results show that the proposed multilevel FBPINNs consistently and significantly outperform PINNs across a range of problems with high frequency and multi-scale solutions. Furthermore, as expected in classical DDMs, we show that multilevel FBPINNs improve the accuracy of FBPINNs when using large numbers of subdomains by aiding global communication between subdomains.

Graph representation learning for hypergraphs can be used to extract patterns among higher-order interactions that are critically important in many real world problems. Current approaches designed for hypergraphs, however, are unable to handle different types of hypergraphs and are typically not generic for various learning tasks. Indeed, models that can predict variable-sized heterogeneous hyperedges have not been available. Here we develop a new self-attention based graph neural network called Hyper-SAGNN applicable to homogeneous and heterogeneous hypergraphs with variable hyperedge sizes. We perform extensive evaluations on multiple datasets, including four benchmark network datasets and two single-cell Hi-C datasets in genomics. We demonstrate that Hyper-SAGNN significantly outperforms the state-of-the-art methods on traditional tasks while also achieving great performance on a new task called outsider identification. Hyper-SAGNN will be useful for graph representation learning to uncover complex higher-order interactions in different applications.

Recent advances in 3D fully convolutional networks (FCN) have made it feasible to produce dense voxel-wise predictions of volumetric images. In this work, we show that a multi-class 3D FCN trained on manually labeled CT scans of several anatomical structures (ranging from the large organs to thin vessels) can achieve competitive segmentation results, while avoiding the need for handcrafting features or training class-specific models. To this end, we propose a two-stage, coarse-to-fine approach that will first use a 3D FCN to roughly define a candidate region, which will then be used as input to a second 3D FCN. This reduces the number of voxels the second FCN has to classify to ~10% and allows it to focus on more detailed segmentation of the organs and vessels. We utilize training and validation sets consisting of 331 clinical CT images and test our models on a completely unseen data collection acquired at a different hospital that includes 150 CT scans, targeting three anatomical organs (liver, spleen, and pancreas). In challenging organs such as the pancreas, our cascaded approach improves the mean Dice score from 68.5 to 82.2%, achieving the highest reported average score on this dataset. We compare with a 2D FCN method on a separate dataset of 240 CT scans with 18 classes and achieve a significantly higher performance in small organs and vessels. Furthermore, we explore fine-tuning our models to different datasets. Our experiments illustrate the promise and robustness of current 3D FCN based semantic segmentation of medical images, achieving state-of-the-art results. Our code and trained models are available for download: //github.com/holgerroth/3Dunet_abdomen_cascade.