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Under ray-optical light transport, the classical ray serves as a local and linear "point query" of light's behaviour. Such point queries are useful, and sophisticated path tracing and sampling techniques enable efficiently computing solutions to light transport problems in complex, real-world settings and environments. However, such formulations are firmly confined to the realm of ray optics, while many applications of interest, in computer graphics and computational optics, demand a more precise understanding of light. We rigorously formulate the generalized ray, which enables local and linear point queries of the wave-optical phase space. Furthermore, we present sample-solve: a simple method that serves as a novel link between path tracing and computational optics. We will show that this link enables the application of modern path tracing techniques for wave-optical rendering, improving upon the state-of-the-art in terms of the generality and accuracy of the formalism, ease of application, as well as performance. Sampling using generalized rays enables interactive rendering under rigorous wave optics, with orders-of-magnitude faster performance compared to existing techniques.

Physics-based deep learning frameworks have shown to be effective in accurately modeling the dynamics of complex physical systems with generalization capability across problem inputs. However, time-independent problems pose the challenge of requiring long-range exchange of information across the computational domain for obtaining accurate predictions. In the context of graph neural networks (GNNs), this calls for deeper networks, which, in turn, may compromise or slow down the training process. In this work, we present two GNN architectures to overcome this challenge - the Edge Augmented GNN and the Multi-GNN. We show that both these networks perform significantly better (by a factor of 1.5 to 2) than baseline methods when applied to time-independent solid mechanics problems. Furthermore, the proposed architectures generalize well to unseen domains, boundary conditions, and materials. Here, the treatment of variable domains is facilitated by a novel coordinate transformation that enables rotation and translation invariance. By broadening the range of problems that neural operators based on graph neural networks can tackle, this paper provides the groundwork for their application to complex scientific and industrial settings.

Physics-informed neural networks have emerged as an alternative method for solving partial differential equations. However, for complex problems, the training of such networks can still require high-fidelity data which can be expensive to generate. To reduce or even eliminate the dependency on high-fidelity data, we propose a novel multi-fidelity architecture which is based on a feature space shared by the low- and high-fidelity solutions. In the feature space, the projections of the low-fidelity and high-fidelity solutions are adjacent by constraining their relative distance. The feature space is represented with an encoder and its mapping to the original solution space is effected through a decoder. The proposed multi-fidelity approach is validated on forward and inverse problems for steady and unsteady problems described by partial differential equations.

With the remarkable empirical success of neural networks across diverse scientific disciplines, rigorous error and convergence analysis are also being developed and enriched. However, there has been little theoretical work focusing on neural networks in solving interface problems. In this paper, we perform a convergence analysis of physics-informed neural networks (PINNs) for solving second-order elliptic interface problems. Specifically, we consider PINNs with domain decomposition technologies and introduce gradient-enhanced strategies on the interfaces to deal with boundary and interface jump conditions. It is shown that the neural network sequence obtained by minimizing a Lipschitz regularized loss function converges to the unique solution to the interface problem in $H^2$ as the number of samples increases. Numerical experiments are provided to demonstrate our theoretical analysis.

Physics-informed neural networks have emerged as a coherent framework for building predictive models that combine statistical patterns with domain knowledge. The underlying notion is to enrich the optimization loss function with known relationships to constrain the space of possible solutions. Hydrodynamic simulations are a core constituent of modern cosmology, while the required computations are both expensive and time-consuming. At the same time, the comparatively fast simulation of dark matter requires fewer resources, which has led to the emergence of machine learning algorithms for baryon inpainting as an active area of research; here, recreating the scatter found in hydrodynamic simulations is an ongoing challenge. This paper presents the first application of physics-informed neural networks to baryon inpainting by combining advances in neural network architectures with physical constraints, injecting theory on baryon conversion efficiency into the model loss function. We also introduce a punitive prediction comparison based on the Kullback-Leibler divergence, which enforces scatter reproduction. By simultaneously extracting the complete set of baryonic properties for the Simba suite of cosmological simulations, our results demonstrate improved accuracy of baryonic predictions based on dark matter halo properties, successful recovery of the fundamental metallicity relation, and retrieve scatter that traces the target simulation's distribution.

Deep learning surrogate models are being increasingly used in accelerating scientific simulations as a replacement for costly conventional numerical techniques. However, their use remains a significant challenge when dealing with real-world complex examples. In this work, we demonstrate three types of neural network architectures for efficient learning of highly non-linear deformations of solid bodies. The first two architectures are based on the recently proposed CNN U-NET and MAgNET (graph U-NET) frameworks which have shown promising performance for learning on mesh-based data. The third architecture is Perceiver IO, a very recent architecture that belongs to the family of attention-based neural networks--a class that has revolutionised diverse engineering fields and is still unexplored in computational mechanics. We study and compare the performance of all three networks on two benchmark examples, and show their capabilities to accurately predict the non-linear mechanical responses of soft bodies.

During an infectious disease outbreak, public health decision-makers require real-time monitoring of disease transmission to respond quickly and intelligently. In these settings, a key measure of transmission is the instantaneous time-varying reproduction number, $R_t$. Estimation of this number using a Time-Since-Infection model relies on case-notification data and the distribution of the serial interval on the target population. However, in practice, case-notification data may contain measurement error due to variation in case reporting while available serial interval estimates may come from studies on non-representative populations. We propose a new data-driven method that accounts for particular forms of case-reporting measurement error and can incorporate multiple partially representative serial interval estimates into the transmission estimation process. In addition, we provide practical tools for automatically identifying measurement error patterns and determining when measurement error may not be adequately accounted for. We illustrate the potential bias undertaken by methods that ignore these practical concerns through a variety of simulated outbreaks. We then demonstrate the use of our method on data from the COVID-19 pandemic to estimate transmission and explore the relationships between social distancing, temperature, and transmission.

Recently, graph neural networks have been gaining a lot of attention to simulate dynamical systems due to their inductive nature leading to zero-shot generalizability. Similarly, physics-informed inductive biases in deep-learning frameworks have been shown to give superior performance in learning the dynamics of physical systems. There is a growing volume of literature that attempts to combine these two approaches. Here, we evaluate the performance of thirteen different graph neural networks, namely, Hamiltonian and Lagrangian graph neural networks, graph neural ODE, and their variants with explicit constraints and different architectures. We briefly explain the theoretical formulation highlighting the similarities and differences in the inductive biases and graph architecture of these systems. We evaluate these models on spring, pendulum, gravitational, and 3D deformable solid systems to compare the performance in terms of rollout error, conserved quantities such as energy and momentum, and generalizability to unseen system sizes. Our study demonstrates that GNNs with additional inductive biases, such as explicit constraints and decoupling of kinetic and potential energies, exhibit significantly enhanced performance. Further, all the physics-informed GNNs exhibit zero-shot generalizability to system sizes an order of magnitude larger than the training system, thus providing a promising route to simulate large-scale realistic systems.

Understanding causality helps to structure interventions to achieve specific goals and enables predictions under interventions. With the growing importance of learning causal relationships, causal discovery tasks have transitioned from using traditional methods to infer potential causal structures from observational data to the field of pattern recognition involved in deep learning. The rapid accumulation of massive data promotes the emergence of causal search methods with brilliant scalability. Existing summaries of causal discovery methods mainly focus on traditional methods based on constraints, scores and FCMs, there is a lack of perfect sorting and elaboration for deep learning-based methods, also lacking some considers and exploration of causal discovery methods from the perspective of variable paradigms. Therefore, we divide the possible causal discovery tasks into three types according to the variable paradigm and give the definitions of the three tasks respectively, define and instantiate the relevant datasets for each task and the final causal model constructed at the same time, then reviews the main existing causal discovery methods for different tasks. Finally, we propose some roadmaps from different perspectives for the current research gaps in the field of causal discovery and point out future research directions.

This paper introduces a new fundamental characteristic, \ie, the dynamic range, from real-world metric tools to deep visual recognition. In metrology, the dynamic range is a basic quality of a metric tool, indicating its flexibility to accommodate various scales. Larger dynamic range offers higher flexibility. In visual recognition, the multiple scale problem also exist. Different visual concepts may have different semantic scales. For example, ``Animal'' and ``Plants'' have a large semantic scale while ``Elk'' has a much smaller one. Under a small semantic scale, two different elks may look quite \emph{different} to each other . However, under a large semantic scale (\eg, animals and plants), these two elks should be measured as being \emph{similar}. %We argue that such flexibility is also important for deep metric learning, because different visual concepts indeed correspond to different semantic scales. Introducing the dynamic range to deep metric learning, we get a novel computer vision task, \ie, the Dynamic Metric Learning. It aims to learn a scalable metric space to accommodate visual concepts across multiple semantic scales. Based on three types of images, \emph{i.e.}, vehicle, animal and online products, we construct three datasets for Dynamic Metric Learning. We benchmark these datasets with popular deep metric learning methods and find Dynamic Metric Learning to be very challenging. The major difficulty lies in a conflict between different scales: the discriminative ability under a small scale usually compromises the discriminative ability under a large one, and vice versa. As a minor contribution, we propose Cross-Scale Learning (CSL) to alleviate such conflict. We show that CSL consistently improves the baseline on all the three datasets. The datasets and the code will be publicly available at //github.com/SupetZYK/DynamicMetricLearning.