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The goal of this paper is to encode a 3D scene into an extremely compact representation from 2D images and to enable its transmittance, decoding and rendering in real-time across various platforms. Despite the progress in NeRFs and Gaussian Splats, their large model size and specialized renderers make it challenging to distribute free-viewpoint 3D content as easily as images. To address this, we have designed a novel 3D representation that encodes the plenoptic function into sinusoidal function indexed dense volumes. This approach facilitates feature sharing across different locations, improving compactness over traditional spatial voxels. The memory footprint of the dense 3D feature grid can be further reduced using spatial decomposition techniques. This design combines the strengths of spatial hashing functions and voxel decomposition, resulting in a model size as small as 150 KB for each 3D scene. Moreover, PPNG features a lightweight rendering pipeline with only 300 lines of code that decodes its representation into standard GL textures and fragment shaders. This enables real-time rendering using the traditional GL pipeline, ensuring universal compatibility and efficiency across various platforms without additional dependencies.

The framework of deep operator network (DeepONet) has been widely exploited thanks to its capability of solving high dimensional partial differential equations. In this paper, we incorporate DeepONet with a recently developed policy iteration scheme to numerically solve optimal control problems and the corresponding Hamilton--Jacobi--Bellman (HJB) equations. A notable feature of our approach is that once the neural network is trained, the solution to the optimal control problem and HJB equations with different terminal functions can be inferred quickly thanks to the unique feature of operator learning. Furthermore, a quantitative analysis of the accuracy of the algorithm is carried out via comparison principles of viscosity solutions. The effectiveness of the method is verified with various examples, including 10-dimensional linear quadratic regulator problems (LQRs)., Comment: 24 pages, 5 figures

The latest regularized Neural Radiance Field (NeRF) approaches produce poor geometry and view extrapolation for large scale sparse view scenes, such as ETH3D. Density-based approaches tend to be under-constrained, while surface-based approaches tend to miss details. In this paper, we take a density-based approach, sampling patches instead of individual rays to better incorporate monocular depth and normal estimates and patch-based photometric consistency constraints between training views and sampled virtual views. Loosely constraining densities based on estimated depth aligned to sparse points further improves geometric accuracy. While maintaining similar view synthesis quality, our approach significantly improves geometric accuracy on the ETH3D benchmark, e.g. increasing the F1@2cm score by 4x-8x compared to other regularized density-based approaches, with much lower training and inference time than other approaches.

This work explores the application of deep operator learning principles to a problem in statistical physics. Specifically, we consider the linear kinetic equation, consisting of a differential advection operator and an integral collision operator, which is a powerful yet expensive mathematical model for interacting particle systems with ample applications, e.g., in radiation transport. We investigate the capabilities of the Deep Operator network (DeepONet) approach to modelling the high dimensional collision operator of the linear kinetic equation. This integral operator has crucial analytical structures that a surrogate model, e.g., a DeepONet, needs to preserve to enable meaningful physical simulation. We propose several DeepONet modifications to encapsulate essential structural properties of this integral operator in a DeepONet model. To be precise, we adapt the architecture of the trunk-net so the DeepONet has the same collision invariants as the theoretical kinetic collision operator, thus preserving conserved quantities, e.g., mass, of the modeled many-particle system. Further, we propose an entropy-inspired data-sampling method tailored to train the modified DeepONet surrogates without requiring an excessive expensive simulation-based data generation., Comment: 12 pages, 8 figures

Scientific computing using deep learning has seen significant advancements in recent years. There has been growing interest in models that learn the operator from the parameters of a partial differential equation (PDE) to the corresponding solutions. Deep Operator Network (DeepONet) and Fourier Neural operator, among other models, have been designed with structures suitable for handling functions as inputs and outputs, enabling real-time predictions as surrogate models for solution operators. There has also been significant progress in the research on surrogate models based on graph neural networks (GNNs), specifically targeting the dynamics in time-dependent PDEs. In this paper, we propose GraphDeepONet, an autoregressive model based on GNNs, to effectively adapt DeepONet, which is well-known for successful operator learning. GraphDeepONet exhibits robust accuracy in predicting solutions compared to existing GNN-based PDE solver models. It maintains consistent performance even on irregular grids, leveraging the advantages inherited from DeepONet and enabling predictions on arbitrary grids. Additionally, unlike traditional DeepONet and its variants, GraphDeepONet enables time extrapolation for time-dependent PDE solutions. We also provide theoretical analysis of the universal approximation capability of GraphDeepONet in approximating continuous operators across arbitrary time intervals., Comment: 25 pages, 11 figures

We investigate whether region-based representations are effective for recognition. Regions were once a mainstay in recognition approaches, but pixel and patch-based features are now used almost exclusively. We show that recent class-agnostic segmenters like SAM can be effectively combined with strong unsupervised representations like DINOv2 and used for a wide variety of tasks, including semantic segmentation, object-based image retrieval, and multi-image analysis. Once the masks and features are extracted, these representations, even with linear decoders, enable competitive performance, making them well suited to applications that require custom queries. The compactness of the representation also makes it well-suited to video analysis and other problems requiring inference across many images., Comment: CVPR 2024 Camera Ready; website: https://regionreps.web.illinois.edu/

Fast and accurate predictions for complex physical dynamics are a significant challenge across various applications. Real-time prediction on resource-constrained hardware is even more crucial in real-world problems. The deep operator network (DeepONet) has recently been proposed as a framework for learning nonlinear mappings between function spaces. However, the DeepONet requires many parameters and has a high computational cost when learning operators, particularly those with complex (discontinuous or non-smooth) target functions. This study proposes HyperDeepONet, which uses the expressive power of the hypernetwork to enable the learning of a complex operator with a smaller set of parameters. The DeepONet and its variant models can be thought of as a method of injecting the input function information into the target function. From this perspective, these models can be viewed as a particular case of HyperDeepONet. We analyze the complexity of DeepONet and conclude that HyperDeepONet needs relatively lower complexity to obtain the desired accuracy for operator learning. HyperDeepONet successfully learned various operators with fewer computational resources compared to other benchmarks., Comment: 26 pages, 13 figures. Published as a conference paper at Eleventh International Conference on Learning Representations (ICLR 2023)

Large Language Models (LLMs) have demonstrated strong natural language processing and code synthesis capabilities, which has led to their rapid adoption in software engineering applications. However, details about LLM training data are often not made public, which has caused concern as to whether existing bug benchmarks are included. In lieu of the training data for the popular GPT models, we examine the training data of the open-source LLM StarCoder, and find it likely that data from the widely used Defects4J benchmark was included, raising the possibility of its inclusion in GPT training data as well. This makes it difficult to tell how well LLM-based results on Defects4J would generalize, as for any results it would be unclear whether a technique's performance is due to LLM generalization or memorization. To remedy this issue and facilitate continued research on LLM-based SE, we present the GitHub Recent Bugs (GHRB) dataset, which includes 76 real-world Java bugs that were gathered after the OpenAI data cut-off point.

Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena across numerous fields, including physics, engineering, and finance. However, solving parametric PDEs is a complex task that necessitates efficient numerical methods. In this paper, we propose a novel approach for solving parametric PDEs using a Finite Element Operator Network (FEONet). Our proposed method leverages the power of deep learning in conjunction with traditional numerical methods, specifically the finite element method, to solve parametric PDEs in the absence of any paired input-output training data. We performed various experiments on several benchmark problems and confirmed that our approach has demonstrated excellent performance across various settings and environments, proving its versatility in terms of accuracy, generalization, and computational flexibility. Our FEONet framework shows potential for application in various fields where PDEs play a crucial role in modeling complex domains with diverse boundary conditions and singular behavior. Furthermore, we provide theoretical convergence analysis to support our approach, utilizing finite element approximation in numerical analysis., Comment: 23 pages, 11 figures

Our long term goal is to use image-based depth completion to quickly create 3D models from sparse point clouds, e.g. from SfM or SLAM. Much progress has been made in depth completion. However, most current works assume well distributed samples of known depth, e.g. Lidar or random uniform sampling, and perform poorly on uneven samples, such as from keypoints, due to the large unsampled regions. To address this problem, we extend CSPN with multiscale prediction and a dilated kernel, leading to much better completion of keypoint-sampled depth. We also show that a model trained on NYUv2 creates surprisingly good point clouds on ETH3D by completing sparse SfM points.

Multilayer perceptrons (MLPs) learn high frequencies slowly. Recent approaches encode features in spatial bins to improve speed of learning details, but at the cost of larger model size and loss of continuity. Instead, we propose to encode features in bins of Fourier features that are commonly used for positional encoding. We call these Quantized Fourier Features (QFF). As a naturally multiresolution and periodic representation, our experiments show that using QFF can result in smaller model size, faster training, and better quality outputs for several applications, including Neural Image Representations (NIR), Neural Radiance Field (NeRF) and Signed Distance Function (SDF) modeling. QFF are easy to code, fast to compute, and serve as a simple drop-in addition to many neural field representations.

Recent work in multi-view stereo (MVS) combines learnable photometric scores and regularization with PatchMatch-based optimization to achieve robust pixelwise estimates of depth, normals, and visibility. However, non-learning based methods still outperform for large scenes with sparse views, in part due to use of geometric consistency constraints and ability to optimize over many views at high resolution. In this paper, we build on learning-based approaches to improve photometric scores by learning patch coplanarity and encourage geometric consistency by learning a scaled photometric cost that can be combined with reprojection error. We also propose an adaptive pixel sampling strategy for candidate propagation that reduces memory to enable training on larger resolution with more views and a larger encoder. These modifications lead to 6-15% gains in accuracy and completeness on the challenging ETH3D benchmark, resulting in higher F1 performance than the widely used state-of-the-art non-learning approaches ACMM and ACMP.

We propose a hybrid framework opPINN: physics-informed neural network (PINN) with operator learning for approximating the solution to the Fokker-Planck-Landau (FPL) equation. The opPINN framework is divided into two steps: Step 1 and Step 2. After the operator surrogate models are trained during Step 1, PINN can effectively approximate the solution to the FPL equation during Step 2 by using the pre-trained surrogate models. The operator surrogate models greatly reduce the computational cost and boost PINN by approximating the complex Landau collision integral in the FPL equation. The operator surrogate models can also be combined with the traditional numerical schemes. It provides a high efficiency in computational time when the number of velocity modes becomes larger. Using the opPINN framework, we provide the neural network solutions for the FPL equation under the various types of initial conditions, and interaction models in two and three dimensions. Furthermore, based on the theoretical properties of the FPL equation, we show that the approximated neural network solution converges to the a priori classical solution of the FPL equation as the pre-defined loss function is reduced., Comment: 28 pages, 12 figures

Learning the mapping between two function spaces has garnered considerable research attention. However, learning the solution operator of partial differential equations (PDEs) remains a challenge in scientific computing. Fourier neural operator (FNO) was recently proposed to learn solution operators, and it achieved an excellent performance. In this study, we propose a novel \textit{pseudo-differential integral operator} (PDIO) to analyze and generalize the Fourier integral operator in FNO. PDIO is inspired by a pseudo-differential operator, which is a generalized differential operator characterized by a certain symbol. We parameterize this symbol using a neural network and demonstrate that the neural network-based symbol is contained in a smooth symbol class. Subsequently, we verify that the PDIO is a bounded linear operator, and thus is continuous in the Sobolev space. We combine the PDIO with the neural operator to develop a \textit{pseudo-differential neural operator} (PDNO) and learn the nonlinear solution operator of PDEs. We experimentally validate the effectiveness of the proposed model by utilizing Darcy flow and the Navier-Stokes equation. The obtained results indicate that the proposed PDNO outperforms the existing neural operator approaches in most experiments., Comment: 23 pages, 13 figures

DIVeR builds on the key ideas of NeRF and its variants -- density models and volume rendering -- to learn 3D object models that can be rendered realistically from small numbers of images. In contrast to all previous NeRF methods, DIVeR uses deterministic rather than stochastic estimates of the volume rendering integral. DIVeR's representation is a voxel based field of features. To compute the volume rendering integral, a ray is broken into intervals, one per voxel; components of the volume rendering integral are estimated from the features for each interval using an MLP, and the components are aggregated. As a result, DIVeR can render thin translucent structures that are missed by other integrators. Furthermore, DIVeR's representation has semantics that is relatively exposed compared to other such methods -- moving feature vectors around in the voxel space results in natural edits. Extensive qualitative and quantitative comparisons to current state-of-the-art methods show that DIVeR produces models that (1) render at or above state-of-the-art quality, (2) are very small without being baked, (3) render very fast without being baked, and (4) can be edited in natural ways.

The modeling and control of complex physical systems are essential in real-world problems. We propose a novel framework that is generally applicable to solving PDE-constrained optimal control problems by introducing surrogate models for PDE solution operators with special regularizers. The procedure of the proposed framework is divided into two phases: solution operator learning for PDE constraints (Phase 1) and searching for optimal control (Phase 2). Once the surrogate model is trained in Phase 1, the optimal control can be inferred in Phase 2 without intensive computations. Our framework can be applied to both data-driven and data-free cases. We demonstrate the successful application of our method to various optimal control problems for different control variables with diverse PDE constraints from the Poisson equation to Burgers' equation., Comment: 15 pages, 12 figures. Published as a conference paper at Thirty-Sixth AAAI Conference on Artificial Intelligence (AAAI 2022)

Recent learning-based multi-view stereo (MVS) methods show excellent performance with dense cameras and small depth ranges. However, non-learning based approaches still outperform for scenes with large depth ranges and sparser wide-baseline views, in part due to their PatchMatch optimization over pixelwise estimates of depth, normals, and visibility. In this paper, we propose an end-to-end trainable PatchMatch-based MVS approach that combines advantages of trainable costs and regularizations with pixelwise estimates. To overcome the challenge of the non-differentiable PatchMatch optimization that involves iterative sampling and hard decisions, we use reinforcement learning to minimize expected photometric cost and maximize likelihood of ground truth depth and normals. We incorporate normal estimation by using dilated patch kernels, and propose a recurrent cost regularization that applies beyond frontal plane-sweep algorithms to our pixelwise depth/normal estimates. We evaluate our method on widely used MVS benchmarks, ETH3D and Tanks and Temples (TnT), and compare to other state of the art learning based MVS models. On ETH3D, our method outperforms other recent learning-based approaches and performs comparably on advanced TnT., Comment: Accepted to ICCV 2021 for oral presentation

The model reduction of a mesoscopic kinetic dynamics to a macroscopic continuum dynamics has been one of the fundamental questions in mathematical physics since Hilbert's time. In this paper, we consider a diagram of the diffusion limit from the Vlasov-Poisson-Fokker-Planck (VPFP) system on a bounded interval with the specular reflection boundary condition to the Poisson-Nernst-Planck (PNP) system with the no-flux boundary condition. We provide a Deep Learning algorithm to simulate the VPFP system and the PNP system by computing the time-asymptotic behaviors of the solution and the physical quantities. We analyze the convergence of the neural network solution of the VPFP system to that of the PNP system via the Asymptotic-Preserving (AP) scheme. Also, we provide several theoretical evidence that the Deep Neural Network (DNN) solutions to the VPFP and the PNP systems converge to the a priori classical solutions of each system if the total loss function vanishes., Comment: 49 pages, 16 figures

The issue of the relaxation to equilibrium has been at the core of the kinetic theory of rarefied gas dynamics. In the paper, we introduce the Deep Neural Network (DNN) approximated solutions to the kinetic Fokker-Planck equation in a bounded interval and study the large-time asymptotic behavior of the solutions and other physically relevant macroscopic quantities. We impose the varied types of boundary conditions including the inflow-type and the reflection-type boundaries as well as the varied diffusion and friction coefficients and study the boundary effects on the asymptotic behaviors. These include the predictions on the large-time behaviors of the pointwise values of the particle distribution and the macroscopic physical quantities including the total kinetic energy, the entropy, and the free energy. We also provide the theoretical supports for the pointwise convergence of the neural network solutions to the \textit{a priori} analytic solutions. We use the library \textit{PyTorch}, the activation function \textit{tanh} between layers, and the \textit{Adam} optimizer for the Deep Learning algorithm., Comment: 31 pages, 13 figures

Network densification through the deployment of large number of small cells has been considered as the dominant driver for wireless evolution into 5G. However, it has increased the complexity of mobility management, and operators have been facing the technical challenges in handover (HO) parameter optimization. The trade-off between the HO failure (HOF) rate and the ping-pong (PP) rate has further complicated the challenges. In this article, we proposed ZEro handover failure with Unforced and automatic time-to-execute Scaling (ZEUS) HO. ZEUS HO assures HO signaling when a user equipment (UE) is in a good radio link condition and executes the HO at an optimal time. We analyzed the HO performance of Long-Term Evolution (LTE) and ZEUS theoretically using a geometry-based model, considering the most important HO parameter, i.e., HO margin (HOM). We derived the probabilities of HOF and PP from the analysis. The numerical results demonstrated that ZEUS HO can achieve zero HOF rate without increasing the PP rate, solving the trade-off. Furthermore, we showed that the ZEUS HO can accomplish zero HOF rate and zero PP rate simultaneously with an extension of keeping fast moving users out of small cells., Comment: 15 pages, 9 figures. This work has been submitted to IEEE Transactions on Vehicular Technology