An End-to-End Deep Learning Method for Solving Nonlocal Allen-Cahn and Cahn-Hilliard Phase-Field Models

We propose an efficient end-to-end deep learning method for solving nonlocal Allen-Cahn (AC) and Cahn-Hilliard (CH) phase-field models. One motivation for this effort emanates from the fact that discretized partial differential equation-based AC or CH phase-field models result in diffuse interfaces between phases, with the only recourse for remediation is to severely refine the spatial grids in the vicinity of the true moving sharp interface whose width is determined by a grid-independent parameter that is substantially larger than the local grid size. In this work, we introduce non-mass conserving nonlocal AC or CH phase-field models with regular, logarithmic, or obstacle double-well potentials. Because of non-locality, some of these models feature totally sharp interfaces separating phases. The discretization of such models can lead to a transition between phases whose width is only a single grid cell wide. Another motivation is to use deep learning approaches to ameliorate the otherwise high cost of solving discretized nonlocal phase-field models. To this end, loss functions of the customized neural networks are defined using the residual of the fully discrete approximations of the AC or CH models, which results from applying a Fourier collocation method and a temporal semi-implicit approximation. To address the long-range interactions in the models, we tailor the architecture of the neural network by incorporating a nonlocal kernel as an input channel to the neural network model. We then provide the results of extensive computational experiments to illustrate the accuracy, structure-preserving properties, predictive capabilities, and cost reductions of the proposed method.

EfficientGS: Streamlining Gaussian Splatting for Large-Scale High-Resolution Scene Representation

In the domain of 3D scene representation, 3D Gaussian Splatting (3DGS) has emerged as a pivotal technology. However, its application to large-scale, high-resolution scenes (exceeding 4k$\times$4k pixels) is hindered by the excessive computational requirements for managing a large number of Gaussians. Addressing this, we introduce 'EfficientGS', an advanced approach that optimizes 3DGS for high-resolution, large-scale scenes. We analyze the densification process in 3DGS and identify areas of Gaussian over-proliferation. We propose a selective strategy, limiting Gaussian increase to key primitives, thereby enhancing the representational efficiency. Additionally, we develop a pruning mechanism to remove redundant Gaussians, those that are merely auxiliary to adjacent ones. For further enhancement, we integrate a sparse order increment for Spherical Harmonics (SH), designed to alleviate storage constraints and reduce training overhead. Our empirical evaluations, conducted on a range of datasets including extensive 4K+ aerial images, demonstrate that 'EfficientGS' not only expedites training and rendering times but also achieves this with a model size approximately tenfold smaller than conventional 3DGS while maintaining high rendering fidelity.

TransNet: Transferable Neural Networks for Partial Differential Equations

Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information of the target PDEs such as its formulation and/or data of its solution for pre-training. In this work, we propose to construct transferable neural feature spaces from purely function approximation perspectives without using PDE information. The construction of the feature space involves re-parameterization of the hidden neurons and uses auxiliary functions to tune the resulting feature space. Theoretical analysis shows the high quality of the produced feature space, i.e., uniformly distributed neurons. Extensive numerical experiments verify the outstanding performance of our method, including significantly improved transferability, e.g., using the same feature space for various PDEs with different domains and boundary conditions, and the superior accuracy, e.g., several orders of magnitude smaller mean squared error than the state of the art methods.

Style Mixing and Patchwise Prototypical Matching for One-Shot Unsupervised Domain Adaptive Semantic Segmentation

In this paper, we tackle the problem of one-shot unsupervised domain adaptation (OSUDA) for semantic segmentation where the segmentors only see one unlabeled target image during training. In this case, traditional unsupervised domain adaptation models usually fail since they cannot adapt to the target domain with over-fitting to one (or few) target samples. To address this problem, existing OSUDA methods usually integrate a style-transfer module to perform domain randomization based on the unlabeled target sample, with which multiple domains around the target sample can be explored during training. However, such a style-transfer module relies on an additional set of images as style reference for pre-training and also increases the memory demand for domain adaptation. Here we propose a new OSUDA method that can effectively relieve such computational burden. Specifically, we integrate several style-mixing layers into the segmentor which play the role of style-transfer module to stylize the source images without introducing any learned parameters. Moreover, we propose a patchwise prototypical matching (PPM) method to weighted consider the importance of source pixels during the supervised training to relieve the negative adaptation. Experimental results show that our method achieves new state-of-the-art performance on two commonly used benchmarks for domain adaptive semantic segmentation under the one-shot setting and is more efficient than all comparison approaches.

Level set learning with pseudo-reversible neural networks for nonlinear dimension reduction in function approximation

Due to the curse of dimensionality and the limitation on training data, approximating high-dimensional functions is a very challenging task even for powerful deep neural networks. Inspired by the Nonlinear Level set Learning (NLL) method that uses the reversible residual network (RevNet), in this paper we propose a new method of Dimension Reduction via Learning Level Sets (DRiLLS) for function approximation. Our method contains two major components: one is the pseudo-reversible neural network (PRNN) module that effectively transforms high-dimensional input variables to low-dimensional active variables, and the other is the synthesized regression module for approximating function values based on the transformed data in the low-dimensional space. The PRNN not only relaxes the invertibility constraint of the nonlinear transformation present in the NLL method due to the use of RevNet, but also adaptively weights the influence of each sample and controls the sensitivity of the function to the learned active variables. The synthesized regression uses Euclidean distance in the input space to select neighboring samples, whose projections on the space of active variables are used to perform local least-squares polynomial fitting. This helps to resolve numerical oscillation issues present in traditional local and global regressions. Extensive experimental results demonstrate that our DRiLLS method outperforms both the NLL and Active Subspace methods, especially when the target function possesses critical points in the interior of its input domain.

A Comparison of Neural Network Architectures for Data-Driven Reduced-Order Modeling

The popularity of deep convolutional autoencoders (CAEs) has engendered effective reduced-order models (ROMs) for the simulation of large-scale dynamical systems. However, it is not known whether deep CAEs provide superior performance in all ROM scenarios. To elucidate this, the effect of autoencoder architecture on its associated ROM is studied through the comparison of deep CAEs against two alternatives: a simple fully connected autoencoder, and a novel graph convolutional autoencoder. Through benchmark experiments, it is shown that the superior autoencoder architecture for a given ROM application is highly dependent on the size of the latent space and the structure of the snapshot data, with the proposed architecture demonstrating benefits on data with irregular connectivity when the latent space is sufficiently large.

Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver

Partial differential equations are often used to model various physical phenomena, such as heat diffusion, wave propagation, fluid dynamics, elasticity, electrodynamics and image processing, and many analytic approaches or traditional numerical methods have been developed and widely used for their solutions. Inspired by rapidly growing impact of deep learning on scientific and engineering research, in this paper we propose a novel neural network, GF-Net, for learning the Green's functions of linear reaction-diffusion equations in an unsupervised fashion. The proposed method overcomes the challenges for finding the Green's functions of the equations on arbitrary domains by utilizing physics-informed approach and the symmetry of the Green's function. As a consequence, it particularly leads to an efficient way for solving the target equations under different boundary conditions and sources. We also demonstrate the effectiveness of the proposed approach by experiments in square, annular and L-shape domains.

Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations

A dimension reduction method based on the "Nonlinear Level set Learning" (NLL) approach is presented for the pointwise prediction of functions which have been sparsely sampled. Leveraging geometric information provided by the Implicit Function Theorem, the proposed algorithm effectively reduces the input dimension to the theoretical lower bound with minor accuracy loss, providing a one-dimensional representation of the function which can be used for regression and sensitivity analysis. Experiments and applications are presented which compare this modified NLL with the original NLL and the Active Subspaces (AS) method. While accommodating sparse input data, the proposed algorithm is shown to train quickly and provide a much more accurate and informative reduction than either AS or the original NLL on two example functions with high-dimensional domains, as well as two state-dependent quantities depending on the solutions to parametric differential equations.

DANNet: A One-Stage Domain Adaptation Network for Unsupervised Nighttime Semantic Segmentation

Semantic segmentation of nighttime images plays an equally important role as that of daytime images in autonomous driving, but the former is much more challenging due to poor illuminations and arduous human annotations. In this paper, we propose a novel domain adaptation network (DANNet) for nighttime semantic segmentation without using labeled nighttime image data. It employs an adversarial training with a labeled daytime dataset and an unlabeled dataset that contains coarsely aligned day-night image pairs. Specifically, for the unlabeled day-night image pairs, we use the pixel-level predictions of static object categories on a daytime image as a pseudo supervision to segment its counterpart nighttime image. We further design a re-weighting strategy to handle the inaccuracy caused by misalignment between day-night image pairs and wrong predictions of daytime images, as well as boost the prediction accuracy of small objects. The proposed DANNet is the first one stage adaptation framework for nighttime semantic segmentation, which does not train additional day-night image transfer models as a separate pre-processing stage. Extensive experiments on Dark Zurich and Nighttime Driving datasets show that our method achieves state-of-the-art performance for nighttime semantic segmentation.

Interactive Binary Image Segmentation with Edge Preservation

Binary image segmentation plays an important role in computer vision and has been widely used in many applications such as image and video editing, object extraction, and photo composition. In this paper, we propose a novel interactive binary image segmentation method based on the Markov Random Field (MRF) framework and the fast bilateral solver (FBS) technique. Specifically, we employ the geodesic distance component to build the unary term. To ensure both computation efficiency and effective responsiveness for interactive segmentation, superpixels are used in computing geodesic distances instead of pixels. Furthermore, we take a bilateral affinity approach for the pairwise term in order to preserve edge information and denoise. Through the alternating direction strategy, the MRF energy minimization problem is divided into two subproblems, which then can be easily solved by steepest gradient descent (SGD) and FBS respectively. Experimental results on the VGG interactive image segmentation dataset show that the proposed algorithm outperforms several state-of-the-art ones, and in particular, it can achieve satisfactory edge-smooth segmentation results even when the foreground and background color appearances are quite indistinctive.