Hyung Ju Hwang, Jin Woo Jang, Hyeontae Jo, Jae Yong Lee
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.
Jae Yong Lee, Chuhang Zou, Derek Hoiem
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.
Sung Woong Cho, Jae Yong Lee, Hyung Ju Hwang
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.
Jae Yong Lee, Yuqun Wu, Chuhang Zou, Derek Hoiem, Shenlong Wang
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.
Jin Woo Jang, Jae Yong Lee, Liu Liu, Zhenyi Zhu
We present a deep learning approach for computing multi-phase solutions to the semiclassical limit of the Schrödinger equation. Traditional methods require deriving a multi-phase ansatz to close the moment system of the Liouville equation, a process that is often computationally intensive and impractical. Our method offers an efficient alternative by introducing a novel two-stage neural network framework to close the $2N\times 2N$ moment system, where $N$ represents the number of phases in the solution ansatz. In the first stage, we train neural networks to learn the mapping between higher-order moments and lower-order moments (along with their derivatives). The second stage incorporates physics-informed neural networks (PINNs), where we substitute the learned higher-order moments to systematically close the system. We provide theoretical guarantees for the convergence of both the loss functions and the neural network approximations. Numerical experiments demonstrate the effectiveness of our method for one- and two-dimensional problems with various phase numbers $N$ in the multi-phase solutions. The results confirm the accuracy and computational efficiency of the proposed approach compared to conventional techniques.
Jinsil Lee, Youngjoon Hong, Seungchan Ko, Jae Yong Lee
Recent advances in machine learning (ML) have opened new possibilities for solving partial differential equations (PDEs), yet robust performance in challenging regimes remains limited. In particular, singularly perturbed differential equations exhibit sharp boundary or interior layers with rapid transitions, where standard ML surrogates often fail without extensive resolution. Generating training data for such problems is also costly, as accurate reference solutions typically require massive adaptive mesh refinement. In this work, we propose eFEONet, an enriched Finite Element Operator Network tailored to singularly perturbed problems. Guided by classical singular perturbation theory, eFEONet augments the operator-learning framework with specialized enrichment basis functions that encode the asymptotic structure of layer solutions. This design enables accurate approximation of sharp transitions without relying on large datasets, and can operate with minimal supervision-or even in a data-free manner under appropriate settings. We further provide a rigorous convergence analysis of the proposed method and demonstrate its effectiveness through extensive experiments on representative problems featuring both boundary and interior layers.
Jae Yong Lee, Sungmin Kang, Shin Yoo
Large Language Models (LLMs) are machine learning models that have seen widespread adoption due to their capability of handling previously difficult tasks. LLMs, due to their training, are sensitive to how exactly a question is presented, also known as prompting. However, prompting well is challenging, as it has been difficult to uncover principles behind prompting -- generally, trial-and-error is the most common way of improving prompts, despite its significant computational cost. In this context, we argue it would be useful to perform `predictive prompt analysis', in which an automated technique would perform a quick analysis of a prompt and predict how the LLM would react to it, relative to a goal provided by the user. As a demonstration of the concept, we present Syntactic Prevalence Analyzer (SPA), a predictive prompt analysis approach based on sparse autoencoders (SAEs). SPA accurately predicted how often an LLM would generate target syntactic structures during code synthesis, with up to 0.994 Pearson correlation between the predicted and actual prevalence of the target structure. At the same time, SPA requires only 0.4\% of the time it takes to run the LLM on a benchmark. As LLMs are increasingly used during and integrated into modern software development, our proposed predictive prompt analysis concept has the potential to significantly ease the use of LLMs for both practitioners and researchers.
Jae Yong Lee, Daniel Scharstein, Akash Bapat, Hao Hu, Andrew Fu, Haoru Zhao, Paul Sammut, Xiang Li, Stephen Jeapes, Anik Gupta, Lior David, Saketh Madhuvarasu, Jay Girish Joshi, Jason Wither
We present Ego-1K, a large-scale collection of time-synchronized egocentric multiview videos designed to advance neural 3D video synthesis and dynamic scene understanding. The dataset contains nearly 1,000 short egocentric videos captured with a custom rig with 12 synchronized cameras surrounding a 4-camera VR headset worn by the user. Scene content focuses on hand motions and hand-object interactions in different settings. We describe rig design, data processing, and calibration. Our dataset enables new ways to benchmark egocentric scene reconstruction methods, an important research area as smart glasses with multiple cameras become omnipresent. Our experiments demonstrate that our dataset presents unique challenges for existing 3D and 4D novel view synthesis methods due to large disparities and image motion caused by close dynamic objects and rig egomotion. Our dataset supports future research in this challenging domain. It is available at https://huggingface.co/datasets/facebook/ego-1k.
Jin Young Shin, Jae Yong Lee, Hyung Ju Hwang
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.
Jae Yong Lee
The composite little Higgs model, a UV completion for the $SU(5)/SO(5)$ little Higgs model, incorporates supersymmetry into strong gauge dynamics. We extend the study of flavor physics in the model, and find that it is similar to the bosonic technicolor model. Lepton flavor violations and neutrino mass matrix arise once R-parity violating superpotential is introduced to the model, as in the MSSM. We identify various low-energy effective $ΔL=2$ lepton flavor violating operators, and find that most of them are similar to those of the R-parity violating MSSM. There is a new operator which involves only leptons and the pseudo-Nambu Goldstone bosons of the little Higgs model. We further study a possibility that this operator gives a dominant contribution to the neutrino mass matrix.
Dong-Won Jung, Jae Yong Lee
We implement a one-loop analysis of the $ρ$ parameter in the Left Right Twin Higgs model, including the logarithmically enhanced contributions from both heavy fermion and scalar loops. Numerical analysis indicates that the one-loop corrections are dominant over the tree-level contributions in most regions of parameter space. The experimentally allowed values of the $ρ$-parameter divide the allowed parameter space into two regions; less than $670 {\rm GeV}$ and larger than $1100 {\rm GeV}$ roughly, for the symmetry breaking scale $f$. Therefore our result significantly reduces the parameter space which are favorably accessible to the LHC.
Kyoungjin Jung, Jae Yong Lee, Dongwook Shin
Neural operators have shown remarkable performance in approximating solutions of partial differential equations. However, their convergence behavior under grid refinement is still not well understood from the viewpoint of numerical analysis. In this work, we propose a numerically informed convolutional operator network, called NICON, that explicitly couples classical finite difference and finite element methods with operator learning through residual-based training loss functions. We introduce two types of networks, FD-CON and FE-CON, which use residual-based loss functions derived from the corresponding numerical methods. We derive error estimates for FD-CON and FE-CON using finite difference and finite element analysis. These estimates show a direct relation between the convergence behavior and the decay rate of the training loss. From these analyses, we establish training strategies that guarantee optimal convergence rates under grid refinement. Several numerical experiments are presented to validate the theoretical results and show performance on fine grids.
Jae Yong Lee, Jin Woo Jang, Hyung Ju Hwang
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.
Jae Yong Lee, Joseph DeGol, Chuhang Zou, Derek Hoiem
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.
Jae Yong Lee
The simplest little Higgs model based on a SU(3) global symmetry contains a $SU(3)_{weak}$ triplet and a singlet per a generation in the lepton sector. A neutral component of the triplet and the singlet turn into a neutral vector-like $SU(2)_L$ singlet after electroweak symmetry breaking while the other neutral component of the triplet is the SM neutrino. At tree level, Yukawa couplings of the lepton sector not only allow the neutral vector-like lepton to couple to the SM neutrino, but also give them a Dirac mass. Majorana mass terms for the SM neutrinos and their partners arise at one loops, leading to neutrino flavor mixing in addition to neutrino-heavy neutral lepton mixing.
Jae Yong Lee, Seungchan Ko, Youngjoon Hong
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. While our method is not meshless, the 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.
Rakhoon Hwang, Jae Yong Lee, Jin Young Shin, Hyung Ju Hwang
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.
Jae Yong Lee, Yuqun Wu, Chuhang Zou, Shenlong Wang, Derek Hoiem
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.
Jae Yong Lee
The N=1 supergravity coupled to chiral and Yang-Mills superfields is regarded as the ultraviolet theory of the supersymmetric Standard Model. The gravitino acquires a mass from the hidden sector and participate in mediating supersymmetry breaking to the visible sector. We consider the quantum effects of the gravitino and evaluate its contribution to the gaugino masses at one loop level, which turns out to be irrespective of the SM gauge couplings. Due to the universality of gravitino, its contribution to the gaugino masses can significantly modify the gaugino mass relations of other supersymmetry breaking mechanisms depending on the gravitino mass. For a given gaugino mass the gravitino mass cannot be arbitrarily large even when other supersymmetry breaking mechanisms are dominant.
Jae Yong Lee
The Littlest Higgs model contains a new vector-like heavy quark in the up sector. There are two interesting features of its existence. One is that it extends the $3\times3$ CKM matrix in the Standard Model to a $4\times3$ matrix and the other is that it allows Z-mediated flavor changing neutral currents at tree level in the up sector but not in the down sector. We examine a few of possible windows in which the Z-mediated flavor changing neutral currents in the Littlest Higgs Model can be tested.