arXiv subjects

Jae Yong Lee

Publications and source records attributed to Jae Yong Lee.

At least 19 recordsLinked to original sources

arXiv2026

Discovering Ordinary Differential Equations with LLM-Based Qualitative and Quantitative Evaluation

Discovering governing differential equations from observational data is a fundamental challenge in scientific machine learning. Existing symbolic regression approaches rely primarily on quantitative metrics; however, real-world differential equation modeling also requires incorporating domain knowledge to ensure physical plausibility. To address this gap, we propose DoLQ, a method for discovering ordinary differential equations with LLM-based qualitative and quantitative evaluation. DoLQ employs a multi-agent architecture: a Sampler Agent proposes dynamic system candidates, a Parameter Optimizer refines equations for accuracy, and a Scientist Agent leverages an LLM to conduct both qualitative and quantitative evaluations and synthesize their results to iteratively guide the search. Experiments on multi-dimensional ordinary differential equation benchmarks demonstrate that DoLQ achieves superior performance compared to existing methods, not only attaining higher success rates but also more accurately recovering the correct symbolic terms of ground truth equations. Our code is available at https://github.com/Bon99yun/DoLQ.

cs.AI↗

arXiv2026

Deep Learning-based Surrogate Modelling of the LOD Method for Multiscale Problems

Multiscale problems are notoriously difficult to tackle using traditional numerical methods, as accurately resolving fine-scale features often requires prohibitively fine discretizations. This challenge is particularly pronounced in applications such as materials science, fluid dynamics, climate systems, chemical processes, and complex networks. Recent neural operator models provide a promising data-driven alternative, but frequently struggle to achieve sufficient accuracy in the presence of strongly heterogeneous or oscillatory coefficients. In this work, we focus on the solution of elliptic PDEs with rough and high-contrast inputs. The Localized Orthogonal Decomposition (LOD) method is a well-established numerical approach for such problems, but it comes, however, at a substantial computational cost. We investigate the performance of popular neural operator architectures on these challenging multiscale problems and identify key limitations in their ability to resolve fine-scale structure. To overcome these challenges, we introduce LOD-MSNO (LOD-Multiscale Neural Operator), a hybrid approach that leverages the LOD method as a strong multiscale prior by building on its representation of the solution as a linear combination of problem-adapted basis functions, while addressing its main computational bottlenecks through data-driven operator learning. We further provide theoretical error estimates for the proposed coefficient-learning framework. Lastly, we demonstrate the potential of our proposed method to outperform current neural operator baselines in terms of accuracy for challenging multiscale inputs, while mainly retaining the computational efficiency of neural operator models.

math.NA↗

arXiv2026

A residual-based finite element surrogate solver for elliptic partial differential equations

We propose a residual-based finite element surrogate solver for elliptic partial differential equations. The method combines convolutional neural networks with classical finite element discretization in a data-free setting, where the loss function is defined directly from the finite element residual. This enables the approximation of the solution operator without requiring paired input-output data. A key feature of the proposed approach is that it can be analyzed using standard finite element theory under mesh refinement. We establish a relationship between the training loss and the error in the $H^1$-seminorm, and derive training criteria that ensure optimal convergence rates. To improve efficiency, we introduce a decomposition strategy that separates the contributions of different input components. This allows the model to learn simpler sub-operators. Numerical experiments demonstrate that the proposed method achieves stable convergence under grid refinement, remains robust on complex geometries and oscillatory solutions, and extends naturally to nonlinear equations.

math.NA↗

arXiv2026

Unbiased and Second-Order-Free Training for High-Dimensional PDEs

Deep learning methods based on backward stochastic differential equations (BSDEs) have emerged as competitive alternatives to physics-informed neural networks (PINNs) for solving high-dimensional partial differential equations (PDEs). By leveraging probabilistic representations, BSDE approaches can avoid the curse of dimensionality and often admit second-order-free training objectives that do not require explicit Hessian evaluations. It has recently been established that the commonly used Euler-Maruyama (EM) time discretization induces an intrinsic bias in BSDE training losses. While high-order schemes such as Heun can fully eliminate this bias, such schemes re-introduce second-order spatial derivatives and incur substantial computational overhead. In this work, we provide a principled analysis of EM-induced loss bias and propose an unbiased, second-order-free training framework that preserves the computational advantages of BSDE methods. Our code is available at https://github.com/seojaemin22/Un-EM-BSDE.

cs.LG↗

arXiv2026

Ego-1K -- A Large-Scale Multiview Video Dataset for Egocentric Vision

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.

cs.CV↗

arXiv2026

FourierSpecNet: Neural Collision Operator Approximation Inspired by the Fourier Spectral Method for Solving the Boltzmann Equation

The Boltzmann equation, a fundamental model in kinetic theory, describes the evolution of particle distribution functions through a nonlinear, high-dimensional collision operator. However, its numerical solution remains computationally demanding, particularly for inelastic collisions and high-dimensional velocity domains. In this work, we propose the Fourier Neural Spectral Network (FourierSpecNet), a hybrid framework that integrates the Fourier spectral method with deep learning to approximate the collision operator in Fourier space efficiently. FourierSpecNet achieves resolution-invariant learning and supports zero-shot super-resolution, enabling accurate predictions at unseen resolutions without retraining. Beyond empirical validation, we establish a consistency result showing that the trained operator converges to the spectral solution as the discretization is refined. We evaluate our method on several benchmark cases, including Maxwellian and hard-sphere molecular models, as well as inelastic collision scenarios. The results demonstrate that FourierSpecNet offers competitive accuracy while significantly reducing computational cost compared to traditional spectral solvers. Our approach provides a robust and scalable alternative for solving the Boltzmann equation across both elastic and inelastic regimes.

cs.LG↗

arXiv2026

Data-Free Asymptotics-Informed Operator Networks for Singularly Perturbed PDEs

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.

math.NA↗

arXiv2026

Discontinuous Galerkin finite element operator network for solving non-smooth PDEs

We introduce Discontinuous Galerkin Finite Element Operator Network (DG--FEONet), a data-free operator learning framework that combines the strengths of the discontinuous Galerkin (DG) method with neural networks to solve parametric partial differential equations (PDEs) with discontinuous coefficients and non-smooth solutions. Unlike traditional operator learning models such as DeepONet and Fourier Neural Operator, which require large paired datasets and often struggle near sharp features, our approach minimizes the residual of a DG-based weak formulation using the Symmetric Interior Penalty Galerkin (SIPG) scheme. DG-FEONet predicts element-wise solution coefficients via a neural network, enabling data-free training without the need for precomputed input-output pairs. We provide theoretical justification through convergence analysis and validate the model's performance on a series of one- and two-dimensional PDE problems, demonstrating accurate recovery of discontinuities, strong generalization across parameter space, and reliable convergence rates. Our results highlight the potential of combining local discretization schemes with machine learning to achieve robust, singularity-aware operator approximation in challenging PDE settings.

math.NA↗

arXiv2025

Fully stabilized 25 GHz frequency comb for frequency calibration of optical spectrum analyzer

Optical spectrometers are widely used in scientific and industrial applications, and precise frequency calibration is essential for ensuring their reliable performance. Traditionally, spectrometers have been calibrated using reference gas cells or reference lamps. However, such conventional methods are not enough to meet the demands for high accuracy and stability. Although frequency-stabilized lasers offer excellent frequency uncertainty, they provide only a single calibration point at a fixed frequency, which is unsuitable for wide-range spectrometer calibration. In this work, we demonstrate a fully stabilized, high-repetition rate electro-optic frequency comb (EO comb) as an absolute frequency reference providing multiple calibration points over a broad spectral range. Our 25 GHz EO comb provides a broadband spectrum spanning from 189.5 THz to 196 THz (corresponding to 1530 nm to 1582 nm), traceable to a frequency standard with a relative standard uncertainty of 10^-13. The well-defined and evenly spaced comb modes can be spectrally resolved by conventional optical spectrometers, enabling wide-range, high-precision frequency calibration. We directly calibrated a commercial spectrometer by referencing its measured frequency values to our well-defined comb modes, thereby evaluating the frequency error with a standard uncertainty of 20 MHz (or relative standard uncertainty of 10^-7), which is limited by the Type A uncertainty (repeatability) of the spectrometer. The proposed method is simple to implement and provides multiple calibration points referenced to the frequency standards over a broad spectral range. This approach improves both the calibration and performance evaluation of spectrometers, and it contributes to the advancement of optical metrology.

physics.optics↗

arXiv2025

Deep learning-based moment closure for multi-phase computation of semiclassical limit of the Schrödinger equation

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.

math.NA↗

arXiv2025

Predictive Prompt Analysis

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.

cs.SE↗

arXiv2025

Finite Element Operator Network for Solving Elliptic-type parametric PDEs

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.

math.NA↗

arXiv2024

Plenoptic PNG: Real-Time Neural Radiance Fields in 150 KB

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.

cs.CV↗

arXiv2024

MonoPatchNeRF: Improving Neural Radiance Fields with Patch-based Monocular Guidance

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.

cs.CV↗

arXiv2024

Hamilton-Jacobi Based Policy-Iteration via Deep Operator Learning

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).

math.OC↗

arXiv2024

Region-Based Representations Revisited

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.

cs.CV↗

arXiv2024

Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations

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.

cs.LG↗

arXiv2024

Structure-Preserving Operator Learning: Modeling the Collision Operator of Kinetic Equations

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.

math.NA↗