The Mathematical AI and Scientific Computing (MAISC) Lab at Chung-Ang University focuses on research related to AI based on mathematical theory (Mathematical AI) and the utilization of AI for scientific problem-solving (Scientific Computing), grounded in mathematical principles.

중앙대학교 수학적 인공지능 및 과학계산(MAISC) 연구실은 수학적 이론에 기반한 인공지능(수학적 인공지능) 연구와 수학적 원리에 근거하여 과학적 문제 해결을 위한 인공지능 활용(과학계산) 연구에 중점을 둡니다.

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

Spectral-guided Physical Dynamics Distillation

Conformal mapping based Physics-informed neural networks for designing neutral inclusions

Error analysis for finite element operator learning methods for solving parametric second-order elliptic PDEs

Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular grids

Physics-Informed Neural Operators for Tissue Elasticity Reconstruction

Hamilton-Jacobi Based Policy-Iteration via Deep Operator Learning

Isometric Regularization for Manifolds of Functional Data

Finite Element Operator Network for Solving Elliptic-type parametric PDEs

Numerically Informed Convolutional Operator Network with Subproblem Decomposition for Poisson Equations

Neural advection-diffusion equation for long-term climate dynamics

Discontinuous Galerkin Finite Element Operator Network for Solving Non-Smooth PDEs

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

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

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

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

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

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

물리 정보 기계학습 의 발전 및 응용

opPINN: Physics-Informed Neural Network with operator learning to approximate solutions to the Fokker-Planck-Landau equation

HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetwork

Solving PDE-constrained Control Problems using Operator Learning

The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the Deep Neural Network Approach

Trend to Equilibrium for the Kinetic Fokker-Planck Equation via the Neural Network Approach

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.

FourierSpecNet: Neural collision operator approximation inspired by the Fourier spectral method for solving the Boltzmann equation

While conventional data are represented as discrete vectors, Implicit Neural Representations (INRs) utilize neural networks to represent data points as continuous functions. By incorporating a shared network that maps latent vectors to individual functions, one can model the distribution of functional data, which has proven effective in many applications, such as learning 3D shapes, surface reflectance, and operators.
However, the infinite-dimensional nature of these representations makes them prone to overfitting, necessitating sufficient regularization. Naïve regularization methods -- those commonly used with discrete vector representations -- may enforce smoothness to increase robustness but result in a loss of data fidelity due to improper handling of function coordinates. 
To overcome these challenges, we start by interpreting the mapping from latent variables to INRs as a parametrization of a Riemannian manifold. We then recognize that preserving geometric quantities -- such as distances and angles -- between the latent space and the data manifold is crucial. As a result, we obtain a manifold with minimal intrinsic curvature, leading to robust representations while maintaining high-quality data fitting. Our experiments on various data modalities demonstrate that our method effectively discovers a well-structured latent space, leading to robust data representations even for challenging datasets, such as those that are small or noisy.

Mikyoung Lim

Professor

Jae Yong Lee

Mathematical AI and Scientific Computing (MAISC) Lab

교수님 소개

임미경

연구실 소개

연구실 키워드

알럼나이가 간 회사

연구실 지원 방법

대우 조건

연도별 논문 인용수

저널 & 학회 논문 출판 수

연구실 논문 목록

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

Spectral-guided Physical Dynamics Distillation

Conformal mapping based Physics-informed neural networks for designing neutral inclusions

연구실 보유 특허

대학별 입시끈

우리 연구실 모집

교수님 소개

임미경

연구실 키워드

알럼나이가 간 회사

연구실 지원 방법

대우 조건

연도별 논문 인용수

저널 & 학회 논문 출판 수

연구실 논문 목록

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

Spectral-guided Physical Dynamics Distillation

Conformal mapping based Physics-informed neural networks for designing neutral inclusions

연구실 보유 특허

최근 3년간 연구실적 높은 연구실

대학별 입시끈

우리 연구실 모집