Model Reduction and Neural Networks

Derniers papiers

• MC$^2$: Monte Carlo Correction for Fast Elliptic PDE Solving

  Ethan Hsu, Hong Meng Yam, Ivan Ge · 12 mai 2026

  Partial differential equation (PDE) solvers underpin scientific computing, but real-world deployment is bounded by compute. Classical Monte Carlo solvers such as Walk-on-Spheres (WoS) are unbiased and geometry-agnostic but are slow. Learned solvers are fast but biased and brittle under distribution …

• M$^3$: Reframing Training Measures for Discretized Physical Simulations

  Yuan Mei, Xingyu Song, Xiaowen Song, Naoya Takeishi · 12 mai 2026

  Neural surrogate models for physical simulations are trained on discretized samples of continuous domains, where the induced empirical measure leads to uneven supervision, biasing optimization and causing spatial inconsistencies in physical fidelity. To mitigate this measure-induced bias, we propose…

• Can We Formally Verify Neural PDE Surrogates? SMT Compilation of Small Fourier Neural Operators

  Ali Baheri, David Millard, Ignacio Laguna Peralta · 12 mai 2026

  Fourier Neural Operators (FNOs) can greatly accelerate PDE simulation, but they are often used without formal guarantees that they preserve basic physical structure. We show that, once the trained weights and grid are fixed, the spectral convolution in an FNO is a linear map. As a result, the full f…

• Semi-Supervised Neural Super-Resolution for Mesh-Based Simulations

  Jiyeon Kim, Youngjoon Hong, Won-Yong Shin · 12 mai 2026

  Mesh-based simulations provide high-fidelity solutions to partial differential equations (PDEs), but achieving such accuracy typically requires fine meshes, leading to substantial computational overhead. Super-resolution techniques aim to mitigate this cost by reconstructing high-resolution (HR), hi…

• CATO: Charted Attention for Neural PDE Operators

  Chun-Wun Cheng, Sifan Wang, Carola-Bibiane Sch\"onlieb, Angelica I. Aviles-Rivero · 12 mai 2026

  Neural operators have emerged as powerful data-driven solvers for PDEs, offering substantial acceleration over classical numerical methods. However, existing transformer-based operators still face critical challenges when modeling PDEs on complex geometries: directly processing over massive mesh poi…

• Recovering Physical Dynamics from Discrete Observations via Intrinsic Differential Consistency

  Yuxiang Luo, Andrew Perrault · 12 mai 2026

  Recovering continuous-time dynamics from discrete observations is difficult because local supervision (e.g., pointwise regression targets, derivative approximations, or equation residuals) loses fidelity as the observation interval grows. We replace local supervision with a global structural constra…

• Adaptive Data Harvesting for Efficient Neural Network Learning with Universal Constraints

  Siteng Kang, Xinhua Zhang · 12 mai 2026

  Training neural networks to satisfy universal constraints over continuous domains poses unique challenges. Common examples include Lyapunov Neural Networks (Lyapunov NNs) and Physics-Informed Neural Networks (PINNs), where analytical solutions are generally either unavailable or overly restrictive. …

• When Attention Beats Fourier: Multi-Scale Transformers for PDE Solving on Irregular Domains

  Brandon Yee, Pairie Koh, Jack Rodriguez, Mihir Tekal · 12 mai 2026

  We study the problem of \emph{architecture selection} for deep learning models trained to solve partial differential equations (PDEs), asking when transformer-based architectures with learned attention outperform Fourier-domain neural operators. We introduce the \textbf{Multi-Scale Attention Transfo…

• A meshfree exterior calculus for generalizable and data-efficient learning of physics from point clouds

  Benjamin D. Shaffer, Brooks Kinch, M. Ani Hsieh, Nathaniel Trask · 12 mai 2026

  We introduce a meshfree exterior calculus (MEEC) for learning structure-preserving descriptions of physics on point clouds, and use it to build MEEC-Net, a data-efficient surrogate that transfers across resolutions, geometries, and physical parameters. MEEC equips an $\varepsilon$-ball graph with vi…

• Learned Lagrangian Models of PDEs via Euler-Lagrange Residual Minimization

  Lyra Zhornyak, Eric Forgoston, M. Ani Hsieh · 11 mai 2026

  We present the first method to directly use a learned continuous Lagrangian to forecast the dynamics of systems governed by partial differential equations, exploiting the inherent conservative structure to achieve stable long-range predictions. We develop an optimization-based integrator that minimi…

• Stabilized neural Hamilton--Jacobi--Bellman solvers: Error analysis and applications in model-based reinforcement learning

  Minseok Kim, Yeongjong Kim, Namkyeong Cho, Yeoneung Kim · 11 mai 2026

  Physics-informed neural solvers offer a promising route to model-based reinforcement learning in continuous time, where optimal feedback synthesis is governed by Hamilton--Jacobi--Bellman (HJB) equations. Practical implementations often occupy a regime that is neither a classical grid method nor a c…

• On the Role of Strain and Vorticity in Numerical Integration Error for Flow Matching

  Chenxi Tao, Seung-Kyum Choi · 11 mai 2026

  Flow matching generates data by integrating a learned velocity field, where the number of integration steps (NFE) directly determines inference cost. We analyze which properties of the velocity field govern integration error by decomposing the velocity Jacobian into its symmetric part S (strain rate…

• CarCrashNet: A Large-Scale Dataset and Hierarchical Neural Solver for Data-Driven Structural Crash Simulation

  Mohamed Elrefaie, Dule Shu, Matthew Klenk, Faez Ahmed · 11 mai 2026

  Crash simulation is a cornerstone of modern vehicle development because it reduces the need for costly physical prototypes, accelerates safety-driven design iteration, and increasingly supports virtual testing workflows. At the same time, modeling structural crash mechanics remains exceptionally cha…

• Physics-based Digital Twins for Integrated Thermal Energy Systems Using Active Learning

  Umme Mahbuba Nabila, Paul Seurin, Linyu Lin, Majdi I. Radaideh · 11 mai 2026

  Real-time supervisory control of thermal energy distribution systems requires digital twins that are accurate, interpretable, and uncertainty-aware, yet remain data and computationally efficient. High-fidelity simulations alone are costly, while purely data-driven surrogates often lack robustness. T…

• Geometric Kolmogorov--Arnold Network (GeoKAN)

  Abhijit Sen, Bikram Keshari Parida, Giridas Maiti, Mahima Arya, Denys I. Bondar · 11 mai 2026

  We introduce Geometric Kolmogorov--Arnold Networks (GeoKANs), a family of geometry-aware KAN-type models in which approximation is carried out in learned, geometry-adapted coordinates rather than in fixed Euclidean input coordinates. GeoKAN achieves this by learning a diagonal Riemannian metric that…

• On the Robustness of Distribution Support under Diffusion Guidance

  Ruijia Cao, Yuchen Wu, Nisha Chadramoorthy · 11 mai 2026

  Diffusion guidance is a powerful technique that enables controllable and high-fidelity sample generation with diffusion models. At a high level, it modifies the score function by incorporating a guidance term that steers the generative process toward a desired condition. Despite its empirical succes…

• Persistent-Transient Policy Evaluation for Markov Chains via Minimal Peripheral Quotients

  Yang Xu, Vaneet Aggarwal · 11 mai 2026

  We study fixed-policy evaluation for finite Markov chains that may be reducible and periodic. Classical evaluation methods with gain and bias decomposition are not always diagnostic: the gain records only invariant Ces\`aro averages, while persistent phase-dependent behavior is absorbed into the bia…

• Identifiability Challenges in Sparse Linear Ordinary Differential Equations

  Cecilia Casolo, S\"oren Becker, Niki Kilbertus · 11 mai 2026

  Dynamical systems modeling is a core pillar of scientific inquiry across natural and life sciences. Increasingly, dynamical system models are learned from data, rendering identifiability a paramount concept. For systems that are not identifiable from data, no guarantees can be given about their beha…

• NSPOD: acceleratingthe convergence ofKrylov-based iterative linearsolvers via approximated PODs

  Francesc Levrero-Florencio, Youngkyu Lee, Jay Pathak, George Em Karniadakis · 11 mai 2026

  The convergence of Krylov-based linear iterative solvers applied to parametric partial differential equations (PDEs) is often highly sensitive to the domain, its discretization, the location/values of the applied Dirichlet/Neumann boundary conditions, body forces and material properties, among other…

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

  Sum Kyun Song, Bong Gyun Shin, Jae Yong Lee · 11 mai 2026

  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 knowle…

• Sparse Random-Feature Neural Networks with Krylov-Based SVD for Singularly Perturbed ODE

  Kevin Kurian Thomas Vaidyan, Siddharth Rout · 11 mai 2026

  Random-feature neural networks (RFNNs), including architectures with fixed hidden layers and analytically determined output weights, offer fast training but often suffer from issues due to dense representations of the hidden layer activation. Their reliance on dense feature mappings and least square…

• Adaptive Domain Decomposition Physics-Informed Neural Networks for Traffic State Estimation with Sparse Sensor Data

  Eunhan Ka, Ludovic Leclercq, Satish V. Ukkusuri · 11 mai 2026

  Traffic state estimation from sparse fixed sensors is challenging because physics-informed neural networks (PINNs) tend to over-smooth the shockwaves admitted by the Lighthill-Whitham-Richards (LWR) model. This study proposes Adaptive Domain Decomposition Physics-Informed Neural Networks (ADD-PINN),…

• Functional-prior-based Bayesian PDE-constrained inversion using PINNs

  Ryoichiro Agata, Tomohisa Okazaki · 11 mai 2026

  Physics-informed neural networks (PINNs) provide a mesh-free framework for solving PDE-constrained inverse problems, but their extension to Bayesian inversion still faces a fundamental difficulty: prior distributions are typically defined in the weight space of neural networks, whereas physically me…

• The E$\Delta$-MHC-Geo Transformer: Adaptive Geodesic Operations with Guaranteed Orthogonality

  Arash Shahmansoori · 11 mai 2026

  We present the E$\Delta$-MHC-Geo Transformer, a novel architecture that unifies Manifold-Constrained Hyper-Connections (mHC), Deep Delta Learning (DDL), and the Cayley transform to obtain input-adaptive, unconditionally orthogonal residual connections. Unlike DDL, whose Householder operator is ortho…

• Physics-Informed Reduced-Order Operator Learning for Hyperelasticity in Continuum Micromechanics

  Hamidreza Eivazi, Henning Wessels · 11 mai 2026

  Physics-informed operator learning is an attractive candidate for surrogate modeling of microstructures, especially in multiscale finite-element simulations. Its practical use, however, is often limited by the high cost of loss evaluation. We address this bottleneck by combining the Equilibrium Neur…