Model Reduction and Neural Networks

Latest papers

• Learning Where to Simulate: Generative Active Sampling for Online PDE Surrogate Training

  Pierre Cesar (DATAMOVE), Sofya Dymchenko (DATAMOVE), Abhishek Purandare (DATAMOVE), Bruno Raffin (DATAMOVE) · 10 June 2026

  Data-driven PDE surrogates are trained with data produced by numerical PDE solvers. However, when the surrogate's goal is to generalize across a wide range of PDE configurations (e.g., initial conditions and physical coefficients), generating a representative training set is non-trivial. Uniform sam…

• PL-KKT-hPINN: Enforcing Nonlinear Equality Constraints on Neural Networks via Piecewise-Linear Projection

  Fateme Mohammad Mohammadi, Hector Budman, Joshua L. Pulsipher · 10 June 2026

  While physics-informed neural networks (PINNs) have shown strong potential for process modeling, physical equations are only enforced as soft constraints during training, and thus, they do not guarantee constraint satisfaction at inference. We propose a framework, called piecewise-linear Karush--Kuh…

• Non-linear mechanical field reconstruction coupling recurrent neural networks with physics-informed graph neural networks

  Manuel Ricardo Guevara Garban, Yves Chemisky, \'Etienne Pruli\`ere, Micha\"el Cl\'ement, Martin Abendroth, Bj\"orn Kiefer · 10 June 2026

  Reconstructing local stress fields in heterogeneous microstructures under non-linear, history-dependent loading remains a major computational bottleneck in multi-scale simulations. We propose a coupled LSTM-GNN framework that links the temporal and spatial aspects of local stress field reconstructio…

• A Constrained Natural-Language Interface for Variational Multi-Physics Finite Element Simulations in FEniCS

  Nilay Upadhyay, Wesley F. Reinhart · 10 June 2026

  Large language models can reduce the manual effort required to set up finite element simulations, but they introduce reliability risks when generated solver code lies on the critical path. We present a constrained natural-language interface for multi-physics finite element analysis in which the LLM …

• AutoPDE: Reliable Agentic PDE Solving via Explicitly Represented Solver Strategies

  Huanshuo Dong, Keyao Zhang, Hong Wang, Zhezheng Hao, Zhiwei Zhuang, Ziyan Liu, Jiacong Wang, Gengyuan Liu, Xin Jin · 10 June 2026

  Numerical solvers for partial differential equations (PDEs) are core computational tools in science and engineering. Building reliable PDE solvers requires not only executable code, but a numerical solver strategy, a set of decisions about discretization, stabilization, solver configuration, and res…

• Conformal Prediction for Neural Operators: Distribution-Free Uncertainty Quantification in Physics Simulation

  Michael Chin · 10 June 2026

  Neural operators such as the Fourier Neural Operator (FNO) have emerged as powerful surrogates for solving partial differential equations (PDEs), achieving speedups of several orders of magnitude over traditional numerical solvers. However, deploying these models in safety-critical engineering appli…

• Uncertainty-aware Multi-fidelity Closure via Conditional Normalizing Flows

  Jice Zeng, Shady E. Ahmed, David Barajas-Solano, Panos Stinis · 10 June 2026

  Reduced-order models (ROMs) provide an efficient surrogate for complex multiscale systems, but their predictive accuracy is often compromised by truncation errors and the inadequate representation of interactions between resolved and unresolved scales. The missing effect of truncated (unresolved) sc…

• Population-Aware Physics-Informed Neural Particle Flow for Bayesian Update

  Batu Candan, Simone Servadio · 10 June 2026

  Physics-informed neural particle flow (PINPF) learns a deterministic transport field that moves particles from a prior distribution toward a Bayesian posterior while enforcing the governing probability-evolution equation. However, the standard PINPF velocity model processes particles independently a…

• Structure from Reasoning, Numbers from Search: On-Premise Open LLMs as Structural Priors for Coupled MIMO Controller Tuning

  Jiaxuan Chen, Haonan Li, Yang Shu · 10 June 2026

  Tuning controllers for strongly coupled multi-input multi-output (MIMO) industrial processes is hard: decentralized classical auto-tuning ignores loop interaction, and local numerical optimization from natural initializations stalls in the resulting non-convex cost landscape. We ask whether on-premi…

• Divide-and-Conquer Modeling for the CTF-4-Science Lorenz Benchmark

  Shundong Li · 10 June 2026

  This work presents a divide-and-conquer modeling strategy for the CTF-4-Science Lorenz benchmark, which evaluates chaotic-system prediction across twelve hidden scores and five scenario families: clean forecasting, noisy reconstruction, noisy-input forecasting, few-shot learning, and parametric gene…

• Geometry-Aware Anisotropic Boundary Correction for Aerodynamic Simulation

  Xin Zhang, Yipeng Huang, Shu Jiang, Zhenzhong Wang, Min Jiang · 10 June 2026

  Aerodynamic simulation is a key component of engineering shape design, where core quantities such as the surface pressure coefficient strongly depend on flow dynamics near solid boundaries. Neural operators provide an efficient alternative to expensive Computational Fluid Dynamics (CFD) solvers. How…

• GENERIC-FNO: Embedding Energy Conservation and Entropy Production into Fourier Neural Operators

  Jason Sulskis, Sathya Ravi · 9 June 2026

  We introduce GENERIC-FNO, the first neural operator to embed the full GENERIC (metriplectic) structure of nonequilibrium thermodynamics -- reversible, energy-conserving dynamics and irreversible, entropy-producing dynamics coupled through the degeneracy conditions -- directly in function space. Exis…

• From inverse problems to neural operators: prediction, mechanism, and generalization of data-driven models

  Conor Rowan · 9 June 2026

  Scientists have historically relied on mathematical models based on differential equations to relate system inputs -- forces, fluxes, or heat sources -- to outputs, such as displacement, velocity, concentration, and temperature. These models rely on deep domain knowledge to determine the form of the…

• Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime

  Weinan Wang, Bowen Gang, Hao Deng · 9 June 2026

  In this paper, we propose a perturbation-based conformal prediction framework for uncertainty quantification in operator learning, with a focus on the 2D Navier--Stokes equations. While neural operators provide fast surrogates for expensive PDE solvers, they do not by themselves provide calibrated u…

• Mitigating the Contractivity Trap in Diffusion ODEs via Stein Stabilization

  Shigui Li, Delu Zeng · 9 June 2026

  A fundamental tension exists in the large-step inference of diffusion models via their deterministic probability flow ordinary differential equation (PF-ODE) trajectories, which we identify as the contractivity trap: efficient inference favors large step sizes, while aggressive steps and highly expr…

• A Geometry-Aware Triplane Field Network for Vehicle Aerodynamic Prediction

  Kangkang Qi, Huiyu Yang, Keqi Ding, Yunpeng Wang, Yuntian Chen, Yuanwei Bin, Rikui Zhang, Jianchun Wang · 9 June 2026

  High-fidelity computational fluid dynamics (CFD) is crucial to vehicle aerodynamic analysis, but its cost still constrains early-stage design exploration. Machine-learning-based surface-field prediction offers a faster alternative if the model can efficiently capture both global flow context and loc…

• Operator learning for solving Fokker-Planck equations with various initial conditions

  Li Zeng, Xiaoliang Wan, Yaobin Wang, Fabio Nobile, Tao Zhou · 9 June 2026

  The Fokker-Planck equation (FPE) plays a pivotal role in describing the time evolution of probability density functions (PDFs) for systems governed by stochastic dynamics. In this work, we propose a conditional normalizing flow-based physics-informed neural network (PINN) framework for efficiently a…

• Graph Mamba Operator: A Latent Simulator for Interacting Particle Systems

  Karn Tiwari, Niladri Dutta, N M Anoop Krishnan, Prathosh A P · 9 June 2026

  Modeling interacting dynamical systems requires capturing spatial interactions alongside long-range temporal dependencies. Graph neural networks (GNNs) provide a natural representation but typically rely on autoregressive rollouts and treat spatial and temporal dynamics separately, leading to error …

• In-Context Learning of Stochastic Differential Equations with Foundation Inference Models

  Patrick Seifner, Kostadin Cvejoski, David Berghaus, Cesar Ojeda, Ramses J. Sanchez · 9 June 2026

  Stochastic differential equations (SDEs) describe dynamical systems where deterministic flows, governed by a drift function, are superimposed with random fluctuations, dictated by a diffusion function. The accurate estimation (or discovery) of these functions from data is a central problem in machin…

• Stochastic Dimension Implicit Functional Projections for Global Integral Conservation in High-Dimensional PINNs

  Zhangyong Liang, Huanhuan Gao · 9 June 2026

  Enforcing prescribed global integral constraints in mesh-free neural PDE solvers is challenging in high-dimensional domains. Existing projection methods for spatial integrals are often tied to fixed grids or uniform quadrature, which can conflict with randomly sampled physics-informed neural network…

• Data-driven discovery of governing differential equations across physical systems

  Siyu Lou, Hao Xu, Wenguan Wang, Lu Lu, Hao Sun, Yang Liu, Linfeng Zhang, Dongxiao Zhang, Yuntian Chen · 9 June 2026

  Differential equations play a critical role in scientific discovery because they provide a mathematical framework to describe the behaviour of physical phenomena. As a promising alternative to traditional first principles, data-driven differential equation discovery has attracted increasing attentio…

• Fourier Neural Operators with rank-1 lattice points and hyperbolic cross

  Jakob Dilen, Alexander Keller, Frances Y. Kuo, Dirk Nuyens · 9 June 2026

  The \emph{Fourier neural operator} (FNO) is a neural network architecture that learns mappings between function spaces. Its efficient implementation is based on the multi-dimensional Fourier transform. By deriving general regularity bounds for the FNO with respect to both the spatial and parametric …

• LFNO: Bridging Laplace and Fourier via Transient-Steady Decomposition

  Jeongun Ha, Sanga Yoon, Donghun Lee · 9 June 2026

  We introduce the Laplace-Fourier Neural Operator (LFNO), a unified framework for modeling dynamical systems across transient and steady-state regimes by integrating the spectral advantages of Laplace and Fourier Neural Operators. LFNO employs a dual-branch architecture that explicitly decomposes sys…

• Mesh Graph Neural Network Framework for Accelerating Finite Element Simulation for Arbitrary Geometries

  Josiah D. Kunz, Kamal Choudhary · 9 June 2026

  Finite element analysis (FEA) is essential for structural design but remains computationally expensive, particularly when evaluating multiple design iterations or load scenarios. Machine learning surrogate models offer a promising alternative, yet most approaches struggle with a critical limitation:…

• Reachability and asymptotics of Gaussian Transformer dynamics

  Albert Alcalde, Zhengping Ji, Enrique Zuazua · 9 June 2026

  We formulate data propagation through the Transformer, the machine learning architecture powering large language models, as a nonlinear control system on the space of probability measures. For the mean-field Transformer model with self-attention and affine feed-forward layers, we prove that Gaussian…