Model Reduction and Neural Networks

Neueste Paper

• RINS: Residual-Image Neural Subspace Solvers for Large Sparse Linear Systems

  Zhongyan Ouyang, Weixin Liao, Mingquan Feng, Yehui Tang, Junchi Yan · 5. Oktober 2026

  Large sparse linear systems from PDE discretizations require correction subspaces whose operator images explain the current residual. We study this residual-image viewpoint and propose Gate-RINS, a neural subspace solver that generates polynomial correction bases from cached residual probes and modu…

• Simulation-Free Learning of Population Dynamics with Wasserstein Lagrangian Residuals

  Fedor Sergeev, Markus Heinonen, Daniel Waxman, Tim Cooijmans, Ricardo Baptista, Dmitry Batenkov, Eli Bingham · 5. Oktober 2026

  The dynamics of cells, organisms, and fluids are often modeled as probability distributions evolving over time. Reconstructing and extrapolating this evolution from unpaired snapshots requires assumptions about the underlying process. Wasserstein gradient flows are a common choice, but they cannot d…

• Operator-informed initialization for Fourier features physics-informed neural networks

  Juan Molina, Paris Perdikaris, Mircea Petrache, Mat\'ias Courdurier, Francisco Sahli Costabal · 5. Oktober 2026

  Physics-Informed Neural Networks (PINNs) typically exhibit spectral bias, where some frequencies of the target function converge more slowly than others. In this work, we analyze the training dynamics of Fourier Feature PINNs in the Neural Tangent Kernel regime to address this limitation. We derive …

• Dirac-Interconnected Neural Elements: Discovering Modularity in Physical Systems Without Reduction

  Reiho Li, Razmik Arman Khosrovian, Takaharu Yaguchi, Hiroaki Yoshimura, Takashi Matsubara · 5. Oktober 2026

  Deep learning has shown remarkable success in the data-driven modeling of dynamical systems. Much of its success is attributed not to the flexibility of neural networks but to inductive biases based on physical prior knowledge, such as energy conservation and symplecticity. However, existing methods…

• Hyperparameter selection for equation learning with biologically-informed neural networks

  William Lavery, Jodie A. Cochrane, John T. Nardini, Sara Hamis · 5. Oktober 2026

  Biologically-informed neural networks (BINNs) have emerged as a flexible subclass of physics-informed neural networks (PINNs) for learning terms in partial differential equations from data. BINNs are particularly suited for biological systems, where the governing equations are highly nonlinear and o…

• SPEAR: A Spectral-Disentangled MoE Neural Operator with Knowledge-Guided Expert Aggregation for Large-Scale PDE Pretraining

  Dengdi Sun, Xiaoya Zhou, Xiao Wang, Wanli Lyu, Jin Tang, Bin Luo · 5. Oktober 2026

  Large-scale pre-training has improved the generalization of neural operators across diverse PDEs. However, existing PDE foundation models still struggle with heterogeneous dynamics, where shared representations may cause knowledge interference, while mixture-of-experts (MoE) architectures suffer fro…

• Geometry Meets Physics: Data-Efficient Pre-Training for Unstructured Neural PDE Solvers

  Luis Medrano-Navarro, Giacomo Baldan, Qiang Liu, Benjamin Holzschuh, Jan Hagnberger, Mathias Niepert, Nils Thuerey · 5. Oktober 2026

  Neural surrogate models for Partial Differential Equations (PDEs) on unstructured 3D geometries are often limited by poor generalization and the high cost of generating large-scale training datasets. Consequently, pre-training on massive datasets of related PDE dynamics has emerged as a critical alt…

• Cluster Attention Neural Operators for Solving Parametric Partial Differential Equations

  Ming Zhong, Antonio Colanera, Gianluigi Rozza, Zhenya Yan · 1. Oktober 2026

  Traditional simulations of parametric partial differential equations (PDEs) rely on repetitive computations for each parameter, which makes high-fidelity design impractical. Neural operators address this issue by learning solution operators, accelerating parameter-space mapping by orders of magnitud…

• PTNO: Training Neural Operators with Noisy Monte Carlo Estimates for Particle Transport Problems

  Yubo Cao, Xi Deng, Mengqi Xia, Vignesh Gopakumar, Ander Gray, Anima Anandkumar · 1. Oktober 2026

  Particle transport under multiple scattering is central to radiative transfer and plasma physics, yet high-fidelity Monte Carlo (MC) simulations must trace prohibitively many particles. Learning-based surrogates can amortize this cost, but typically train on expensive, well-converged MC solutions. W…

• Identifying ODEs from Unstructured Data with Causal Representation Learning

  Alessandro Trenta, Riccardo Massidda, Davide Bacciu, Sara Magliacane · 30. September 2026

  We study the problem of recovering the governing ODE of a dynamical system from unstructured, high-dimensional observations such as images. Existing methods for ODE discovery typically assume direct measurements of the variables, or do not provide theoretical guarantees on the learned variables and …

• Neural networks for spectral optimization

  Alexis de Villeroch\'e, Beniamin Bogosel, St\'ephane Breuils, Dorin Bucur, Jacques-Olivier Lachaud · 30. September 2026

  Given a functional dependent on the spectrum of a differential operator, we address the problem of finding a domain which optimizes this functional. PDE solvers might be used to tackle this optimization. It is however computationally expensive. We propose two neural network models which learn the sp…

• PE-EK-PINN: Physics Embedding with Evolving Kernel for Scalable Physics-Informed Neural Networks

  Huiwen Zhang, Feng Ye, Chu Ma · 30. September 2026

  Physics-Informed Neural Networks (PINNs) embed governing equations into deep learning, but enforce them only through loss residuals, leaving highly oscillatory wave behavior to be discovered by optimization. As a result, methods that achieve relative $L_2$ errors below $10^{-3}$ on standard manufact…

• Transolver-$\sigma$: Joint Spectral-Physical Subspace Modeling for Neural PDE Solving

  Haonan Shangguan, Hang Zhou, Haixu Wu, Yuezhou Ma, Jianmin Wang, Mingsheng Long · 30. September 2026

  Neural solvers offer efficient surrogates for numerical simulation of partial differential equations (PDEs). For time-dependent problems, strong one-step accuracy does not necessarily translate into reliable autoregressive rollout. We observe that a solver based only on physical-state modeling can a…

• When Known Physics Helps Neural PDE Models: Residual Constraints Out-Regularize Generic Priors for Nonlinear Dynamics

  Zahra Farazpay, Aniruddha Bora · 29. September 2026

  Neural PDE surrogates increasingly incorporate structural priors, yet it is often unclear whether their gains arise from physics-specific information or simply from regularization and training choices. We evaluate several such priors under a common protocol against a matched from-scratch neural oper…

• PI-NOMT: Physics-Informed Neural Optimal Mass Transport for Brain Fluid Dynamics

  Mehmet Emin Acar, Vahit Bugra Yesilkaynak, Helene Benveniste, Gozde Unal · 29. September 2026

  Recovering hidden transport mechanisms from sparse spatiotemporal observations is a fundamental inverse problem in scientific machine learning. In brain tracer imaging, dynamic contrast-enhanced MRI (DCE-MRI) provides time-resolved measurements of tracer concentration, while the underlying velocity …

• Dynamic Kuramoto-Hodge Operators for PDEs on Complex Geometries and Topologies

  Xiang Li, Yue Song · 29. September 2026

  Learning PDE operators on complex domains requires capturing interactions among fields on vertices, edges, and faces, alongside global responses shaped by topology. Existing neural operators accommodate irregular geometries but often overlook these distinct field supports or their condition-dependen…

• Source Anchoring for Physical Consistency in Flow Matching Models

  Giulia Romoli, Filippo Ruffini, Paolo Soda · 29. September 2026

  Deep generative models are used to solve partial differential equations and model distributions of physical system states, but ensuring that the generated samples satisfy the governing laws remains challenging. Projection-based flow-matching methods enforce physics by correcting the flow from an unc…

• SMORE: Stability-Promoting Mesh-Agnostic Model Reduction for Time-Dependent PDEs

  Yangyuan Li, Weichao Li, Shaowu Pan · 29. September 2026

  High-fidelity simulations of time-dependent partial differential equations (PDEs) are computationally expensive, motivating data-driven reduced-order surrogates for many-query tasks such as uncertainty quantification, design optimization, data assimilation, and optimal control. However, existing sur…

• Efficient Message Passing for Partial Differential Equation Priors

  Anna Kazachkova, Leonhard Hennicke, Rainer Schlosser, Ralf Herbrich · 29. September 2026

  Prior information for real-world physical quantities is most elegantly expressed via partial differential equations (PDEs). In this paper, we propose a novel way to solve PDEs using probabilistic inference on a factor graph. In general, factor graphs provide a natural way to encode prior knowledge i…

• AutoPDEBench: Benchmarking LLM Auto-Research for Neural PDE Solver Design

  Ruoyan Li, Wei Wang, Yizhou Sun · 29. September 2026

  Partial differential equations (PDEs) are essential for modeling complex physical systems, and neural solvers have recently emerged as powerful data-driven tools for numerically solving them. However, existing neural solvers struggle with domain-specific challenges, such as varying parameters and hi…

• PDE-JEPA: Predictive Representation Learning of Latent Dynamics Modeling for Parametric PDEs

  Zhentao Tan, Jianrong Zhang, Ruijie Quan, Yi Yang · 29. September 2026

  Physical trajectories contain more than snapshots of a system: they also reveal how its states evolve under governing conditions. However, representation learning for parametric partial differential equations (PDEs) has largely relied on reconstruction-based objectives that emphasize recovering obse…

• PINNMorph: Evolving Online Adaptation Policies for Physics-Informed Neural Networks

  Xu Yang, Mingyang Yu, Jun Zhang, Keqian Li, Jing Xu · 29. September 2026

  Physics-informed neural networks (PINNs) provide a learning-based framework for solving partial differential equations (PDEs), yet their training behavior can change substantially throughout optimization. Residual distributions, gradient interactions, regional learning difficulty, and model-capacity…

• Why Directly Learning Periodic Trajectories Can Fail

  Kaixin Zheng, Anita Layton · 29. September 2026

  Operator learning of periodic solutions requires deciding how simulation data should be recorded and represented. A natural choice is to integrate long enough for transients to decay and record a window wide enough to contain at least one full period of all trajectories. We find that these conservat…

• Deep-Learning Solvers and Surrogates for Infinity and p-Laplace Problems

  Tak Shing Au Yeung, Ka Chun Cheung, Hannah Potgieter, Steven J. Ruuth, Simon See · 28. September 2026

  We investigate the use of neural network solvers for infinity and $p$-Laplace problems, which are fundamental in nonlinear analysis and have practical applications. Our approach employs Physics-Informed Neural Networks (PINNs) and Deep Operator Networks (DeepONets) to address computational challenge…

• NEXT: Physics-Informed Neuro-Spectral Exponential Time Differencing Architectures

  M\'arcio Marques, Leonardo Mendon\c{c}a, Leonardo M. Moreira, Christian J\'unior de Oliveira, Vitor Balestro, Tiago Novello, Daniel Yukimura, Pavel Petrov, Lucas Nissenbaum · 28. September 2026

  Physics-Informed Neural Networks (PINNs) build neural representations of time-dependent PDE solutions, naturally incorporating physics knowledge and observational data, which makes them well suited to both forward and inverse PDE problems. PINNs, however, are known to suffer from spectral bias and l…