Model Reduction and Neural Networks

Latest papers

• Cluster Attention Neural Operators for Solving Parametric Partial Differential Equations

  Ming Zhong, Antonio Colanera, Gianluigi Rozza, Zhenya Yan · 1 October 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 October 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…

• Gradient Surgery for Physics-Informed Neural Networks

  Thomas Borsani, Giuseppe Di Fatta · 28 September 2026

  Physics-Informed Neural Networks (PINNs) are trained by optimising a composite objective that combines data fitting with physics-based constraints, typically resulting in a highly imbalanced multi-task optimisation problem. Under these conditions, existing optimisation strategies are affected by con…

• Learning coarse-step dynamics and internal mechanical response with graph networks

  Vinay Sharma, Olga Fink · 28 September 2026

  Modern sensing records the motion of physical systems, but often leaves the forces and mechanical response governing that motion unobserved. Inferring these quantities from discretely sampled trajectories is especially difficult at coarse time scales, when mechanical response evolves between observa…

• Elucidating the Conformal Structure of the Brinkman Penalisation Method for Geometry-Adapted, Structure-Preserving Operator Learning of Hamiltonian PDEs

  Teo Deveney, Baige Xu, Takaharu Yaguchi · 25 September 2026

  The Brinkman penalisation method embeds boundary-value problems on complex domains into a simple computational box by modeling the solid region as a strongly dissipative medium, avoiding body-fitted mesh generation. We show that multi-symplectic Hamiltonian PDEs regularised by Brinkman-type penalisa…

• AFT Neural Function Approximators for 1D Nonlinear Force Laws

  Miriam Goldack, Johann Gro{\ss}, Malte Krack, Merten Stender · 25 September 2026

  Nonlinear contacts and friction strongly influence the vibration response of assembled structures, but their accurate numerical treatment is computationally demanding. The harmonic balance method is widely used to compute periodic steady-state responses, yet the required alternating frequency-time s…

• Functional dynamic mode decomposition: Learning infinite-dimensional systems from data

  Stefan Klus, Eirini Ioannou · 25 September 2026

  Dynamic mode decomposition (DMD) is a data-driven method that computes the best linear approximation of the underlying dynamical system and decomposes the dynamics into a superposition of characteristic spatiotemporal patterns. Originally introduced by the fluid dynamics community, DMD and its exten…

• Beyond Compression: Training Latent Representations for Stable Long-Horizon Rollout in Neural Surrogate Solvers

  Andreas E. Robertson, Ashley T. Lenau, John D. Shimanek, Benjamin A. Jasperson, Vivek Oommen, David L. Damm, Krishna Garikipati, Remi Dingreville · 25 September 2026

  Latent neural surrogate solvers, or latent dynamics models, accelerate simulations of time-dependent physical systems by evolving a compressed latent space rather than resolving full-resolution fields directly. In principle this reduces computational cost and simplifies learning, but in practice err…

• Physics and Data Driven Transformer-Mamba Framework for Flow Field

  Zhuo Zhang, Shun Zou, Canqun Yang, Xi Yang · 25 September 2026

  While deep learning accelerates expensive partial differential equation solving in computational fluid dynamics (CFD), existing methods like PINNs and FNOs often struggle with generalization, noise robustness, and physical consistency. We introduce the Transformer-Mamba for Flow Field (TM4FF) framew…