Model Reduction and Neural Networks

Latest papers

• Self-explainable Operator Learning for Discovering Spatial Patterns in Functional Data

  Mojgan Alishiri, Amirhossein Arzani · 3 July 2026

  Operator learning has emerged as a powerful tool for modeling complex physical systems in functional spaces. However, their neural network-based architectures make them opaque models, obscuring the reasoning behind their predictions. In this work, we introduce a self-explainable operator learning fr…

• Conditional Co-Ablation: Recovering Self-Repair Backups in Transformer Circuits

  Zhiren Gong, Zihao Zeng, Chau Yuen, Wei Yang Bryan Lim · 3 July 2026

  Mechanistic interpretability often relies on component-level interventions to discover how a model produces a behavior. This guides attribution, capability knockout, and model pruning downstream to operate by scoring each unit by the effect of ablation in isolation. Such first-order scoring is natur…

• Fourier Neural Operators for Rayleigh-B\'enard Convection

  Chelsea Maria John, Thibaut Lunet, Sebastian G\"otschel, Andreas Herten, Stefan Kesselheim, Daniel Ruprecht · 3 July 2026

  We propose an improved Fourier Neural Operator (FNO) for modeling two-dimensional Rayleigh-B\'enard convection by predicting time increments instead of full solutions, achieving higher accuracy than a standard FNO baseline. The resulting model is compact (314k parameters, 1.26 MB) and fast (7 ms inf…

• An Optimisation Framework for the Well-Conditioned Training of Physics-Informed Neural Networks

  Joseph Webb, Sadok Jerad, Coralia Cartis · 3 July 2026

  Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers. The obstacle is increasingly understood to be one of optimisation, owing to the severely ill-conditioned loss lands…

• ART for Diffusion Sampling: Continuous-Time Control and Actor-Critic Learning

  Yilie Huang, Wenpin Tang, Xun Yu Zhou · 3 July 2026

  We study timestep allocation for score-based diffusion sampling, where a learned reverse-time dynamics is discretized on a finite grid. Uniform and hand-crafted schedules are standard choices, but they rely on fixed prescriptions and can therefore be suboptimal. To address this limitation, we propos…

• Koopman operator theory: fundamentals, control, and applications

  Igor Mezi\'c, Jorge Cort\'es, Karl Worthmann, Mircea Lazar, Armin Lederer · 3 July 2026

  The Koopman operator has gained considerable attention due to its ability to provide a global linear representation of highly complex dynamical systems. The operator describes nonlinear dynamics in a linear way through the lens of real- or complex-valued observable functions. Recently proposed data-…

• McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation

  Jiwei Jia, Xinliang Liu, Juntao Wang, Jinchao Xu · 3 July 2026

  Solving heterogeneous Helmholtz equations at high wavenumbers remains challenging because the discretized operator is indefinite, pollution degrades phase accuracy, and scalar coarse-grid correction can discard the local phase and propagation-direction information carried by oscillatory errors. We p…

• Geometry-Aware R-Structured Kolmogorov-Arnold Networks

  Sergei Kucherenko, Nilay Shah · 3 July 2026

  We propose a novel hybrid neural architecture, the Geometry-aware R-Structured Kolmogorov-Arnold Network (GRS-KAN), which integrates V.L.Rvachev's R-functions into the Kolmogorov-Arnold Network (KAN) framework. The proposed approach combines two complementary modeling mechanisms: smooth nonlinear st…

• TRIE: An Evaluation Framework for Stochastic PDE Surrogates

  Bharat Srikishan, Javier E. Santos, Nikhil Muralidhar, Charles D. Young · 2 July 2026

  Many scientific systems exhibit uncertainty from stochastic forcing, unresolved degrees of freedom, or imperfect observations, making reliable surrogate forecasting fundamentally distributional rather than pointwise. For such systems, deterministic neural surrogates fail to capture statistical measu…

• GPU-Parallel Linearization Error Bounds for Real-Time Robust Optimal Control of Nonlinear and Neural Network Dynamics

  Jeffrey Fang, Keyi Shen, Anutam Srinivasan, Glen Chou · 2 July 2026

  This paper studies real-time robust optimal control for uncertain nonlinear systems, where linear time-varying (LTV) approximations make planning tractable but require sound linearization error bounds (LEBs) to guarantee robust constraint satisfaction. We develop tight, differentiable, GPU-parallel …

• From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators

  Nisha Chandramoorthy, Daniel Sanz-Alonso, Nathan Waniorek · 2 July 2026

  We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-$T$ solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stabl…

• Learning dynamical systems from noisy data with Weak-form Kernel Ridge Regression

  Max Kreider, John Harlim, Daning Huang · 2 July 2026

  Accurate prediction of complex dynamical systems from noisy measurements remains a significant challenge in scientific computing. Kernel ridge regression learning strategies are often effective when applied to clean data, but have limited success with noisy data. Recent work has observed that a weak…

• Goal-oriented learning of stochastic differential equations using error bounds on path-space observables

  Joanna Zou, Han Cheng Lie, Youssef Marzouk · 2 July 2026

  Stochastic differential equations (SDEs), which serve as the governing equations for dynamical systems in a broad range of applications, can become cost-prohibitive for numerical simulation at scales necessary for quantifying key properties. Surrogate models of the drift function of an SDE, learned …

• GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems

  Meenakshi Krishnan, Pranav Pulijala, Ke Chen, Haizhao Yang, Ramani Duraiswami · 2 July 2026

  Operator learning for partial differential equations (PDEs) on arbitrary geometries builds fast neural surrogates for large-scale simulation. Although recent geometry-adaptive neural operators have made substantial progress, they are mainly designed for forward problems in which inputs and outputs s…

• Learning Cardiac Motion Priors for Implicit Neural Representations

  Andrew Bell, George Webber, Andrew P King, Steffen E Petersen, Muhummad Sohaib Nazir, Alistair Young · 2 July 2026

  Implicit neural representations (INRs) are well suited to cardiac motion estimation, providing continuous, compact representations of motion fields. However, fitting an INR to each image sequence is time-consuming and sensitive to the optimisation trajectory. Learned priors can help guide optimisati…

• Generative Model Proposal based Particle Filtering for Data Assimilation

  Chandni Nagda, Mayank Shrivastavam Gudrun Thorkelsdottir, Gan Zhang, Morteza Mardani, Arindam Banerjee · 2 July 2026

  Data assimilation models state dynamics conditioned on sequential observations, and has wide-ranging scientific applications. In the filtering setting, the goal is to model the posterior over the current state given all observations so far. Classical solutions typically make simplifying distribution…

• A Multi-Resolution Finite-Volume Inspired Deep Learning Framework for Spatiotemporal Dynamics Prediction

  Xin-Yang Liu, Xiantao Fan, Jian-Xun Wang · 2 July 2026

  Predicting complex spatiotemporal dynamics in physical processes often demands computationally expensive numerical methods or data-driven neural networks that suffer from high training costs, error accumulation, and limited generalizability to unseen parameters. An effective approach to address thes…

• Real vs. Complex Spectral Bases for Neural Operators: The Role of Green's Function Alignment

  Jason Sulskis, Sathya Ravi · 1 July 2026

  Fourier Neural Operators (FNO) learn solution operators of partial differential equations by parameterizing global convolutions in the complex Fourier domain. For real-valued PDE solutions, the complex FFT carries representational redundancy through conjugate symmetry. We introduce the Hartley Neura…

• Mind the Residual Gap: Probabilistic Downscaling under Real-World Bias

  Yujin Kim, Nidhi Soma, Sarah Dean · 1 July 2026

  Probabilistic downscaling is the task of modeling the conditional distribution of high-resolution fields given coarse inputs, and is a central challenge to atmospheric science, climate modeling, and other multiscale physical systems. A widely used paradigm decomposes the problem into a deterministic…

• Domain-Decomposed Randomized Neural Networks for Partial Differential Equations in Unbounded Domains

  Haixin Wang, Haoning Dang, Fei Wang, Shimin Guo · 1 July 2026

  Partial differential equations on unbounded domains are challenging because the exterior region must be represented without excessive truncation error. Truncation-based methods often require problem-dependent artificial boundary conditions, while global spectral bases may be inefficient for localize…

• Offline Reinforcement Learning for Fluid Controls: Data-based Multi-observational Policy Extraction

  Deepak Akhare, Luning Sun, Xin-Yang Liu, Xiantao Fan, Timo Bremer, Ben Zhu, Jian-Xun Wang · 1 July 2026

  Active flow control is a fundamental application in engineering. Recent advances in deep reinforcement learning have made progress in this field. However, the classical online RL approaches require extensive real-time interactions with the high fidelity environment, while each sensor configuration c…

• Patch-PODiff-ViT: Structured Latent Diffusion with Patchwise POD for Super-Resolution and Uncertainty Quantification

  Onkar Jadhav, Tim French, Matthew Rayson, Nicole L. Jones · 1 July 2026

  Diffusion models enable probabilistic super-resolution and conditional generation, but pixel-space methods are computationally expensive and learned latent spaces often lack interpretable uncertainty quantification. We introduce Patch-PODiff-ViT, a structured latent diffusion framework in which the …

• Joint discovery of governing partial differential equations from multi-source datasets by competitive optimization

  Hao Xu, Siyu Lou, Yuntian Chen, Dongxiao Zhang · 1 July 2026

  Discovering governing equations directly from observational data is a key step towards interpretable scientific machine learning. Current data-driven approaches typically operate on a single dataset, inherently limiting their performance when faced with restricted observations. In practice, multiple…

• A Bayesian Filtering Approach for Learning Lagrangian Dynamics from Noisy Measurements

  Kundan Kumar, Shreya Das, Simo S\"arkk\"a · 1 July 2026

  This paper proposes a Bayesian filtering-based approach for learning the dynamics of a physical system from partial, noisy measurements. We model the system dynamics using a Lagrangian mechanics formulation. As in Lagrangian neural networks (LNNs), we parameterize the kinetic and potential energies …

• The HydroGym Reinforcement Learning Platform for Fluid Dynamics

  Christian Lagemann, Sajeda Mokbel, Miro Gondrum, Mario R\"uttgers, Yuning Wang, Pol Su\'arez, Ludger Paehler, Deniz A. Bezgin, Aaron B. Buhendwa, Jared L. Callaham, Samuel Ahnert, Nicholas Zolman, Xiao Shao, Jean-Christophe Loiseau, Nikolaus Adams, Matthias Meinke, Wolfgang Schr\"oder, Kai Lagemann, Esther Lagemann, Ricardo Vinuesa, Steven L. Brunton · 1 July 2026

  Modeling and controlling fluids is critical across science and engineering. Effective flow control can increase lift, reduce drag, enhance mixing, and attenuate noise, potentially unlocking new technologies. Yet controlling fluids is hard: the dynamics are high-dimensional, nonlinear, and multiscale…