Model Reduction and Neural Networks

Latest papers

• Sobolev Approximation by Fixed-Size Neural Networks with Arbitrary Accuracy

  Baicheng Li, Haizhao Yang, Shijun Zhang · 16 June 2026

  In this work, we investigate new activation functions for achieving arbitrary-accuracy Sobolev approximation by fixed-size neural networks. We first show that any function in $W^{2,\infty}((a,b)^d)$ can be approximated with arbitrary accuracy, measured in the $W^{1,\infty}$-norm, by a fixed-size neu…

• Graphical conditional generative modeling for digital twin modeling

  Zongren Zou, Th\'eo Bourdais, Ricardo Baptista, Houman Owhadi · 16 June 2026

  Digital twin modeling, including control and data assimilation under model uncertainty, often faces an open-ended fidelity problem: adding variables, data streams, and time scales can indefinitely increase model complexity, ultimately producing systems that are difficult to maintain, validate, inter…

• ANCHOR: Error-Controlled Adaptive Numerical Correction for Neural Operator Time Marching

  Rajyasri Roy, Dibyajyoti Nayak, Somdatta Goswami · 16 June 2026

  Numerical simulation of time-dependent partial differential equations (PDEs) is central to scientific and engineering applications, but high-fidelity solvers are often prohibitively expensive for long-horizon or time-critical settings. Neural operator (NO) surrogates offer fast inference across para…

• Multi-Grade Deep Learning for Partial Differential Equations with Applications to the Burgers Equation

  Yuesheng Xu, Taishan Zeng · 16 June 2026

  Deep neural networks (DNNs) show great promise for solving partial differential equations (PDEs), but their deep architectures introduce complex, large-scale, non-convex optimization challenges. Nonlinear PDEs, like the viscous Burgers' equation, compound these difficulties due to steep gradients an…

• Physics-conforming Latent Twins

  Matthias Chung, Yutong Bu, Deepanshu Verma · 16 June 2026

  Surrogate models are central to scientific machine learning, where they enable fast prediction, simulation, inference, and control for complex physical systems. For time-dependent problems, however, accurate interpolation of training trajectories is not sufficient: reliable surrogates should also re…

• The Algebra of Units: From Buckingham's Pi-grec Theorem to Latent-Variable Learning

  Mauro Valorani · 16 June 2026

  Engineers often measure many quantities-speed, pressure, temperature, length-expressed in different physical units. The Buckingham Pi-grec theorem states that these variables can always be combined into a smaller set of dimensionless numbers whose values fully determine the system's behaviour. Ide…

• Factorized Neural Operators Decompose Dynamic and Persistent Responses

  Hao Tang, Yuechen Duan, Jiongyu Zhu, Zimeng Feng, Hao Li, Chao Li · 16 June 2026

  Physical systems often exhibit heterogeneous mechanisms, where rapidly evolving dynamics coexist with persistent structures. Capturing such multiscale physical behavior remains challenging for existing neural operators, which typically rely on single dominant inductive bias and therefore couple dist…

• Transformers Learn the Mestre-Nagao Heuristic

  Pranav Venkata Konda · 16 June 2026

  We train a two-layer transformer encoder to classify rational elliptic curves $E/\mathbb{Q}$ of conductor $\leq 10000$ as either rank 0 or rank 1 from the first 128 normalized Frobenius traces. We achieve >99% accuracy on both classes, and accuracy is essentially unchanged on test curves with no iso…

• Separable Neural Architectures as Physical World Models: from Mathematical Theory to Applications

  Reza T Batley, Andrew Kichline, Sourav Saha · 16 June 2026

  This work introduces the Separable Neural Architecture (SNA), a function representational class combining neural approximation with tensor decomposition. The SNA decouples localized coordinate functions (atoms) from global interactions governed by a sparse, low-rank interaction object. This architec…

• MR-GVNO: A Geometry-Aware Variational Physics-Informed Neural Operator for Mindlin-Reissner Plates on Irregular Domains

  Siqi Wang, Daobo Sun, Yizheng Wang, Yilong Zhang, Yabin Jin, Xiaoying Zhuang, Timon Rabczuk · 16 June 2026

  Plate and shell structures are widely used in engineering, making rapid response prediction under varying geometries, materials, and loads highly desirable. However, conventional finite element methods require repeated modeling and solution, resulting in high computational costs. This study proposes…

• Multi-Fidelity SINDy: Sparse Discovery of Nonlinear Dynamical Systems with Fidelity-Weighted Measurements

  Filippo Zacchei, Ana Larra\~naga, Attilio Frangi, Andrea Manzoni, Steven L. Brunton · 16 June 2026

  Data from simulations and experiments are rarely noise-free and often exhibit heterogeneous levels of fidelity. Measurement uncertainty may vary across repeated observations, sensing devices, or even within a single experiment. This work addresses the problem of discovering nonlinear dynamical syste…

• RepNet: Tackling spectral bias in deep neural networks via parameter reparameterization

  Yong Wang, Tao Zhou, Xuhui Meng · 16 June 2026

  Deep neural networks (DNNs) have achieved remarkable success in scientific computing, yet they often suffer from spectral bias in capturing oscillatory and multiscale behaviors. In this study, we investigate this limitation by examining the failure of shallow ReLU neural networks in fitting high-fre…

• Dual-Network PINNs for Optimal Control: A Reproducible Benchmark on the Mass-Spring-Damper System

  Abdeladhim Tahimi, Rinaldo Vieira da Silva Junior · 16 June 2026

  This work presents a transparent and reproducible benchmark study of a direct dual-network Physics-Informed Neural Network (PINN) formulation for the optimal control of a mass-spring-damper system. The classical linear-quadratic optimal control problem is solved by two independent classical methods …

• Recursively Trained Diffusion Models: Limiting Collapse Distribution and Spectral Characterization

  Na\"il B. Khelifa, Richard E. Turner, Ramji Venkataramanan · 15 June 2026

  Recursive training of generative models on their own outputs can lead to model collapse, a compounding drift away from the true data distribution. Existing theoretical works bound finite-round error accumulation in the context of diffusion models, but two questions remain open:~what distribution doe…

• Robin-Neumann Coupling of PINN and FEM Solvers: A Steklov-Poincar\'e View, with Application to Fluid-Structure Interaction with Contact

  Mikel Landajuela · 15 June 2026

  Physics-informed neural networks (PINNs) are meshless and carry moving geometry and topology change through resampling of collocation points; the finite-element method (FEM) is the workhorse for boundary-fitted discretisations. Coupling the two across a shared interface promises the best of both, ye…

• EqCollide: Equivariant and Collision-Aware Deformable Objects Neural Simulator

  Qianyi Chen, Tianrun Gao, Chenbo Jiang, Tailin Wu · 15 June 2026

  Simulating collisions of deformable objects is a fundamental yet challenging task due to the complexity of modeling solid mechanics and multi-body interactions. Existing data-driven methods often suffer from lack of equivariance to physical symmetries, inadequate handling of collisions, and limited …

• A Unified Framework for Structured Flow Modeling: From Representation to Verification and Model Discovery

  Diego Casadei · 15 June 2026

  Many dynamical systems can be described in terms of structured flows combining source/sink behavior, cyclic dynamics, and topology-constrained transport. These features arise across a wide range of physical, engineered, and data-driven systems. The objective of this work is to establish a unified pe…

• A fully GPU-based workflow for building physics emulators of hypersonic flows

  Fabian Paischer, Dylan Rubini, Deniz A. Bezgin, Aaron B. Buhendwa, David Hauser, Florian Sestak, Johannes Brandstetter, Sebastian Kaltenbach, Nikolaus A. Adams · 15 June 2026

  The ability to resolve complex physical phenomena with high fidelity and at low computational cost is central to addressing key challenges in modern engineering. A prime example lies in hypersonic flows, where the precise prediction of the full flowfield topology, in particular with respect to shock…

• CANN-EUCLID: unsupervised constitutive artificial neural network model discovery from full-field data

  Benjamin Alheit, Siddhant Kumar, Mathias Peirlinck · 15 June 2026

  Constitutive artificial neural networks (CANNs) provide interpretable material model discovery, but have so far been used in stress-supervised settings based on apparent stress-strain data from homogeneous tests. Because each test samples only a narrow loading path and provides homogenized rather th…

• Graph Diffusion Residuals for Control-Function Instrumental Variables

  Rui Wu, Zongyuan Chen, Hong Xie, Defu Lian, Enhong Chen · 15 June 2026

  Control-function instrumental variable estimators need a first-stage residual, not merely a first-stage prediction. High-capacity first stages can interpolate treatment and leave too little residual information for the outcome equation. We study Adaptive Anisotropic Instrumental Heat Flow (A-IHF), a…

• Zero-shot generalization of transformer neural operators to larger domains

  Armand de Villeroch\'e, Sibo Cheng, Vincent Le Guen, Marc Bocquet, Rem-Sophia Mouradi, Patrick Armand, Alban Farchi, Patrick Massin · 15 June 2026

  Transformer-based neural operators have shown remarkable performance for approximating solution operators of partial differential equations on complex geometries. However, existing approaches implicitly assume a fixed domain size, which limits their ability to generalize at inference. In this work, …

• On Approximating the Dynamic Response of Synchronous Generators via Operator Learning: A Step Towards Building Deep Operator-based Power Grid Simulators

  Christian Moya, Amirhossein Mollaali, Guang Lin, Meng Yue · 12 June 2026

  This paper develops an Operator Learning framework for approximating the dynamic response of synchronous generators. The framework can be used to (i) build a neural network-based generator model that interacts with a power grid simulator or (ii) shadow the true generator's transient response. First,…

• Structure-Preserving Neural Surrogates with Tractable Uncertainty Quantification

  Handi Zhang, Adrienne M. Propp, Brooks Kinch, Houman Owhadi, Nathaniel Trask · 11 June 2026

  Recent advances in scientific machine learning provide a means of near-real-time solution to partial differential equations (PDEs), but lack the theoretical underpinnings of conventional simulators that support contemporary verification and validation. In this work, we construct data-driven reduced-…

• How Low Can You Go? Active Learning for Sparse Model Discovery in the Ultra-Low-Data Limit

  Ana Larra\~naga, Urban Fasel, Steven L. Brunton · 11 June 2026

  Identifying the governing equations of complex dynamical systems remains a fundamental challenge across science and engineering. While early approaches relied on empirical data and heuristics, modern data-driven methods offer greater flexibility and fewer assumptions. However, data acquisition in re…

• Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems

  Zhen Zhang, Alessandro Alla, George Em Karniadakis · 11 June 2026

  Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative. Their relative performance remains difficult to asse…