Model Reduction and Neural Networks

Latest papers

• Data-Driven Energy-Based Learning via Gibbs Measures on Hierarchical Structures

  L. U. Abdullaev, F. Herrera, U. A. Rozikov, M. V. Velasco · 30 June 2026

  We introduce a data-driven probabilistic framework for learning systems based on Gibbs measures on hierarchical structures. Unlike standard empirical risk minimization, where a dataset is used to identify a single optimal parameter, our approach transforms the empirical loss function into an interac…

• Weak Dominant Balance for Robust Identification of Dynamically Consistent Fluid Flow Structure

  Samuel Ahnert, Esther Lagemann, H. Jane Bae, Kunihiko Taira, Ricardo Vinuesa, Christian Lagemann, Steven L. Brunton · 30 June 2026

  Extracting interpretable, localized physical mechanisms from complex spatiotemporal data is a foundational challenge across physics, biology, and engineering, but has remained out of reach on real measurements. The central obstacle is obtaining high-quality gradients of data via numerical differenti…

• A Bayesian latent Gaussian process framework for aerodynamic uncertainty quantification

  Geoffrey Davis, Ashwin Renganathan · 30 June 2026

  Predicting the aerodynamic performance (e.g. lift, drag, and moment coefficients) of an aircraft is challenging -- computational models are biased and direct simulations are prohibitive. A pragmatic way to overcome this limitation is by calibrating low-fidelity computational predictions with experim…

• Implementation of Hyperelastic Physics-Augmented Neural Networks in the Explicit Finite Element Codes Simcenter Radioss and OpenRadioss with Applications to Impact Events

  Lukas Maurer, Sascha Eisentr\"ager, Marian Bulla, Daniel Juhre · 30 June 2026

  Data-driven material modeling techniques have gained significant attention due to their ability to capture complex constitutive behaviors beyond the limitations of classical material models. Physics-augmented neural networks (PANNs), which embed physical constraints directly into their architecture,…

• Fast Equivariant Imaging: Accelerating Unsupervised Learning and Model Adaptation via Inexact Splitting

  Guixian Xu, Jinglai Li, Junqi Tang · 30 June 2026

  In this work, we propose Fast Equivariant Imaging (FEI), a novel unsupervised learning framework to rapidly and efficiently train deep imaging networks without ground-truth data. FEI reformulates the EI objective through an inexact variable-splitting scheme, decoupling network training from an auxil…

• Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Models

  Xinghao Dong, Huchen Yang, Jin-Long Wu · 30 June 2026

  We propose a latent score-based generative AI framework for learning stochastic, non-local closure models and constitutive laws in nonlinear dynamical systems of computational mechanics. This work addresses a key challenge of modeling complex multiscale dynamical systems without a clear scale separa…

• Gravitational Duals from Equations of State II: Large Hierarchies and False Vacua

  Raul Jimenez, David Mateos, Pavlos Protopapas, Pau Sol\'e-Vilar\'o, Pedro Taranc\'on-\'Alvarez, Pablo Tejerina-P\'erez · 30 June 2026

  We investigate the reconstruction of holographic duals for strongly coupled quantum field theories in regimes characterized by large hierarchies and the presence of false vacua. Within the gauge/gravity duality, these features translate into non-trivial thermodynamic behaviour and exotic renormaliza…

• Bidirectional Autoregressive Latent Diffusion for Forward and Inverse Magnetohydrodynamics

  Alexander Scheinker · 30 June 2026

  This work presents a new bidirectional autoregressive latent diffusion approach for predicting the evolution of multiple fields (mass density, pressure, velocity, and magnetic field components) for magnetohydrodynamics. We show that this bidirectional flow can be used as a self-supervised consistenc…

• PCGD: Physics-Guided Conditional Graph Diffusion for TCAD Device Simulation

  Yihan Zhang, Zhiteng Zhang, Kun Chen, Chen Wang · 30 June 2026

  Technology computer-aided design (TCAD) semiconductor device simulation is fundamentally constrained by the high computational cost of iteratively solving coupled drift-diffusion equations. Existing ML surrogates either reduce internal physics to macroscopic scalar regressions, or rely on single-ste…

• Kriging and neural network models for pressure losses across perforated plates

  Shuai Li · 30 June 2026

  In this paper, two novel data-driven models based on kriging and neural networks (NN) are proposed to predict pressure losses across perforated plates with circular perforations in turbulent flows. The models are developed using two sets of experimental data available in the literature. The predicti…

• Extrapolating from Regularised Solutions for Solving Ill-Conditioned Linear Systems in Machine Learning

  Disha Hegde, Jon Cockayne, Chris. J. Oates · 30 June 2026

  Rapid prototyping of algorithms is a critical step in modern machine learning. Most algorithms exploit linear algebra, creating a need for lightweight numerical routines which -- while potentially sub-optimal for the task at hand -- can be rapidly implemented. For the numerical solution of ill-condi…

• Entropy-Regularized Reinforcement Learning for Linear-Quadratic Stackelberg Differential Games in Regime-Switching Diffusion Models

  Congde Hu, Danping Li, Lin Xu, Wenying Xu · 30 June 2026

  Stackelberg differential games (SDGs) provide a powerful framework for hierarchical decision-making in stochastic and continuous-time environments, yet their solution remains computationally challenging due to the complexity of traditional dynamic programming and Hamilton-Jacobi-Bellman-Isaacs (HJBI…

• Physics-Informed Distillation of Diffusion Models for PDE-Constrained Generation

  Yi Zhang, Peng Wang, Difan Zou · 30 June 2026

  Modeling physical systems in a generative manner offers several advantages, including the ability to handle partial observations, generate diverse solutions, and address both forward and inverse problems. Recently, diffusion models have gained increasing attention in the modeling of physical systems…

• A Trainable-by-Parts Operator Learning Framework: Bridging DeepONet and Karhunen-Loeve Expansions for Large-Scale Applications

  Christian Munoz, Alexandre Tartakovsky · 30 June 2026

  Training operator-learning models for large-scale problems governed by partial differential equations (PDEs) is challenging due to the curse of dimensionality, memory constraints, and limited training data. These challenges arise in many scientific and engineering applications, including subsurface …

• Fourier Neural Operators with Least-Squares Readout Refit for Learning Random Obstacle-to-Solution Maps

  Chenhui Zhu, Fei Wang · 30 June 2026

  We study operator learning for random obstacle-to-solution maps arising from elliptic variational inequalities with finite-band self-affine random obstacle fields. Instead of introducing an explicit truncated stochastic parametrization of the random input, we learn the map directly from sampled obst…

• Residual-Guided Dictionary Learning for Spectrally Accurate Koopman Approximation

  George Coote, Matthew J. Colbrook · 30 June 2026

  Koopman theory promises linear structure in nonlinear dynamics, but numerical Koopman spectra are easy to compute and hard to trust. A finite EDMD matrix always has eigenvalues; the problem is that many of them may have nothing to do with the infinite-dimensional operator. In this paper we make spec…

• Operator Learning for Cubic Nonlinear Schr\"odinger Equation on Periodic Domains

  Emmanuel E. Oguadimma, Victory C. Obieke, Xueying Yu · 29 June 2026

  We consider the cubic nonlinear Schr\"odinger (NLS) equation on two-dimensional flat tori with varying aspect ratios. In this formulation, the choice of aspect ratio governs the Fourier resonance structure, so rational and irrational geometries can exhibit different high-frequency cascade behaviors.…

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

  Jason Sulskis, Sathya Ravi · 29 June 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…

• Disentangling Continuous-Time Latent Dynamics: Identifiability of Latent SDEs via Diffusion Shifts

  Yuanyuan Wang, Wenjie Wang, Haoxuan Li, Mingming Gong, Kun Zhang · 29 June 2026

  Causal representation learning for time series has developed strong identifiability results in discrete-time latent causal models, but identifiability in continuous-time latent stochastic differential equation (SDE) models remains largely open. We address this gap using environment-induced shifts in…

• PAC-Bayesian Certificates for Quadratic Closed-Loop Control

  Domagoj Herceg · 29 June 2026

  PAC-Bayesian bounds provide finite-sample guarantees for data-dependent randomized predictors, but applying them to learning-based control is difficult because the natural objective is a quadratic trajectory cost. Such losses are unbounded, non-Lipschitz , and lead to response-dependent Chernoff ter…

• Deep Residual Networks Learn the Geodesic Curve in the Wasserstein Space

  Kuo Gai, Shihua Zhang · 29 June 2026

  Recent studies revealed the mathematical connection between deep neural networks (DNNs) and dynamic systems. However, the specific dynamics that DNNs, especially deep residual networks (ResNets), tend to learn during training remain insufficiently characterized. To this end, we model the forward pro…

• Recovering Sharp Conductivity Features in the Finite-Data Calder\'on Problem with Physics-Informed Neural Networks

  Ali AlHadi Kalout, Pablo Tejerina-P\'erez, Konstantin Karchev, Pedro Taranc\'on-\'Alvarez, Leonid Sarieddine, Raul Jimenez, Max Engelstein, Guy David · 29 June 2026

  Physics-informed neural networks (PINNs) have recently emerged as a promising framework for addressing the Calder\'on inverse problem from limited boundary data. In this work, we revisit neural Calder\'on inversion by introducing multiscale boundary excitations based on randomized wavelet functions …

• Deep Neural Networks Inspired by Differential Equations

  Yongshuai Liu, Lianfang Wang, Kuilin Qin, Qinghua Zhang, Faqiang Wang, Li Cui, Jun Liu, Yuping Duan, Tieyong Zeng · 29 June 2026

  Deep learning has become a pivotal technology in fields such as computer vision, scientific computing, and dynamical systems, significantly advancing these disciplines. However, neural Networks persistently face challenges related to theoretical understanding, interpretability, and generalization. T…

• Mosaic: A Benchmark Suite for Differentiable Physics Solvers

  Andrin Rehmann, Heiko Zimmermann, Dion H\"afner · 29 June 2026

  Differentiable partial differential equation (PDE) solvers underpin solver-in-the-loop ML training, gradient-based optimal control, and inverse problems, yet the practical cost of obtaining correct, usable gradients from a given solver on a given problem is largely undocumented. Integration effort, …

• LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries

  Ren\'e P. Klausen, Ivan Timofeev, Jonas Naujoks, Johannes Frank, Thomas Wiegand, Sebastian Lapuschkin, Wojciech Samek · 29 June 2026

  Initial-boundary value problems (IBVPs) provide the essential framework for modelling a wide range of phenomena in physics and engineering. We introduce a novel method for efficiently solving IBVPs using Lie symmetries to enforce the associated partial differential equation (PDE) exactly by construc…