Model Reduction and Neural Networks

Derniers papiers

• 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 juin 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…

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

  Kuo Gai, Shihua Zhang · 29 juin 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…

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

  Yuanyuan Wang, Wenjie Wang, Haoxuan Li, Mingming Gong, Kun Zhang · 29 juin 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 juin 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…

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

  Emmanuel E. Oguadimma, Victory C. Obieke, Xueying Yu · 29 juin 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 juin 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…

• Mosaic: A Benchmark Suite for Differentiable Physics Solvers

  Andrin Rehmann, Heiko Zimmermann, Dion H\"afner · 29 juin 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, …

• 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 juin 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…

• Recovering Governing Equations from Solution Data: Identifiability Bounds for Linear and Nonlinear ODEs

  Yang Pan, Helmut B\"olcskei · 26 juin 2026

  Learning governing equations from observed solution data is a fundamental challenge in scientific machine learning \cite{bruntonDiscoveringGoverningEquations2016,kovachkiNeuralOperatorLearning2023,longPDENetLearningPDEs2018,rudyDatadrivenDiscoveryPartial2017,raonicConvolutionalNeuralOperators2023}, …

• Generative Models on Analog Hardware with Dynamics

  Yu-Neng Wang, Sara Achour · 26 juin 2026

  Analog hardware platforms such as coupled oscillators and Analog Ising Machines naturally solve differential equations at a fraction of the energy cost of digital computation, making them attractive for low-power generative modeling, yet a fundamental mismatch exists: modern generative models assume…

• Kolmogorov Arnold networks (KAN) for aerodynamic prediction: a comparison with MLPs and GNNs

  Miguel Jaraiz, Fermin Gutierrez, Pablo Yeste, Miguel S\'anchez-Dom\'inguez, Eusebio Valero, Gonzalo Rubio, Lucas Lacasa · 26 juin 2026

  Kolmogorov Arnold networks (KAN) have recently been introduced as a (deep) neural network architecture whose trainable parameters adapt the activation functions, instead of the coefficients of the affine transformations at the core of traditional architectures such as deep multilayer perceptrons (ML…

• ResilPhase: Plug-and-Play Phase Mapping and Noise-Resilient Macro-Trajectory Extrapolation for Diffusion Acceleration

  Qicheng Zhao, Yu Li, Qi Sun, Zheyu Yan · 26 juin 2026

  The adoption of powerful diffusion models is hindered by their significant inference latency. Recent ``cache-then-forecast'' schemes alleviate this issue by accelerating DiTs using derivative-based polynomials, but they suffer from severe quality degradation at high acceleration ratios. Our analysis…

• Effective Covariance Dynamics in Solvable High-Dimensional GANs

  Andrew Bond, Zafer Do\u{g}an · 26 juin 2026

  We study a solvable high-dimensional model of generative adversarial network (GAN) training in which a linear generator learns a low-dimensional subspace from data with structured latent covariance. Prior solvable GAN analyses assume unconditional signals with diagonal latent covariance; we extend t…

• Error-Conditioned Neural Solvers

  Haina Jiang, Liam Wang, Peng-Chen Chen, Min Seop Kwak, Seungryong Kim, Brian Bell, Jeong Joon Park · 26 juin 2026

  Neural surrogate models offer fast approximate mappings from PDE parameters to solutions, but they typically treat solving as a purely statistical task: once trained, they struggle to correct their own constraint violations and extrapolate beyond the training distribution. Recent hybrid methods prom…

• Symplectic Neural Networks for learning Generalized Hamiltonians

  Harsh Choudhary, Vyacheslav Kungurtsev, Chandan Gupta, Melvin Leok, Georgios Korpas · 26 juin 2026

  Hamiltonian Neural Networks (HNNs) integrate physical priors into neural models by learning a system's Hamiltonian, improving generalization and sample efficiency. Identifying the system Hamiltonian from noisy observations of state variables is a challenging task. For simulations to faithfully refle…

• Silent Failures in Physics-Informed Neural Networks: Parameter Poisoning and the Limits of Loss-Based Validation

  David McShannon, Nicholas Dietrich · 25 juin 2026

  Physics-informed neural networks (PINNs) embed governing equations in their loss function, enabling mesh-free solutions to partial differential equations. Low training loss is treated as evidence that the learned solution is physically correct. This paper shows that assumption breaks down when encod…

• LLM-ACES: Closed-Loop Discovery of Dynamical Systems with LLM-Guided Adaptive Search

  Nikhil Abhyankar, Sha Li, Sanchit Kabra, Naren Ramakrishnan, Yulia Gel, Chandan K. Reddy · 25 juin 2026

  Recovering governing Ordinary Differential Equations (ODEs) from data is a central challenge in modeling dynamical systems across scientific domains. Existing approaches cast discovery as a static inference problem over fixed datasets, assuming that the observed trajectories are sufficiently informa…

• A Flow-rate-conserving CNN-based Domain Decomposition Method for Blood Flow Simulations

  Simon Klaes, Axel Klawonn, Natalie Kubicki, Martin Lanser, Kengo Nakajima, Takashi Shimokawabe, Janine Weber · 25 juin 2026

  This work aims to predict blood flow with non-Newtonian viscosity in stenosed arteries using convolutional neural network (CNN) surrogate models. An alternating Schwarz domain decomposition method is proposed which uses CNN-based subdomain solvers. A universal subdomain solver (USDS) is trained …

• When Do Conservation Laws Survive Learned Representations? Certified Horizons for Latent World Models

  Hongbo Wang · 25 juin 2026

  We ask a representation-learning question about physical world models: when does a conservation law remain certifiable after a model learns a latent representation? A certified horizon bounds -- in advance, from measurable model defects -- how many steps a rollout provably stays on a physical invari…

• A Zeroth-Order Deep Learning Method for Fully Nonlinear Parabolic Partial Differential Equations with Unknown Coefficients

  Yanwei Jia, Du Ouyang, Huy\^en Pham, Xun Yu Zhou · 25 juin 2026

  High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging. Existing deep learning solvers often rely on repeated…

• Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs

  Mat\'ias G\'omez-Aedo, Behzad Azmi, Yuyang Huang, Dante Kalise, Karl Kunisch · 24 juin 2026

  A data-driven method is developed for approximating value functions in deterministic optimal control problems with nonlinear control-affine dynamics. The Pontryagin Maximum Principle optimality system is solved from multiple initial conditions to generate training data consisting of values, gradient…

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

  Jason Sulskis, Sathya Ravi · 24 juin 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…

• Dirac-Frenkel dynamics with inertia for nonlinearly parametrized solutions of evolution problems

  Matteo Raviola, Benjamin Peherstorfer · 24 juin 2026

  Even when Dirac-Frenkel dynamics determine a well-defined evolution in function space, the corresponding parameter dynamics can be non-unique or ill-conditioned for redundant nonlinear parametrizations such as neural networks or mixture models. We propose to add inertia to the Dirac-Frenkel dynamics…

• Structural Kolmogorov-Arnold Convolutions: Learnable Function on the Values or the Filter Shape as Parameter-Efficient Alternative to Per-Edge Convolutional KANs

  Stefano Mereu, Oleksandr Kuznetsov, Gabriele Marchello, Alessandro Galdelli, Emanuele Frontoni, Adriano Mancini, Ferdinando Cannella · 24 juin 2026

  Convolutional Kolmogorov--Arnold Networks (KANs) replace the fixed weights of a convolutional kernel with learnable univariate functions. The dominant formulation attaches one such function to every kernel entry and lets it act on pixel values, expressive but parameter-heavy and prone to overfitting…