Model Reduction and Neural Networks

Derniers papiers

• Physics-Informed Neural Embeddings of PDE Solution Families

  Raul Jimenez, Svitlana Mayboroda, Pavlos Protopapas, Leonid Sarieddine, David N. Spergel, Pedro Taranc\'on-\'Alvarez · 8 juillet 2026

  We introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations. The method uses a multihead Physics-Informed Neural Network in which a shared body learns a latent manifold representing the solution space, while linear heads…

• A Physics-Informed Neural Network Framework for Elastodynamic Wave Propagation in Bimaterial Systems

  Sonal Ankush Chibire, Jenn-Terng Gau, Bo Zhang · 8 juillet 2026

  Physics-informed neural networks (PINNs) provide a promising framework for solving partial differential equations while embedding the underlying physical laws directly into the learning process. This study presents a PINN-based framework for modeling transient elastodynamic wave propagation in bimat…

• LRX-PINN: A Layer-Resolving XNet Physics-Informed Neural Network with Integrated Cauchy Activations for Convection-Dominated Problems

  Zihao Guo, Xin Li, Zhihong Xia · 7 juillet 2026

  Convection-dominated convection-diffusion problems often develop thin layers, where the solution has sharp transition profiles and its derivatives are highly localized. This creates a structural mismatch for standard physics-informed neural networks (PINNs), whose trial spaces are not designed to ma…

• Advances in Neural Controlled Differential Equations

  Benjamin Walker · 7 juillet 2026

  Many real-world systems evolve continuously, yet most machine learning models interpret time series as discrete sequences. Continuous-time approaches instead treat time series as samples from an underlying input path, a formulation that naturally accommodates irregularly sampled or oversampled data.…

• Convergence of Stochastic Gradient Methods for Wide Two-Layer Physics-Informed Neural Networks for the Poisson Equation

  Bangti Jin, Longjun Wu · 7 juillet 2026

  Physics informed neural networks (PINNs) represent a very popular class of neural solvers for partial differential equations. In practice, one often employs stochastic gradient descent type algorithms to train the neural network. Therefore, the convergence guarantee of stochastic gradient descent is…

• Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability

  Ronald Katende · 7 juillet 2026

  We study operator relevance in data-driven partial differential equation (PDE) discovery. Sparse residual methods can select terms that improve residual fit, but residual contribution is not the same as functional necessity. We formalize this distinction through counterfactual operator interventions…

• LiNO: Lifting based multiresolution neural operator

  Himanshu Pandey, Subham Patel, Ratikanta Behera · 7 juillet 2026

  Recently, neural operators have shown promising outcomes for learning solution operators of differential equations directly from data. This framework learns a functional mapping from the parameter field to the solution field, enabling the prediction of an entire class of solutions rather than a spec…

• Quadrature-Aware Complex-Linear Neural Operator for Boundary-to-Field Prediction in Resonant Acoustics

  Muhammad Idrees Khan, Hua-Dong Yao · 7 juillet 2026

  Repeated prediction of acoustic fields from spatially distributed boundary excitation is computationally expensive when each source realization requires a new wave simulation. This work introduces a quadrature-aware complex-linear boundary operator (CLBO) that maps complex normal velocity on a vibra…

• Conditional Clifford-Steerable CNNs for PDE Modeling

  B\'alint L\'aszl\'o Szarvas, Maksim Zhdanov · 7 juillet 2026

  We introduce Conditional Clifford-Steerable CNNs (C-CSCNNs), a unified framework that incorporates equivariance to arbitrary pseudo-Euclidean groups and significantly improves the expressivity of standard CSCNNs. We show that the kernel basis of the standard formulation is incomplete, limiting model…

• Target-Guided Selective Reweighting for Physics-Informed Neural Network Inverse Problems: A Transfer Learning Approach

  Qian Hu, Bin Fan, Yao Xiao, Zhicheng Lin, Meixin Xiong · 7 juillet 2026

  Physics-informed neural networks (PINNs) encounter ill-posed optimization, loss competition, and parameter compensation in partial differential equation (PDE) inverse problems. Transfer learning can reuse representations from source tasks, but direct fine-tuning may introduce negative transfer when …

• Learning rate adaptive stochastic gradient descent optimization methods: numerical simulations for deep learning methods for partial differential equations and convergence analyses

  Steffen Dereich, Arnulf Jentzen, Adrian Riekert · 7 juillet 2026

  The standard stochastic gradient descent (SGD) optimization method, as well as adaptive methods such as the Adam optimizer fail to converge if the learning rates do not converge to zero (particularly, in the situation of constant learning rates). In practice, human-tuned deterministic learning rate …

• Generative Inverse Design with Abstention via Diagonal Flow Matching

  Miguel de Campos, Werner Krebs, Hanno Gottschalk · 7 juillet 2026

  Inverse design aims to find design parameters $x$ achieving target performance $y^*$. Generative approaches learn bidirectional mappings between designs and labels, enabling diverse solution sampling. However, standard conditional flow matching (CFM), when adapted to inverse problems by pairing labe…

• In-span learning: adapting reduced-order models using their own predictions

  Amirpasha Hedayat, Laura Balzano, Karthik Duraisamy · 7 juillet 2026

  Reduced-order models compress high-dimensional dynamics into low-dimensional representations that can be evaluated rapidly, but they lose accuracy when online dynamics drift beyond the training data. Adaptive methods address this by updating the subspace online with external, out-of-span information…

• CSympNet-ID: conformal-symplectic map learning for linearly damped Hamiltonian systems

  Jiale Gong (School of Mathematics), Pengzhan Jin (National Engineering Laboratory for Big Data Analysis and Applications, Peking University, Beijing, China), Dongyang Kuang (School of Mathematics), Lu Li (School of Mathematics), Yifa Tang (State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China) · 7 juillet 2026

  Learning dissipative dynamics from discrete observations is essential for reliable long-horizon prediction and physically meaningful parameter identification. For linearly damped Hamiltonian systems, the exact flow is generally not symplectic but conformally symplectic, contracting the canonical sym…

• Reduced-Order Models: The Mother of World Models

  Rajat Ghosh · 7 juillet 2026

  World models -- compressed latent representations of an environment that support action-conditioned prediction and planning -- are typically presented as a product of modern self-supervised learning. This paper argues that the functional anatomy of a world model was independently developed, deployed…

• PDEFlow: Autonomous Agentic PDE Pipelines for Neural Operator Learning and Solver-Free Inference

  Akshat Jani, Prathamesh Gadekar, Sakhinana Sagar Srinivas, Venkataramana Runkana · 7 juillet 2026

  We present PDEFlow, an autonomous agentic framework that turns user-level ODE and PDE descriptions into solver-backed neural-operator pipelines. The workflow links problem specification, data generation, operator training, and checkpoint-based inference. A stateful input graph converts multi-turn na…

• Wasserstein Residuals: Learning Gradient Flows from Population Dynamics

  Markus Heinonen, Yair Shenfeld, Ricardo Baptista, Daniel Waxman, Dmitry Batenkov, Tim Cooijmans, Eli Bingham · 7 juillet 2026

  Reconstructing population dynamics is a central problem in the physical and data sciences. Often, the dynamics are modeled as a Wasserstein gradient flow (WGF): a curve of distributions driven by an energy functional. Though there are multiple mathematical characterizations of a WGF, the dominant al…

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

  Mojgan Alishiri, Amirhossein Arzani · 3 juillet 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 juillet 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…

• Koopman operator theory: fundamentals, control, and applications

  Igor Mezi\'c, Jorge Cort\'es, Karl Worthmann, Mircea Lazar, Armin Lederer · 3 juillet 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-…

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

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

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

  Yilie Huang, Wenpin Tang, Xun Yu Zhou · 3 juillet 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…

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

  Jiwei Jia, Xinliang Liu, Juntao Wang, Jinchao Xu · 3 juillet 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 juillet 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…

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

  Joseph Webb, Sadok Jerad, Coralia Cartis · 3 juillet 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…