Model Reduction and Neural Networks

Neueste Paper

• NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes

  Anastasis Kratsios, Giulia Livieri, Philipp Schmocker · 17. Juli 2026

  We address fundamental challenges in representing and computing $\mathbb{R}^{d}$-valued predictable square-integrable processes over $[0,T]$, collected in the space $\mathcal{H}^2_T(\mathbb{R}^{d})$. These processes are central to continuous-time stochastic control, reinforcement learning, and mathe…

• An Introduction to Sparse Identification of Nonlinear Dynamics for Engineering Applications

  Yao Cheng Li, Ana Larra\~naga, Steven L. Brunton, Urban Fasel · 17. Juli 2026

  Many engineering problems involve phenomena whose governing equations are poorly characterized or only partially known. Surrogate modeling techniques such as neural networks can capture the behavior of these systems, but they typically demand large training datasets that are difficult to obtain in e…

• Sharp Stability Threshold and Certification for Designing Stable Residual Architectures

  Hyemin Gu, Michael Tyrrell, Tuhin Sahai, Markos A. Katsoulakis · 17. Juli 2026

  We propose \emph{the sublinear-growth principle} for deep residual architectures -- a sharp stability threshold on the input-magnitude exponent of every residual block's velocity field: $$\|v(x, t)\| \leq c\,\|x\|^q + b, \qquad q \in [0, 1].$$ The threshold $q = 1$ is established via two independent…

• Subgrid-Scale Parameterization in Burgers' Equation Using Structure-Preserving Neural Networks and Entropy Variables

  Aijaz Nazir, Ilya Timofeyev · 17. Juli 2026

  We present a machine learning approach for developing subgrid-scale (SGS) parametrizations in coarse simulations of partial differential equations. We utilize structure-preserving neural networks and entropy variables to learn subgrid fluxes in coarse simulations of the Burgers' equation. In particu…

• Probabilistic Physics-Informed Neural Networks for Estimating Heterogeneous Elastic Properties from Low-Resolution and Noisy Displacement Data

  Tatthapong Srikitrungruang, Jaesung Lee · 17. Juli 2026

  Estimating spatially heterogeneous elastic properties from low-resolution displacement measurements is a severely ill-posed inverse elasticity problem because low resolution obscures spatial details needed to distinguish heterogeneous property variations, and small measurement perturbations or fitti…

• Optimal Self-Distillation for Rectified Flow via Linear Probing

  Saptarshi Roy, Debepsita Mukherjee, Pratik Patil · 17. Juli 2026

  Modern generative models are increasingly trained using model-generated signals, creating both opportunities for self-improvement and risks of collapse. We study optimal self-distillation (SD) for rectified flow (RF): given a suboptimal teacher velocity field, can a student trained on a mixture of t…

• A Minimal Interpretable Architecture for Zero-Shot Reconstruction of Dynamical Systems

  Christoph J\"urgen Hemmer, Florian Plaswig, Daniel Durstewitz · 17. Juli 2026

  Recent foundation models (FMs) for zero-shot reconstruction of dynamical systems (DS) achieve strong out-of-domain generalization but provide little insight into the mechanisms that underlie their forecasts. Such an understanding could help to strip down overladen FM architectures to their bare esse…

• LIGO-PINN: Learned Initialization via Gated Optimization to Alleviate Convergence Failures in Physics Informed Neural Networks

  Nilay Anurag, Shital Adhikari, Taniya Kapoor, Nikhil Muralidhar · 17. Juli 2026

  Physics-informed neural networks (PINNs) have had a broad research impact in modeling domains governed by partial differential equations (PDE). However, PINNs have been shown to perform poorly, sometimes even converging to trivial solutions, in challenging PDE domains, or when generalizing to unseen…

• Adaptive Runge-Kutta Step Control Buys Training Loss, Not Generalization: An Honest Compute-Matched Study of RK-Adam Optimizers

  Akhilesh Gogikar · 17. Juli 2026

  Interpreting optimizers as gradient-flow discretizations has motivated applying higher-order Runge-Kutta (RK) integrators to neural networks. We build a representative Adam variant (Bogacki-Shampine 3(2) RK pair, FSAL reuse, local-error step control) and evaluate it under a strict compute-matched pr…

• RTS Smoother-Guided Learning of Physics-Based Neural Differential Models

  Ahmet Demirkaya, Georgios Stratis, Tales Imbiriba, Zachary D. Danziger, Deniz Erdogmus · 17. Juli 2026

  Ordinary differential equations (ODEs) are widely used to model dynamical systems in physics, biology, neuroscience, and physiology, but in many applications some equations of the dynamics are unknown and only a subset of the state variables are measured. We propose a hybrid neural--physics framewor…

• Operator-Informed Gaussian Processes for Complex Helmholtz Wavefields: From Synthetic Benchmarks to In Vivo Brain Elastography

  Boyuan Deng, Kshitiz Upadhyay, Michael Shields · 17. Juli 2026

  The Helmholtz equation governs time-harmonic wave propagation, and in dissipative media a complex modulus renders its squared wavenumber $\kappa^2$ complex. Inferring such fields from sparse, noisy data calls for solvers that also quantify their own uncertainty. Physics-informed Gaussian-process (GP…

• Automatic Differentiation from Scratch: How PyTorch Computes Gradients in Physics-Informed Neural Networks

  Abdeladhim Tahimi · 16. Juli 2026

  This paper traces, with explicit numerical values, how PyTorch's automatic differentiation (AD) engine computes gradients for Physics-Informed Neural Network (PINN) training -- a setting that requires two levels of differentiation: computing the physics derivative $\hat{y}'(t)=d\hat{y}/dt$ through t…

• Discrete Diffusion Models: A Unified Framework from Tokenization to Generation

  Ye Yuan, Weien Li, Rui Song, Zeyu Li, Haochen Liu, Xiangyu Kong, Zixuan Dong, Linfeng Du, Zipeng Sun, Weixu Zhang, Jiaxin Huang, Changjiang Han, Yonghan Yang, Zichen Zhao, Xiuyuan Hu, Haolun Wu, Yankai Chen, Fengran Mo, Jikun Kang, Bowei He, Philip S. Yu, Xue Liu · 16. Juli 2026

  Discrete denoising diffusion models (DDMs) have recently emerged as a compelling alternative to autoregressive (AR) modeling for discrete data, offering parallel generation and iterative global refinement capabilities. Unlike continuous diffusion, where the state space is fixed, DDMs are fundamental…

• Automatic Ordinary Differential Equations Discovery For Biological Systems Using Large Language Model Powered Agentic System

  David Krongauz, Arad Zulti, Eran Segal, Teddy Lazebnik · 16. Juli 2026

  Automatic scientific discovery has long been a goal of computational scholars - a machine that can discover nature's secrets on its own, moving computational systems beyond data-fitting tools toward the generation and refinement of mechanistic models of the universe. Recent advances in symbolic regr…

• Spectral-Informed Neural Networks Outperform Spectral Methods in High-dimensional PDEs

  Tianchi Yu, Ivan Oseledets · 16. Juli 2026

  For low-dimensional problems ($d\leq3$), spectral methods can achieve exceptionally high accuracy. For middle-dimensional problems ($4 \leq d \lesssim 10$), spectral methods remain feasible through specific techniques such as sparse grids or hyperbolic cross. However, for high-dimensional problems (…

• Gradient-free learning of a closed-loop wall controller for turbulent drag reduction

  Giorgio Maria Cavallazzi, Miguel P\'erez Cuadrado, Alfredo Pinelli · 15. Juli 2026

  Closed-loop wall control learnt by multi-agent reinforcement learning can lower skin-friction drag in turbulent channels, but these gradient-based policies are trained on small periodic boxes and exhibit reduced performance when carried over to a larger domain. We recently showed that such policies …

• Learning to Discretize: Diffusion-Based Adaptive Mesh with Spectral Guidance

  Zixuan Shen (Central South University), Bingchuan Wang (Central South University), Zhi Wang (Nanjing University), Yong Wang (Central South University) · 15. Juli 2026

  Most neural partial differential equation (PDE) surrogates learn how fields evolve after a grid has already been chosen. However, before any operator is applied, the grid has already determined how modeling capacity is allocated across space, resolution, and spectral bandwidth. We argue that this hi…

• An Agentic AI Scientific Community for Automated Neural Operator Discovery

  Luis Loo, Ulisses Braga-Neto · 15. Juli 2026

  We present an agentic approach to autonomous neural operator discovery based on an AI scientific community, which consists of a swarm of virtual laboratories that interact under a citation-based economy of influence. Highly-cited labs found new labs that follow their research direction and replace n…

• A Shortcut to Statistically Steady-State Turbulence with Flow Matching

  Gianluca Galletti, Gerald Gutenbrunner, William Hornsby, Lorenzo Zanisi, Naomi Carey, Stanislas Pamela, Johannes Brandstetter, Fabian Paischer · 15. Juli 2026

  Many nonlinear physical systems exhibit an initial transient phase in which perturbations grow before nonlinear interactions lead to a statistically steady state. While this saturated regime is of primary interest, direct numerical simulations must resolve the full transient dynamics before reaching…

• Learning Forced Multibody Dynamics on Lie Groups

  Martine Dyring Hansen, Marta Ghirardelli, Elena Celledoni, David Martin de Diego, Brynjulf Owren · 15. Juli 2026

  We propose an architecture for learning the dynamics of mechanical systems based on discrete forced Euler-Lagrange equations on Lie groups using only position data. By formulating the dynamics directly on manifold-valued configuration spaces, the method naturally respects the geometric structure of …

• Cluster-Weighted EDMD

  Lorenzo Tomaz, Judd Rosenblatt, Flavio Kicis, Thomas B. Jones, Diogo Schwerz de Lucena · 15. Juli 2026

  Extended Dynamic Mode Decomposition (EDMD) approximates Koopman operators from data, but a single global operator is inefficient when different state-space regions exhibit distinct local dynamics. We introduce Cluster-Weighted EDMD (CW-EDMD), which jointly learns a soft phase-space partition and a p…

• A Hyperbolic Neural Closure for M1 Radiation Transfer

  Bongseok Kim, Jiahao Zhang, Johannes Krotz, Dinshaw Balsara, Ryan McClarren, Guang Lin · 14. Juli 2026

  In radiation transfer simulations, an M1 method achieves substantial computational savings by replacing the full angular transport equation with a low-order moment system. Because this reduced system is not closed, a closure model is required to represent the unknown higher-order moments using lower…

• A multi-scale feature enhanced graph neural network for fluid dynamics prediction in complex geometries

  Li Xiao, Tianyu Li, Yiye Zou, Mingjie Zhang, Xiaogangd Deng · 14. Juli 2026

  Industrial design in fields such as vehicle and aerospace engineering often relies on large-scale numerical simulations to evaluate fluid dynamics performance, which can incur substantial computational costs. Deep neural networks have shown promise in improving simulation efficiency, especially grap…

• Robustly Invertible Nonlinear Dynamics and the BiLipREN: From Inversion-Based Control to Generative Trajectory Modelling

  Yurui Zhang, Ruigang Wang, Ian R. Manchester · 14. Juli 2026

  This paper proposes a new notion of robust invertibility for nonlinear dynamical systems, and introduces constructive parameterizations of recurrent neural network which are robustly invertible by design. We define robust invertibility as the existence of a causal inverse system such that both the f…

• Neural Discovery of Memory and Nonlocal Kernels in Integro-Differential Equations with Constrained Kolmogorov--Arnold Networks

  Aruzhan Tleubek, Salah A Faroughi · 14. Juli 2026

  Discovering the memory or nonlocal kernel governing an integro-differential equation (IDE) from sparse and noisy observations is an ill-posed inverse problem. Existing identification methods often rely on problem-specific analytical derivations, specialized observation requirements, or restrictive a…