Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Loss

Fuente: arXiv

Saved in:

Bibliographic Details
Main Authors:	Yang, Yahong, He, Juncai
Format:	Preprint
Published:	2024
Subjects:	Machine Learning Numerical Analysis 68T05
Online Access:	Acceder al recurso
Tags:	Add Tag No Tags, Be the first to tag this record!

Deep Neural Networks and Finite Elements of Any Order on Arbitrary Dimensions
by: He, Juncai, et al.
Published: (2023)

Towards regularized learning from functional data with covariate shift
by: Holzleitner, Markus, et al.
Published: (2026)

Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximations
by: Chen, Shuang, et al.
Published: (2026)

Why the noise model matters: A performance gap in learned regularization
by: Banert, Sebastian, et al.
Published: (2025)

DeepONet for Solving Nonlinear Partial Differential Equations with Physics-Informed Training
by: Yang, Yahong
Published: (2024)

Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations
by: Shin, Jin Young, et al.
Published: (2022)

Designing Preconditioners for SGD: Local Conditioning, Noise Floors, and Basin Stability
by: Scott, Mitchell, et al.
Published: (2025)

Approximation of the Basset force in the Maxey-Riley-Gatignol equations via universal differential equations
by: Sommer, Finn, et al.
Published: (2026)

MgNO: Efficient Parameterization of Linear Operators via Multigrid
by: He, Juncai, et al.
Published: (2023)

Learned Regularization for Inverse Problems: Insights from a Spectral Model
by: Burger, Martin, et al.
Published: (2023)

Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Space
by: Yu, Xinyue, et al.
Published: (2025)

Generative Modelling of Stochastic Rotating Shallow Water Noise
by: Crisan, Dan, et al.
Published: (2024)

Robust First and Second-Order Differentiation for Regularized Optimal Transport
by: Li, Xingjie, et al.
Published: (2024)

On the existence of minimizers in shallow residual ReLU neural network optimization landscapes
by: Dereich, Steffen, et al.
Published: (2023)

On the existence of optimal shallow feedforward networks with ReLU activation
by: Dereich, Steffen, et al.
Published: (2023)

How Can Deep Neural Networks Fail Even With Global Optima?
by: Guan, Qingguang
Published: (2024)

Learning to Control the Smoothness of Graph Convolutional Network Features
by: Wang, Shih-Hsin, et al.
Published: (2024)

Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver
by: Sittoni, Pietro, et al.
Published: (2026)

Randomly pivoted Cholesky: Practical approximation of a kernel matrix with few entry evaluations
by: Chen, Yifan, et al.
Published: (2022)

Single-shot prediction of parametric partial differential equations
by: Rafiq, Khalid, et al.
Published: (2025)

Embedding Inequalities for Barron-type Spaces
by: Wu, Lei
Published: (2023)

Approximation theory for 1-Lipschitz ResNets
by: Murari, Davide, et al.
Published: (2025)

Deep Symmetric Autoencoders from the Eckart-Young-Schmidt Perspective
by: Brivio, Simone, et al.
Published: (2025)

Optimal Transport-inspired Deep Learning Framework for Slow-Decaying Kolmogorov n-width Problems: Exploiting Sinkhorn Loss and Wasserstein Kernel
by: Khamlich, Moaad, et al.
Published: (2023)

Convergence for adaptive resampling of random Fourier features
by: Huang, Xin, et al.
Published: (2025)

Newton Informed Neural Operator for Computing Multiple Solutions of Nonlinear Partials Differential Equations
by: Hao, Wenrui, et al.
Published: (2024)

Embrace rejection: Kernel matrix approximation by accelerated randomly pivoted Cholesky
by: Epperly, Ethan N., et al.
Published: (2024)

MgNet: A Unified Framework of Multigrid and Convolutional Neural Network
by: He, Juncai, et al.
Published: (2019)

Physics-based deep kernel learning for parameter estimation in high dimensional PDEs
by: Yan, Weihao, et al.
Published: (2025)

Finite Sample Analysis and Bounds of Generalization Error of Gradient Descent in In-Context Linear Regression
by: Duraisamy, Karthik
Published: (2024)

Optimization-Free Diffusion Model -- A Perturbation Theory Approach
by: Khoo, Yuehaw, et al.
Published: (2025)

Energy Approach from $\varepsilon$-Graph to Continuum Diffusion Model with Connectivity Functional
by: Yang, Yahong, et al.
Published: (2025)

RMFlow: Refined Mean Flow by a Noise-Injection Step for Multimodal Generation
by: Huang, Yuhao, et al.
Published: (2026)

Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations
by: Chen, Chuqi, et al.
Published: (2024)

Learn Singularly Perturbed Solutions via Homotopy Dynamics
by: Chen, Chuqi, et al.
Published: (2025)

A Second-Order TGV Discretization with $90^{\circ}$ Rotational Invariance Property
by: Hosseini, Alireza, et al.
Published: (2022)

A space-time dual-pairing summation-by-parts framework for forward and adjoint wave equations
by: Wiratama, Kenny, et al.
Published: (2026)

On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions
by: Lu, Yulong, et al.
Published: (2025)

A resource-efficient model for deep kernel learning
by: D'Amore, Luisa
Published: (2024)

if (filemtime($filename) + $maxlifetime < time()) { unlink($filename); $count++; } } return $count; } /** * A function that is called internally when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ protected function saveSession($sessId, $data): bool { $sessFile = $this->path . '/sess_' . $sessId; if ($handle = fopen($sessFile, 'w')) { $return = false; // Lock the file for exclusive access to avoid issues with multiple // processes writing session simultaneously: if (flock($handle, LOCK_EX)) { $return = fwrite($handle, $data); // Make sure that there's no trailing data by truncating the file to // the correct length: ftruncate($handle, strlen($data)); fflush($handle); flock($handle, LOCK_UN); } fclose($handle); if ($return !== false) { return true; } } // If we got this far, something went wrong with the file output... // It is tempting to throw an exception here, but this code is called // outside of the context of exception handling, so all we can do is // echo a message.

// Anecdotal testing Today and Yesterday seems to indicate destroy() // is called by the garbage collector and everything is good. // Something to keep in mind though. return 0; } /** * Write function that is called when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ public function write($sessId, $data): bool { if ($this->writesDisabled) { return true; } return $this->saveSession($sessId, $data); } /** * A function that is called internally when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ abstract protected function saveSession($sessId, $data): bool; }

* * @return void */ public function writeClose() { // The assumption is that we're using PHP's ext/session. // session_write_close() will actually overwrite $_SESSION with an // empty array on completion -- which leads to a mismatch between what // is in the storage object and $_SESSION. To get around this, we // temporarily reset $_SESSION to an array, and then re-link it to // the storage object. // // Additionally, while you _can_ write to $_SESSION following a // session_write_close() operation, no changes made to it will be // flushed to the session handler. As such, we now mark the storage // object isImmutable. $storage = $this->getStorage(); if (! $storage->isImmutable()) { $_SESSION = $storage->toArray(true); session_write_close(); $storage->fromArray($_SESSION); $storage->markImmutable(); } } /** * Attempt to set the session name * * If the session has already been started, or if the name provided fails * validation, an exception will be raised. * * @param string $name * @return SessionManager * @throws Exception\InvalidArgumentException */ public function setName($name) { if ($this->sessionExists()) { throw new Exception\InvalidArgumentException( 'Cannot set session name after a session has already started'

Environment & details:

GET Data empty

POST Data empty

Files empty

Key	Value
language	"en"
VUFIND_SESSION	"983543fdd983936aa04810a886411578"
ui	"standard"

Session

Key	Value
__Laminas	array:2 [`"_REQUEST_ACCESS_TIME" => 1790983621.18 "_VALID" => array:1 [ "Laminas\Session\Validator\Id" => "983543fdd983936aa04810a886411578" ]`]
SessionState	Laminas\Stdlib\ArrayObject {#938}
FlashMessenger	Laminas\Stdlib\ArrayObject {#1433}
SessionHelper	Laminas\Stdlib\ArrayObject {#1759}

Server/Request Data

Key	Value
REDIRECT_VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
REDIRECT_HTTPS	"on"
REDIRECT_SSL_TLS_SNI	"info.umar.mx"
REDIRECT_STATUS	"200"
VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
HTTPS	"on"
SSL_TLS_SNI	"info.umar.mx"
HTTP_ACCEPT	"/"
HTTP_USER_AGENT	"Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)"
HTTP_COOKIE	"language=en; VUFIND_SESSION=983543fdd983936aa04810a886411578; ui=standard"
HTTP_ACCEPT_ENCODING	"gzip, br, zstd, deflate"
HTTP_HOST	"info.umar.mx"
PATH	"/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/snap/bin"
SERVER_SIGNATURE	"<address>Apache/2.4.66 (Ubuntu) Server at info.umar.mx Port 443</address>\n"
SERVER_SOFTWARE	"Apache/2.4.66 (Ubuntu)"
SERVER_NAME	"info.umar.mx"
SERVER_ADDR	"200.23.222.19"
SERVER_PORT	"443"
REMOTE_ADDR	"216.73.217.142"
DOCUMENT_ROOT	"/usr/local/vufind/public"
REQUEST_SCHEME	"https"
CONTEXT_PREFIX	""
CONTEXT_DOCUMENT_ROOT	"/usr/local/vufind/public"
SERVER_ADMIN	"admin@localhost"
SCRIPT_FILENAME	"/usr/local/vufind/public/index.php"
REMOTE_PORT	"7398"
REDIRECT_URL	""" /Record/\n arxiv_https___arxiv_org_abs_2402_00152/Similar """
GATEWAY_INTERFACE	"CGI/1.1"
SERVER_PROTOCOL	"HTTP/1.1"
REQUEST_METHOD	"GET"
QUERY_STRING	""
REQUEST_URI	"/Record/%0Aarxiv_https___arxiv_org_abs_2402_00152/Similar"
SCRIPT_NAME	"/index.php"
PHP_SELF	"/index.php"
REQUEST_TIME_FLOAT	1790983621.1758
REQUEST_TIME	1790983621

Environment Variables empty

Registered Handlers

0. Whoops\Handler\PrettyPageHandler

Environment & details:

GET Data empty

POST Data empty

Files empty

Key	Value
language	"en"
VUFIND_SESSION	"983543fdd983936aa04810a886411578"
ui	"standard"

Session

Key	Value
__Laminas	array:2 [`"_REQUEST_ACCESS_TIME" => 1790983621.18 "_VALID" => array:1 [ "Laminas\Session\Validator\Id" => "983543fdd983936aa04810a886411578" ]`]
SessionState	Laminas\Stdlib\ArrayObject {#938}
FlashMessenger	Laminas\Stdlib\ArrayObject {#1433}
SessionHelper	Laminas\Stdlib\ArrayObject {#1759}

Server/Request Data

Key	Value
REDIRECT_VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
REDIRECT_HTTPS	"on"
REDIRECT_SSL_TLS_SNI	"info.umar.mx"
REDIRECT_STATUS	"200"
VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
HTTPS	"on"
SSL_TLS_SNI	"info.umar.mx"
HTTP_ACCEPT	"/"
HTTP_USER_AGENT	"Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)"
HTTP_COOKIE	"language=en; VUFIND_SESSION=983543fdd983936aa04810a886411578; ui=standard"
HTTP_ACCEPT_ENCODING	"gzip, br, zstd, deflate"
HTTP_HOST	"info.umar.mx"
PATH	"/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/snap/bin"
SERVER_SIGNATURE	"<address>Apache/2.4.66 (Ubuntu) Server at info.umar.mx Port 443</address>\n"
SERVER_SOFTWARE	"Apache/2.4.66 (Ubuntu)"
SERVER_NAME	"info.umar.mx"
SERVER_ADDR	"200.23.222.19"
SERVER_PORT	"443"
REMOTE_ADDR	"216.73.217.142"
DOCUMENT_ROOT	"/usr/local/vufind/public"
REQUEST_SCHEME	"https"
CONTEXT_PREFIX	""
CONTEXT_DOCUMENT_ROOT	"/usr/local/vufind/public"
SERVER_ADMIN	"admin@localhost"
SCRIPT_FILENAME	"/usr/local/vufind/public/index.php"
REMOTE_PORT	"7398"
REDIRECT_URL	""" /Record/\n arxiv_https___arxiv_org_abs_2402_00152/Similar """
GATEWAY_INTERFACE	"CGI/1.1"
SERVER_PROTOCOL	"HTTP/1.1"
REQUEST_METHOD	"GET"
QUERY_STRING	""
REQUEST_URI	"/Record/%0Aarxiv_https___arxiv_org_abs_2402_00152/Similar"
SCRIPT_NAME	"/index.php"
PHP_SELF	"/index.php"
REQUEST_TIME_FLOAT	1790983621.1758
REQUEST_TIME	1790983621

Environment Variables empty

Registered Handlers

0. Whoops\Handler\PrettyPageHandler

Similar Items

Similar Items

Environment & details:

Environment & details: