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Found 14 papers, 8 papers with code
Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs
On complex downstream tasks with limited data, such as fluid flow simulations and fluid-structure interactions, we found CoDA-NO to outperform existing methods on the few-shot learning task by over $36\%$.
DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training
Pre-training has been investigated to improve the efficiency and performance of training neural operators in data-scarce settings.
Neural Operators with Localized Integral and Differential Kernels
In this work, we present a principled approach to operator learning that can capture local features under two frameworks by learning differential operators and integral operators with locally supported kernels.
Large Language Models for Mathematicians
Large language models (LLMs) such as ChatGPT have received immense interest for their general-purpose language understanding and, in particular, their ability to generate high-quality text or computer code.
Improved sampling via learned diffusions
Recently, a series of papers proposed deep learning-based approaches to sample from unnormalized target densities using controlled diffusion processes.
Mathematical Capabilities of ChatGPT
We investigate the mathematical capabilities of two iterations of ChatGPT (released 9-January-2023 and 30-January-2023) and of GPT-4 by testing them on publicly available datasets, as well as hand-crafted ones, using a novel methodology.
An optimal control perspective on diffusion-based generative modeling
In particular, we derive a Hamilton-Jacobi-Bellman equation that governs the evolution of the log-densities of the underlying SDE marginals.
Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning
The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions.
Learning ReLU networks to high uniform accuracy is intractable
Statistical learning theory provides bounds on the necessary number of training samples needed to reach a prescribed accuracy in a learning problem formulated over a given target class.
The Modern Mathematics of Deep Learning
We describe the new field of mathematical analysis of deep learning.
Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning
We show that a single deep neural network trained on simulated data is capable of learning the solution functions of an entire family of PDEs on a full space-time region.
How degenerate is the parametrization of neural networks with the ReLU activation function?
Approximation capabilities of neural networks can be used to deal with the latter non-convexity, which allows us to establish that for sufficiently large networks local minima of a regularized optimization problem on the realization space are almost optimal.
Towards a regularity theory for ReLU networks -- chain rule and global error estimates
Although for neural networks with locally Lipschitz continuous activation functions the classical derivative exists almost everywhere, the standard chain rule is in general not applicable.
Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations
It can be concluded that ERM over deep neural network hypothesis classes overcomes the curse of dimensionality for the numerical solution of linear Kolmogorov equations with affine coefficients.
