arXiv:2607. 13042v1 Announce Type: new Abstract: 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 the network, and computing parameter gradients $\nabla_\theta L$ of a loss that itself depends on $\hat{y}'(t)$.
By Abdeladhim Tahimi
arXiv:2606. 16926v1 Announce Type: cross Abstract: Functional optimization problems are typically solved by optimizing the parameters of a fixed representation, such as a neural network, resulting in highly nonconvex losses that complicate both training and theoretical analysis.
By Daniel Csillag, Rodrigo Schuller, Pedro Dall'Antonia, Leonidas Guibas, Luiz Velho, Tiago Novello
arXiv:2604. 07328v3 Announce Type: replace Abstract: How does the choice of training data influence an AI model?
By Sam Gunn
We systematically investigate finite-difference (FD) derivative computation in Physics-Informed Neural Networks (PINNs) as an alternative to automatic differentiation (AD). On three benchmark PDEs we show that, with a properly calibrated step size, FD matches AD in accuracy on every problem while running faster across the full tested batch-size range and using substantially less GPU memory, and that a stochastic variant we propose outperforms AD on a stationary problem.
arXiv:2608. 11020v1 Announce Type: new Abstract: We systematically investigate finite-difference (FD) derivative computation in Physics-Informed Neural Networks (PINNs) as an alternative to automatic differentiation (AD).
By Maciej J. Mikulski, Tadeusz Uhl
arXiv:2607. 10589v1 Announce Type: cross Abstract: In contrast to most studies on neural network approximation theory that characterize results through a single parameter, such as the total number of network parameters, \cite{shen2020deep} pioneered the characterization of approximation rates as a joint function of the width parameter $N$ and the depth parameter $L$, thereby granting greater architectural flexibility.
By Yanming Lai, Defeng Sun, Yang Wang
arXiv:2607. 03148v1 Announce Type: cross Abstract: Activation functions are considered an essential primitive for neural nonlinearity, i.
By Muhammad Sabih, Frank Hannig, J\"urgen Teich
arXiv:2606. 26705v1 Announce Type: cross Abstract: Feedforward neural network (NN) expressivity is typically studied by emulating optimal basis-expansion schemes.
By Anastasis Kratsios, Simone Brugiapaglia, Bum Jun Kim, Gregory Cousins, Haitz S\'aez de Oc\'ariz Borde
arXiv:2606. 16975v1 Announce Type: cross Abstract: In this work, we investigate new activation functions for achieving arbitrary-accuracy Sobolev approximation by fixed-size neural networks.
By Baicheng Li, Haizhao Yang, Shijun Zhang
arXiv:2604. 20219v2 Announce Type: replace Abstract: Depth is widely viewed as a central contributor to the success of deep neural networks, whereas standard neural network approximation theory typically provides guarantees only for the final output and leaves the role of intermediate layers largely unclear.
By Shijun Zhang, Zuowei Shen, Yuesheng Xu
arXiv:2601. 16884v3 Announce Type: replace Abstract: We study multigrade deep learning (MGDL) as a principled framework for structured error refinement in deep neural networks.
By Shijun Zhang, Zuowei Shen, Yuesheng Xu
arXiv:2511. 14953v2 Announce Type: replace-cross Abstract: Discrete structures are currently second-class in differentiable programming.
By Joey Velez-Ginorio, Nada Amin, Konrad Kording, Steve Zdancewic