# Locally adaptive activation functions with slope recovery term for deep and physics-informed neural networks

arXiv: https://arxiv.org/abs/1909.12228 · HTML: https://mlsanity.com/paper/1909.12228/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Sep 2019.

## Built on by (most cited first, 11 of 11)

- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [On the Theory of Implicit Deep Learning: Global Convergence with Implicit Layers](https://mlsanity.com/paper/2102.07346/) (2021)
- [A novel meta-learning initialization method for physics-informed neural networks](https://mlsanity.com/paper/2107.10991/) (2021)
- [Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/2108.13054/) (2021)
- [Inferring incompressible two-phase flow fields from the interface motion using physics-informed neural networks](https://mlsanity.com/paper/2101.09833/) (2021)
- [A Recipe for Global Convergence Guarantee in Deep Neural Networks](https://mlsanity.com/paper/2104.05785/) (2021)
- [An unsupervised latent/output physics-informed convolutional-LSTM network for solving partial differential equations using peridynamic differential operator](https://mlsanity.com/paper/2210.12177/) (2022)
- [Physics-informed neural networks for non-Newtonian fluid thermo-mechanical problems: an application to rubber calendering process](https://mlsanity.com/paper/2201.13389/) (2022)
- [dNNsolve: an efficient NN-based PDE solver](https://mlsanity.com/paper/2103.08662/) (2021)
- [Learning Free-Surface Flow with Physics-Informed Neural Networks](https://mlsanity.com/paper/2111.09705/) (2021)
- [Data-aware customization of activation functions reduces neural network error](https://mlsanity.com/paper/2301.06635/) (2023)

## Abstract (arXiv)

We propose two approaches of locally adaptive activation functions namely, layer-wise and neuron-wise locally adaptive activation functions, which improve the performance of deep and physics-informed neural networks. The local adaptation of activation function is achieved by introducing a scalable parameter in each layer (layer-wise) and for every neuron (neuron-wise) separately, and then optimizing it using a variant of stochastic gradient descent algorithm. In order to further increase the training speed, an activation slope based slope recovery term is added in the loss function, which further accelerates convergence, thereby reducing the training cost. On the theoretical side, we prove that in the proposed method, the gradient descent algorithms are not attracted to sub-optimal critical points or local minima under practical conditions on the initialization and learning rate, and that the gradient dynamics of the proposed method is not achievable by base methods with any (adaptive) learning rates. We further show that the adaptive activation methods accelerate the convergence by implicitly multiplying conditioning matrices to the gradient of the base method without any explicit computation of the conditioning matrix and the matrix-vector product. The different adaptive activation functions are shown to induce different implicit conditioning matrices. Furthermore, the proposed methods with the slope recovery are shown to accelerate the training process.