# Differentiable Implicit Layers

arXiv: https://arxiv.org/abs/2010.07078 · HTML: https://mlsanity.com/paper/2010.07078/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Oct 2020.

## Built on by (most cited first, 6 of 6)

- [JFB: Jacobian-Free Backpropagation for Implicit Networks](https://mlsanity.com/paper/2103.12803/) (2021)
- [Dynamical systems' based neural networks](https://mlsanity.com/paper/2210.02373/) (2022)
- [Structure-preserving Gaussian Process Dynamics](https://mlsanity.com/paper/2102.01606/) (2021)
- [Tighter Confidence Bounds for Sequential Kernel Regression](https://mlsanity.com/paper/2403.12732/) (2024)
- [Parameter Inference with Bifurcation Diagrams](https://mlsanity.com/paper/2106.04243/) (2021)
- [Regularized Optimal Transport Layers for Generalized Global Pooling Operations](https://mlsanity.com/paper/2212.06339/) (2022)

## Abstract (arXiv)

In this paper, we introduce an efficient backpropagation scheme for non-constrained implicit functions. These functions are parametrized by a set of learnable weights and may optionally depend on some input; making them perfectly suitable as a learnable layer in a neural network. We demonstrate our scheme on different applications: (i) neural ODEs with the implicit Euler method, and (ii) system identification in model predictive control.