# Deep-layer limit and stability analysis of the basic forward-backward-splitting induced network (II): learning problems

arXiv: https://arxiv.org/abs/2605.27133 · HTML: https://mlsanity.com/paper/2605.27133/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 45 references that are in the map.
- First posted on arXiv 26 May 2026.

## Builds on (most cited first, 21 of 21)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [Trainable Nonlinear Reaction Diffusion: A Flexible Framework for Fast and Effective Image Restoration](https://mlsanity.com/paper/1508.02848/) (2015)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Understanding and Improving Transformer From a Multi-Particle Dynamic System Point of View](https://mlsanity.com/paper/1906.02762/) (2019)
- [FISTA-Net: Learning A Fast Iterative Shrinkage Thresholding Network for Inverse Problems in Imaging](https://mlsanity.com/paper/2008.02683/) (2020)
- [Reversible Architectures for Arbitrarily Deep Residual Neural Networks](https://mlsanity.com/paper/1709.03698/) (2017)
- [Machine Learning from a Continuous Viewpoint](https://mlsanity.com/paper/1912.12777/) (2019)
- [MgNet: A Unified Framework of Multigrid and Convolutional Neural Network](https://mlsanity.com/paper/1901.10415/) (2019)
- [IMEXnet: A Forward Stable Deep Neural Network](https://mlsanity.com/paper/1903.02639/) (2019)
- [Model Reduction with Memory and the Machine Learning of Dynamical Systems](https://mlsanity.com/paper/1808.04258/) (2018)
- [PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networks](https://mlsanity.com/paper/2307.09039/) (2023)
- [Forward Stability of ResNet and Its Variants](https://mlsanity.com/paper/1811.09885/) (2018)
- [Low-Dose CT with Deep Learning Regularization via Proximal Forward Backward Splitting](https://mlsanity.com/paper/1909.09773/) (2019)
- [Decoupled Relative Learning Rate Schedules](https://mlsanity.com/paper/2507.03526/) (2025)
- [Mathematical Modeling and Convergence Analysis of Deep Neural Networks with Dense Layer Connectivities in Deep Learning](https://mlsanity.com/paper/2510.02049/) (2025)

## Abstract (arXiv)

Deep unfolding neural networks derived from iterative optimization schemes and numerical ordinary/partial differential equations (ODEs/PDEs) have attracted much attention in data science over the last decade. Therein, numerous important network architectures were constructed from the basic forward-backward-splitting (FBS) algorithm. In this paper, we continue our research on the most basic FBS-induced network, an architecture unrolled from the original FBS algorithm by incorporating direct parameter relaxations. Following the difference/differential inclusion formulations in our previous forward system analyses, we here consider some theoretical aspects of corresponding learning problems. Under some mild assumptions, we establish a general convergence property of the training problem of the basic FBS-induced network to the learning problem of the deep-layer limit system, implying a Gamma-convergence argument showing that any cluster point of the optimal learning parameters for the network is a solution to the learning problem of the deep-layer limit system. A qualitative analysis of perturbation stabilities of these learning problems is also presented. A simple numerical experiment is conducted to validate our main general convergence result.