# LLN: Learnable Lens Networks for Parameter-Efficient Long-Horizon Dynamical Prediction

arXiv: https://arxiv.org/abs/2609.34493 · HTML: https://mlsanity.com/paper/2609.34493/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 43 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 30 of 31)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [On Layer Normalization in the Transformer Architecture](https://mlsanity.com/paper/2002.04745/) (2020)
- [Going deeper with Image Transformers](https://mlsanity.com/paper/2103.17239/) (2021)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [High-Performance Large-Scale Image Recognition Without Normalization](https://mlsanity.com/paper/2102.06171/) (2021)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [ReZero is All You Need: Fast Convergence at Large Depth](https://mlsanity.com/paper/2003.04887/) (2020)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [Understanding the Difficulty of Training Transformers](https://mlsanity.com/paper/2004.08249/) (2020)
- [Fixup Initialization: Residual Learning Without Normalization](https://mlsanity.com/paper/1901.09321/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [DeepNet: Scaling Transformers to 1,000 Layers](https://mlsanity.com/paper/2203.00555/) (2022)
- [AntisymmetricRNN: A Dynamical System View on Recurrent Neural Networks](https://mlsanity.com/paper/1902.09689/) (2019)
- [nGPT: Normalized Transformer with Representation Learning on the Hypersphere](https://mlsanity.com/paper/2410.01131/) (2024)
- [Transformers without Normalization](https://mlsanity.com/paper/2503.10622/) (2025)
- [Depthwise Hyperparameter Transfer in Residual Networks: Dynamics and Scaling Limit](https://mlsanity.com/paper/2309.16620/) (2023)
- [Sensitivity-Constrained Fourier Neural Operators for Forward and Inverse Problems in Parametric Differential Equations](https://mlsanity.com/paper/2505.08740/) (2025)
- [Initialization Schemes for Kolmogorov-Arnold Networks: An Empirical Study](https://mlsanity.com/paper/2509.03417/) (2025)
- [KANO: Kolmogorov-Arnold Neural Operator](https://mlsanity.com/paper/2509.16825/) (2025)

## Abstract (arXiv)

Explicit residual connections of the form (x+f(x)), often combined with normalization layers, have become a standard strategy for training very deep neural networks. However, residual addition primarily provides an algebraic shortcut for gradient propagation, while leaving the evolution of feature geometry across layers largely unconstrained. We introduce Learnable Lens Networks (LLN), a physics-inspired architecture that replaces direct feature-space residual accumulation with learnable optical transport in an augmented position-angle phase space. Each layer alternates between free propagation, which provides an implicit transport path, and a learnable lens field that performs nonlinear trajectory transformation and focusing. Theoretically, we establish that LLN transport is globally invertible and volume-preserving for any differentiable lens field, with the implemented coordinate-wise Gaussian transport further satisfying symplecticity. Importantly, these structural constraints do not limit expressivity: with unrestricted embeddings and readouts, LLN retain universal approximation of continuous end-to-end maps. Experiments across diverse dynamical systems demonstrate that LLN improves long-horizon prediction while using substantially fewer parameters than same-depth comparators. Further analysis reveals stable depth-wise gradient transport and interpretable learned dynamics under the coupled propagation and refraction design.