# Branched Latent Neural Maps

arXiv: https://arxiv.org/abs/2308.02599 · HTML: https://mlsanity.com/paper/2308.02599/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 73 references that are in the map.
- First posted on arXiv 4 Aug 2023.

## Builds on (most cited first, 25 of 25)

- [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)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Liquid Time-constant Networks](https://mlsanity.com/paper/2006.04439/) (2020)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [Multiscale Simulations of Complex Systems by Learning their Effective Dynamics](https://mlsanity.com/paper/2006.13431/) (2020)
- [POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition](https://mlsanity.com/paper/2101.11845/) (2021)
- [β-Variational autoencoders and transformers for reduced-order modelling of fluid flows](https://mlsanity.com/paper/2304.03571/) (2023)
- [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022)
- [Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems](https://mlsanity.com/paper/2204.09157/) (2022)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications](https://mlsanity.com/paper/2004.04276/) (2020)
- [A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEs](https://mlsanity.com/paper/2110.13361/) (2021)
- [Deep learning-based reduced order models in cardiac electrophysiology](https://mlsanity.com/paper/2006.03040/) (2020)
- [Learning Reduced-Order Models for Cardiovascular Simulations with Graph Neural Networks](https://mlsanity.com/paper/2303.07310/) (2023)
- [Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models](https://mlsanity.com/paper/2201.10215/) (2022)
- [A comprehensive and biophysically detailed computational model of the whole human heart electromechanics](https://mlsanity.com/paper/2207.12460/) (2022)
- [Modeling cardiac muscle fibers in ventricular and atrial electrophysiology simulations](https://mlsanity.com/paper/2101.10960/) (2021)
- [Adaptive learning of effective dynamics: Adaptive real-time, online modeling for complex systems](https://mlsanity.com/paper/2304.01732/) (2023)
- [Latent Dynamics Networks (LDNets): learning the intrinsic dynamics of spatio-temporal processes](https://mlsanity.com/paper/2305.00094/) (2023)
- [The importance of the pericardium for cardiac biomechanics: From physiology to computational modeling](https://mlsanity.com/paper/1810.05451/) (2018)

## Built on by (most cited first, 3 of 3)

- [Liquid Fourier Latent Dynamics Networks for fast GPU-based numerical simulations in computational cardiology](https://mlsanity.com/paper/2408.09818/) (2024)
- [Full-field surrogate modeling of cardiac function encoding geometric variability](https://mlsanity.com/paper/2504.20479/) (2025)
- [An Optimization Framework to Personalize Passive Cardiac Mechanics](https://mlsanity.com/paper/2404.02807/) (2024)

## Abstract (arXiv)

We introduce Branched Latent Neural Maps (BLNMs) to learn finite dimensional input-output maps encoding complex physical processes. A BLNM is defined by a simple and compact feedforward partially-connected neural network that structurally disentangles inputs with different intrinsic roles, such as the time variable from model parameters of a differential equation, while transferring them into a generic field of interest. BLNMs leverage latent outputs to enhance the learned dynamics and break the curse of dimensionality by showing excellent generalization properties with small training datasets and short training times on a single processor. Indeed, their generalization error remains comparable regardless of the adopted discretization during the testing phase. Moreover, the partial connections significantly reduce the number of tunable parameters. We show the capabilities of BLNMs in a challenging test case involving electrophysiology simulations in a biventricular cardiac model of a pediatric patient with hypoplastic left heart syndrome. The model includes a 1D Purkinje network for fast conduction and a 3D heart-torso geometry. Specifically, we trained BLNMs on 150 in silico generated 12-lead electrocardiograms (ECGs) while spanning 7 model parameters, covering cell-scale and organ-level. Although the 12-lead ECGs manifest very fast dynamics with sharp gradients, after automatic hyperparameter tuning the optimal BLNM, trained in less than 3 hours on a single CPU, retains just 7 hidden layers and 19 neurons per layer. The resulting mean square error is on the order of 10-4 on a test dataset comprised of 50 electrophysiology simulations. In the online phase, the BLNM allows for 5000x faster real-time simulations of cardiac electrophysiology on a single core standard computer and can be used to solve inverse problems via global optimization in a few seconds of computational time.