We introduce Branched Latent Neural Operators (BLNOs) to learn input-output maps encoding complex physical processes. A BLNO 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. BLNOs leverage interpretable 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, in place of a fully-connected structure, significantly reduce the number of tunable parameters. We show the capabilities of BLNOs in a challenging test case involving biophysically detailed electrophysiology simulations in a biventricular cardiac model of a pediatric patient with hypoplastic left heart syndrome. The model includes a purkinje network for fast conduction and a heart-torso geometry. Specifically, we trained BLNOs on 150 in silico generated 12-lead electrocardiograms (ECGs) while spanning 7 model parameters, covering cell-scale, organ-level and electrical dyssynchrony. Although the 12-lead ECGs manifest very fast dynamics with sharp gradients, after automatic hyperparameter tuning the optimal BLNO, trained in less than 3 hours on a single CPU, retains just 7 hidden layers and 19 neurons per layer. The mean square error is on the order of $10^{-4}$ on an independent test dataset comprised of 50 additional electrophysiology simulations. This paper provides a novel computational tool to build reliable and efficient reduced-order models for digital twinning in engineering applications.

翻译：我们提出分支潜变量神经算子（BLNOs）来学习编码复杂物理过程的输入-输出映射。BLNO由一个简单紧凑的前馈部分连接神经网络定义，该网络从结构上解耦具有不同内在角色的输入（例如微分方程中的时间变量与模型参数），同时将其转化为通用的目标场。BLNO通过利用可解释的潜变量输出增强学习动态，并通过在单个处理器上使用小训练数据集和短训练时间展现出色的泛化能力，从而打破维度诅咒。事实上，无论测试阶段采用何种离散化方式，其泛化误差始终保持可比性。此外，部分连接结构替代全连接结构显著减少了可调参数数量。我们在一项具有挑战性的测试案例中展示了BLNO的能力，该案例涉及儿科左心发育不良综合征患者的双心室心脏模型中生物物理详细电生理学模拟，模型包含用于快速传导的浦肯野网络和心脏-躯干几何结构。具体地，我们在覆盖7个模型参数（涵盖细胞尺度、器官水平和电不同步性）的150个计算机生成的12导联心电图（ECG）上训练BLNO。尽管12导联ECG表现出具有尖锐梯度的极快动态特性，但在自动超参数调优后，最优BLNO（在单个CPU上训练时间少于3小时）仅保留7个隐藏层和每层19个神经元。在包含50个额外电生理学模拟的独立测试数据集上，均方误差达到$10^{-4}$量级。本论文提供了一种新颖的计算工具，用于构建工程应用中数字孪生的可靠高效降阶模型。