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.

翻译：我们提出分支潜在神经映射（BLNMs）以学习编码复杂物理过程的有限维输入-输出映射。BLNM由一个简单紧凑的前馈部分连接神经网络定义，该网络通过结构化解耦具有不同内在角色的输入（如微分方程的时间变量与模型参数），并将其转化为通用目标场。BLNM利用潜在输出增强学习动力学，通过在小训练集和单处理器短时训练下展现卓越的泛化特性，突破维度灾难。具体而言，其泛化误差在测试阶段无论采用何种离散化方式均保持可比性。此外，部分连接显著减少了可调参数数量。我们通过涉及一例左心发育不全综合征儿科患者双心室心脏模型电生理模拟的挑战性测试案例，展示了BLNM的能力。该模型包含用于快速传导的一维浦肯野网络和三维心脏-躯干几何结构。具体而言，我们在跨越7个模型参数（涵盖细胞尺度与器官水平）的150个计算机生成12导联心电图（ECGs）上训练BLNM。尽管12导联ECG表现出具有陡峭梯度的极快动力学特征，但在自动超参数调优后，最优BLNM在单CPU上训练时长不足3小时，仅保留7个隐藏层、每层19个神经元。在包含50个电生理模拟的测试数据集上，所得均方误差量级为$10^{-4}$。在线阶段，BLNM在单核标准计算机上实现心脏电生理5000倍加速实时模拟，并可通过全局优化在数秒计算时间内求解逆问题。