We propose Deep Neural Coregionalization, a scalable framework for uncertainty-aware multivariate geostatistics. DNC models multivariate spatial effects through spatially varying latent factors and loadings, assigning deep Gaussian process (DGP) priors to both the factors and the entries of the loading matrix. This joint construction learns shared latent spatial structure together with response-specific, location-dependent mixing weights, enabling flexible nonlinear and space-dependent associations within and across variables. A key contribution is a variational formulation that makes the DGP to deep neural network (DNN) correspondence explicit: maximizing the DGP evidence lower bound (ELBO) is equivalent to training DNNs with weight decay and Monte Carlo (MC) dropout. This yields fast mini-batch stochastic optimization without Markov Chain Monte Carlo (MCMC), while providing principled uncertainty quantification through MC-dropout forward passes as approximate posterior draws, producing calibrated credible surfaces for prediction and spatial effect estimation. Across simulations, DNC is competitive with existing spatial factor models, particularly under strong nonstationarity and complex cross-dependence, while delivering substantial computational gains. In a multivariate environmental case study, DNC captures spatially varying cross-variable interactions, produces interpretable maps of multivariate outcomes, and scales uncertainty quantification to large datasets with orders-of-magnitude reductions in runtime.

翻译：我们提出深度神经协同区域化，一种可扩展的不确定性感知多元地统计学框架。DNC通过空间变化的潜在因子和载荷对多元空间效应进行建模，对因子和载荷矩阵的条目均赋予深度高斯过程先验。这种联合构建能够同时学习共享的潜在空间结构以及响应特定、位置依赖的混合权重，从而在变量内部和变量之间实现灵活的非线性及空间依赖关联。一个关键贡献是提出了一个变分公式，使DGP与深度神经网络的对应关系显式化：最大化DGP证据下界等价于训练带有权重衰减和蒙特卡洛丢弃法的DNN。这实现了无需马尔可夫链蒙特卡洛的快速小批量随机优化，同时通过MC-dropout前向传播作为近似后验采样，提供原则性的不确定性量化，从而为预测和空间效应估计生成经过校准的可信曲面。在多项模拟中，DNC与现有空间因子模型相比具有竞争力，尤其在强非平稳性和复杂交叉依赖条件下表现突出，同时带来显著的计算效率提升。在一个多元环境案例研究中，DNC成功捕捉了空间变化的跨变量相互作用，生成了可解释的多元结果地图，并将不确定性量化扩展到大型数据集，运行时间实现了数量级的缩减。