# E-LMC: Extended Linear Model of Coregionalization for Spatial Field Prediction

arXiv: https://arxiv.org/abs/2203.00525 · HTML: https://mlsanity.com/paper/2203.00525/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 40 references that are in the map.
- First posted on arXiv 1 Mar 2022.

## Builds on (most cited first, 7 of 7)

- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [Data-dependent Initializations of Convolutional Neural Networks](https://mlsanity.com/paper/1511.06856/) (2015)
- [Surpassing Human-Level Face Verification Performance on LFW with GaussianFace](https://mlsanity.com/paper/1404.3840/) (2014)
- [PCA-Initialized Deep Neural Networks Applied To Document Image Analysis](https://mlsanity.com/paper/1702.00177/) (2017)
- [Large-Margin Multiple Kernel Learning for Discriminative Features Selection and Representation Learning](https://mlsanity.com/paper/1903.03364/) (2019)

## Built on by (most cited first, 2 of 2)

- [Scalable and adaptive prediction bands with kernel sum-of-squares](https://mlsanity.com/paper/2505.21039/) (2025)
- [Multi-output Gaussian process prediction of physical fields under linear equality constraints](https://mlsanity.com/paper/2608.25709/) (2026)

## Abstract (arXiv)

Physical simulations based on partial differential equations typically generate spatial fields results, which are utilized to calculate specific properties of a system for engineering design and optimization. Due to the intensive computational burden of the simulations, a surrogate model mapping the low-dimensional inputs to the spatial fields are commonly built based on a relatively small dataset. To resolve the challenge of predicting the whole spatial field, the popular linear model of coregionalization (LMC) can disentangle complicated correlations within the high-dimensional spatial field outputs and deliver accurate predictions. However, LMC fails if the spatial field cannot be well approximated by a linear combination of base functions with latent processes. In this paper, we present the Extended Linear Model of Coregionalization (E-LMC) by introducing an invertible neural network to linearize the highly complex and nonlinear spatial fields so that the LMC can easily generalize to nonlinear problems while preserving the traceability and scalability. Several real-world applications demonstrate that E-LMC can exploit spatial correlations effectively, showing a maximum improvement of about 40% over the original LMC and outperforming the other state-of-the-art spatial field models.