# LVM-GP: Uncertainty-Aware PDE Solver via coupling latent variable model and Gaussian process

arXiv: https://arxiv.org/abs/2507.22493 · HTML: https://mlsanity.com/paper/2507.22493/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 48 references that are in the map.
- First posted on arXiv 30 Jul 2025.

## Builds on (most cited first, 30 of 36)

- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Predictive Uncertainty Estimation via Prior Networks](https://mlsanity.com/paper/1802.10501/) (2018)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Deep Ensembles: A Loss Landscape Perspective](https://mlsanity.com/paper/1912.02757/) (2019)
- [Deep Evidential Regression](https://mlsanity.com/paper/1910.02600/) (2019)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations](https://mlsanity.com/paper/1703.10230/) (2017)
- [Bayesian Model Selection, the Marginal Likelihood, and Generalization](https://mlsanity.com/paper/2202.11678/) (2022)
- [Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systems](https://mlsanity.com/paper/2205.08304/) (2022)
- [Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priors](https://mlsanity.com/paper/2203.03048/) (2022)
- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Machine Learning and Computational Mathematics](https://mlsanity.com/paper/2009.14596/) (2020)
- [AutoIP: A United Framework to Integrate Physics into Gaussian Processes](https://mlsanity.com/paper/2202.12316/) (2022)
- [IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learning](https://mlsanity.com/paper/2302.03271/) (2023)

## Built on by (most cited first, 1 of 1)

- [MAD-NG: Meta-Auto-Decoder Neural Galerkin Method for Solving Parametric Partial Differential Equations](https://mlsanity.com/paper/2512.21633/) (2025)

## Abstract (arXiv)

We propose a novel probabilistic framework, termed LVM-GP, for uncertainty quantification in solving forward and inverse partial differential equations (PDEs) with noisy data. The core idea is to construct a stochastic mapping from the input to a high-dimensional latent representation, enabling uncertainty-aware prediction of the solution. Specifically, the architecture consists of a confidence-aware encoder and a probabilistic decoder. The encoder implements a high-dimensional latent variable model based on a Gaussian process (LVM-GP), where the latent representation is constructed by interpolating between a learnable deterministic feature and a Gaussian process prior, with the interpolation strength adaptively controlled by a confidence function learned from data. The decoder defines a conditional Gaussian distribution over the solution field, where the mean is predicted by a neural operator applied to the latent representation, allowing the model to learn flexible function-to-function mapping. Moreover, physical laws are enforced as soft constraints in the loss function to ensure consistency with the underlying PDE structure. Compared to existing approaches such as Bayesian physics-informed neural networks (B-PINNs) and deep ensembles, the proposed framework can efficiently capture functional dependencies via merging a latent Gaussian process and neural operator, resulting in competitive predictive accuracy and robust uncertainty quantification. Numerical experiments demonstrate the effectiveness and reliability of the method.