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

TL;DR

LVM-GP is a probabilistic PDE solver that combines latent variable modeling with Gaussian processes and neural operators to provide uncertainty-aware solutions for forward and inverse PDE problems with noisy data.

Contribution

The paper introduces a novel framework that integrates a Gaussian process-based latent variable model with neural operators, enabling uncertainty quantification and improved PDE solution accuracy.

Findings

Effective uncertainty quantification demonstrated in experiments

Competitive predictive accuracy against existing methods

Robustness to noisy data in PDE solving

Abstract

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.