{"ID":2888717,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.22493","arxiv_id":"2507.22493","title":"LVM-GP: Uncertainty-Aware PDE Solver via coupling latent variable model and Gaussian process","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.","short_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 predic...","url_abs":"https://arxiv.org/abs/2507.22493","url_pdf":"https://arxiv.org/pdf/2507.22493v1","authors":"[\"Xiaodong Feng\",\"Ling Guo\",\"Xiaoliang Wan\",\"Hao Wu\",\"Tao Zhou\",\"Wenwen Zhou\"]","published":"2025-07-30T09:00:39Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.AI\",\"cs.LG\"]","methods":"[]","has_code":false}