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

2025-07-30

AI Summary

This work addresses forward and inverse partial differential equation (PDE) problems with noisy observational data by proposing an uncertainty-aware probabilistic solving framework. Methodologically, it integrates a latent-variable Gaussian process prior with neural operators within a confidence-aware encoder–probabilistic decoder architecture: the encoder learns a data-driven distribution over latent variables, while the decoder employs a neural operator to output a conditional Gaussian distribution over the solution field; a learnable confidence function dynamically fuses deterministic features with stochastic priors, and physical laws are incorporated via soft constraints. Compared to Bayesian physics-informed neural networks (B-PINNs) and deep ensembles, the framework achieves significantly improved predictive accuracy and better-calibrated uncertainty quantification across multiple numerical benchmarks—while maintaining high modeling efficiency and robustness to data noise.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

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.

Problem

• Uncertainty quantification in solving noisy PDEs
• Stochastic mapping for uncertainty-aware predictions
• Enforcing physical laws as soft constraints

Innovation

• Couples latent variable model with Gaussian process
• Uses confidence-aware encoder and probabilistic decoder
• Enforces physical laws as soft constraints