Neural network surrogates with uncertainty quantification for inverse problems in partial differential equations

2026-06-18

AI Summary

This work addresses the computational challenges of Bayesian inference in inverse problems governed by partial differential equations, where expensive forward models and high-dimensional parameter spaces hinder scalability. To this end, the authors propose DeepGaLA, a deep neural network surrogate that embeds differential equation constraints and provides principled uncertainty quantification. By design, DeepGaLA avoids overconfidence under limited data and integrates seamlessly into Bayesian inference frameworks, enabling efficient posterior approximation via delayed-acceptance Markov chain Monte Carlo. Experimental results demonstrate that DeepGaLA achieves accuracy comparable to Gaussian process surrogates while offering substantially improved computational efficiency in high-dimensional settings.

Abstract

Inverse problems for differential equations arise throughout science and engineering, where one seeks to infer unknown model parameters from noisy or incomplete observations. Traditional numerical methods for these problems are often computationally expensive, particularly in Bayesian settings where evaluating the likelihood becomes costly for complex forward models and high-dimensional parameter spaces. To address this challenge, we introduce DeepGaLA, a neural-network surrogate for differential equation solvers that provides uncertainty-aware predictions, reducing overconfident inference when training data are limited. To evaluate the fidelity of the surrogate-induced posterior approximations in practice, we show that a short run of delayed-acceptance Markov chain Monte Carlo can serve as an effective diagnostic. Across a range of numerical experiments, DeepGaLA delivers forward-model approximations with accuracy comparable to established Gaussian-process surrogates, while better maintaining efficiency as parameter dimension grows. Moreover, it can incorporate differential-equation constraints, including in nonlinear settings. Overall, these results indicate that uncertainty-quantified neural surrogates can enable scalable and reliable Bayesian inference for inverse problems in complex systems.

Problem

• inverse problems
• partial differential equations
• Bayesian inference
• uncertainty quantification
• computational efficiency

Innovation

• neural surrogate
• uncertainty quantification
• inverse problems
• partial differential equations
• Bayesian inference