Multi-Fidelity Physics-Informed Neural Networks with Bayesian Uncertainty Quantification and Adaptive Residual Learning for Efficient Solution of Parametric Partial Differential Equations

TL;DR

This paper introduces MF-BPINN, a multi-fidelity physics-informed neural network framework that efficiently solves parametric PDEs by combining low- and high-fidelity data with Bayesian uncertainty quantification and adaptive residual learning.

Contribution

It proposes a hierarchical neural architecture with adaptive residuals and Bayesian methods to improve efficiency and accuracy in solving parametric PDEs.

Findings

Enhanced computational efficiency for high-fidelity PDE solutions.

Effective uncertainty quantification via Bayesian methods.

Improved accuracy through adaptive residual learning.

Abstract

Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding physical laws directly into neural network training. However, solving high-fidelity PDEs remains computationally prohibitive, particularly for parametric systems requiring multiple evaluations across varying parameter configurations. This paper presents MF-BPINN, a novel multi-fidelity framework that synergistically combines physics-informed neural networks with Bayesian uncertainty quantification and adaptive residual learning. Our approach leverages abundant low-fidelity simulations alongside sparse high-fidelity data through a hierarchical neural architecture that learns nonlinear correlations across fidelity levels. We introduce an adaptive residual network with learnable gating mechanisms that dynamically balances linear and nonlinear fidelity discrepancies. Furthermore, we develop a rigorous Bayesian framework employing Hamiltonian Monte Carlo.