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

arXiv:2602.01176 · cs.LG, cs.NA, math.NA, physics.comp-ph · Submitted 2026-02-01 · Read on arXiv

cs.LG, cs.NA, math.NA, physics.comp-ph

Submitted: 2026-02-01

Updated: 2026-09-06

Comments: The author has identified fundamental analytical errors in the formulation of the adaptive PDE residual loss and Bayesian uncertainty estimation pipeline that invalidate the numerical convergence results and core findings. The author is withdrawing the manuscript

License: http://creativecommons.org/licenses/by/4.0/

The gist: 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.

Terminology

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.

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