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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 ↗First posted 1 Feb 2026cs.LG cs.NA math.NAcited by 0

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Abstract (arXiv)

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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References · 18

  1. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  2. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  3. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 470
  4. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  5. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  6. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 173
  7. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  8. 2021Efficient training of physics-informed neural networks via importance samplingcited by 109
  9. 2021Improved architectures and training algorithms for deep operator networkscited by 85
  10. 2023PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEscited by 84
  11. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  12. 2021Multi-fidelity regression using artificial neural networks: efficient approximation of parameter-dependent output quantitiescited by 31
6 more
  1. 2023Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operatorscited by 28
  2. 2023Stacked networks improve physics-informed training: applications to neural networks and deep operator networkscited by 21
  3. 2024Multi-Fidelity Residual Neural Processes for Scalable Surrogate Modelingcited by 13
  4. 2023Residual Multi-Fidelity Neural Network Computingcited by 5
  5. 2025Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundlescited by 3
  6. 2025Uncertainty Quantification for Physics-Informed Neural Networks with Extended Fiducial Inferencecited by 2

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