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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

Olaf Yunus Laitinen Imanov

TL;DR

MF-BPINN addresses the computational bottleneck of parametric PDE solving by integrating multi-fidelity learning with physics-informed constraints and Bayesian uncertainty quantification. The framework learns a four-network hierarchy including a low-fidelity predictor, linear and nonlinear correction modules, and a learnable gating mechanism to adapt fidelity contributions across $(x,t,bmu)$. A structured residual formulation separates linear and nonlinear discrepancies, while HMC-based posterior sampling yields calibrated predictive intervals and uncertainty decomposition. Experimental results on Burgers, heat conduction, and Navier–Stokes demonstrate substantial speedups (~7x) and a sixfold improvement in HF sample efficiency, with well-calibrated 95% intervals and robust generalization to new parameter configurations. These findings suggest MF-BPINN provides a practical, uncertainty-aware surrogate framework for efficient parametric PDE solving in engineering contexts.

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.

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

TL;DR

MF-BPINN addresses the computational bottleneck of parametric PDE solving by integrating multi-fidelity learning with physics-informed constraints and Bayesian uncertainty quantification. The framework learns a four-network hierarchy including a low-fidelity predictor, linear and nonlinear correction modules, and a learnable gating mechanism to adapt fidelity contributions across . A structured residual formulation separates linear and nonlinear discrepancies, while HMC-based posterior sampling yields calibrated predictive intervals and uncertainty decomposition. Experimental results on Burgers, heat conduction, and Navier–Stokes demonstrate substantial speedups (~7x) and a sixfold improvement in HF sample efficiency, with well-calibrated 95% intervals and robust generalization to new parameter configurations. These findings suggest MF-BPINN provides a practical, uncertainty-aware surrogate framework for efficient parametric PDE solving in engineering contexts.

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.
Paper Structure (33 sections, 17 equations, 4 figures, 6 tables)

This paper contains 33 sections, 17 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Physics-Informed Neural Networks
  4. Multi-Fidelity Modeling
  5. Bayesian Uncertainty Quantification
  6. Methodology
  7. Problem Formulation
  8. Multi-Fidelity Architecture
  9. Modeling Fidelity Discrepancy as a Structured Residual
  10. Learnable Gating as a Mixture-of-Experts Mechanism
  11. Coupling Strategy and Training Stability
  12. Connection to Composite and Residual Multi-Fidelity Networks
  13. Physics-Informed Loss Function
  14. Loss Terms, Collocation Strategy, and Weighting
  15. Bayesian Uncertainty Quantification
  16. ...and 18 more sections

Figures (4)

  • Figure 1: Computational cost distribution comparing (a) MF-BPINN (6.9 hours total) with balanced stages and (b) PINN-HF (48.7 hours) dominated by training. The 7.1$\times$ speedup stems from leveraging abundant low-fidelity data.
  • Figure 2: Learned gating parameter $\alpha$ distribution for (a) Burgers equation, (b) heat conduction, and (c) Navier--Stokes.
  • Figure 3: Out-of-distribution parameter generalization for (a) Burgers equation (viscosity $\nu$) and (b) heat conduction (conductivity $k$). Shaded regions indicate training ranges.
  • Figure 4: Solution quality vs. computational cost trade-off for 2D heat conduction. MF-BPINN achieves best accuracy-cost balance: 5.8$\times$ faster than FEM-256 with comparable accuracy to Spectral method. Pareto frontier (dashed line) connects non-dominated methods.