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

• URL: http://arxiv.org/abs/2602.01176v1
• Date: Sun, 01 Feb 2026 12:01:31 GMT
• Title: Multi-Fidelity Physics-Informed Neural Networks with Bayesian Uncertainty Quantification and Adaptive Residual Learning for Efficient Solution of Parametric Partial Differential Equations
• Authors: Olaf Yunus Laitinen Imanov,
• Abstract summary: MF-BPINN is a novel multi-fidelity framework for solving partial differential equations.<n>We introduce an adaptive residual network with learnable gating mechanisms.<n>We also develop a rigorous Bayesian framework employing Hamiltonian Monte Carlo.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Forked Physics Informed Neural Networks for Coupled Systems of Differential equations [1.7158443622461192]
  We propose a Forked PINN framework designed for coupled systems of differential equations (DEs)<n>We demonstrate the effectiveness of FPINN in simulating non-Markovian open quantum dynamics governed by coupled DEs.<n>For the spin-boson and XXZ models, FPINN accurately captures non-Markovian features, such as quantum coherence revival and information backflow.
  arXiv  Detail & Related papers  (2026-02-16T08:32:59Z)
• Neural Port-Hamiltonian Differential Algebraic Equations for Compositional Learning of Electrical Networks [21.117540483724603]
  We develop compositional learning algorithms for coupled dynamical systems, with a particular focus on electrical networks.<n>We introduce neural port-Hamiltonian differential algebraic equations (N-PHDAEs), which use neural networks to parameterize unknown terms in both the differential and algebraic components of a port-Hamiltonian DAE.<n>We show that the proposed N-PHDAE model achieves an order of magnitude improvement in prediction accuracy and constraint satisfaction when compared to a baseline N-ODE over long prediction time horizons.
  arXiv  Detail & Related papers  (2024-12-15T15:13:11Z)
• HyResPINNs: Hybrid Residual Networks for Adaptive Neural and RBF Integration in Solving PDEs [22.689531776611084]
  We introduce HyResPINNs, a novel class of PINNs featuring adaptive hybrid residual blocks that integrate standard neural networks and radial basis function networks.<n>A distinguishing characteristic of HyResPINNs is the use of adaptive combination parameters within each residual block, enabling dynamic weighting of the neural and RBF network contributions.
  arXiv  Detail & Related papers  (2024-10-04T16:21:14Z)
• Unified theoretical guarantees for stability, consistency, and convergence in neural PDE solvers from non-IID data to physics-informed networks [0.0]
  We establish a unified theoretical framework addressing the stability, consistency, and convergence of neural networks under realistic training conditions.<n>For standard supervised learning with dependent data, we derive uniform stability bounds for gradient-based methods.<n>In federated learning with heterogeneous data, we quantify model inconsistency via curvature-aware aggregation and information-theoretic divergence.
  arXiv  Detail & Related papers  (2024-09-08T08:48:42Z)
• Enhancing lattice kinetic schemes for fluid dynamics with Lattice-Equivariant Neural Networks [79.16635054977068]
  We present a new class of equivariant neural networks, dubbed Lattice-Equivariant Neural Networks (LENNs) Our approach develops within a recently introduced framework aimed at learning neural network-based surrogate models Lattice Boltzmann collision operators. Our work opens towards practical utilization of machine learning-augmented Lattice Boltzmann CFD in real-world simulations.
  arXiv  Detail & Related papers  (2024-05-22T17:23:15Z)
• Efficient and Flexible Neural Network Training through Layer-wise Feedback Propagation [49.44309457870649]
  Layer-wise Feedback feedback (LFP) is a novel training principle for neural network-like predictors.<n>LFP decomposes a reward to individual neurons based on their respective contributions.<n>Our method then implements a greedy reinforcing approach helpful parts of the network and weakening harmful ones.
  arXiv  Detail & Related papers  (2023-08-23T10:48:28Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• Physics-aware deep learning framework for linear elasticity [0.0]
  The paper presents an efficient and robust data-driven deep learning (DL) computational framework for linear continuum elasticity problems. For an accurate representation of the field variables, a multi-objective loss function is proposed. Several benchmark problems including the Airimaty solution to elasticity and the Kirchhoff-Love plate problem are solved.
  arXiv  Detail & Related papers  (2023-02-19T20:33:32Z)
• ConCerNet: A Contrastive Learning Based Framework for Automated Conservation Law Discovery and Trustworthy Dynamical System Prediction [82.81767856234956]
  This paper proposes a new learning framework named ConCerNet to improve the trustworthiness of the DNN based dynamics modeling. We show that our method consistently outperforms the baseline neural networks in both coordinate error and conservation metrics.
  arXiv  Detail & Related papers  (2023-02-11T21:07:30Z)
• Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations [44.89798007370551]
  This work proposes Neural Galerkin schemes based on deep learning that generate training data with active learning for numerically solving high-dimensional partial differential equations. Neural Galerkin schemes build on the Dirac-Frenkel variational principle to train networks by minimizing the residual sequentially over time. Our finding is that the active form of gathering training data of the proposed Neural Galerkin schemes is key for numerically realizing the expressive power of networks in high dimensions.
  arXiv  Detail & Related papers  (2022-03-02T19:09:52Z)
• Large-scale Neural Solvers for Partial Differential Equations [48.7576911714538]
  Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. Recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. We examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) We discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.
  arXiv  Detail & Related papers  (2020-09-08T13:26:51Z)
• Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism [1.6114012813668932]
  Physics-Informed Neural Networks (PINNs) have emerged as a promising application of deep neural networks to the numerical solution of nonlinear partial differential equations (PDEs) We propose a fundamentally new way to train PINNs adaptively, where the adaptation weights are fully trainable and applied to each training point individually. In numerical experiments with several linear and nonlinear benchmark problems, the SA-PINN outperformed other state-of-the-art PINN algorithm in L2 error.
  arXiv  Detail & Related papers  (2020-09-07T04:07:52Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.