用多精度神经网络加速参数化偏微分方程求解，同时量化不确定性。
Multi-Fidelity Physics-Informed Neural Networks with Bayesian Uncertainty Quantification and Adaptive Residual Learning for Efficient Solution of Parametric Partial Differential Equations
- 融合低/高精度模拟数据，通过自适应残差网络学习不同精度间非线性关联。
- 采用贝叶斯框架与哈密顿蒙特卡洛，实现对预测不确定性的严格量化。
- 适合需要高效且可信求解的科学计算场景，如气候建模与工程仿真。
物理信息神经网络（PINNs）通过将物理定律嵌入训练过程，成为求解偏微分方程（PDEs）的强大方法。然而，求解高精度PDEs仍面临计算瓶颈，尤其在需对多种参数配置进行多次评估的参数化系统中。本文提出MF-BPINN，一种结合物理信息神经网络、贝叶斯不确定性量化与自适应残差学习的新型多精度框架。该方法通过分层神经架构，利用大量低精度仿真数据与少量高精度数据，学习跨精度层级的非线性相关性。我们引入带有可学习门控机制的自适应残差网络，动态平衡线性与非线性精度差异。此外，开发了基于哈密顿蒙特卡洛的严谨贝叶斯框架，用于不确定性量化。
原文摘要 · Abstract (English)
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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