[论文解读] Bayesian neural networks for weak solution of PDEs with uncertainty quantification

Xiaoxuan Zhang, Krishna Garikipati|arXiv (Cornell University)|Jan 13, 2021

Model Reduction and Neural Networks参考文献 64被引用 17

一句话总结

该论文提出了一种物理约束的贝叶斯神经网络（BNN）框架，通过基于向量化和卷积算子的实现方式，利用弱形式离散化求解偏微分方程（PDEs）。通过从离散残差构建损失函数，并采用编码器-解码器架构结合确定性与贝叶斯层，该方法实现了高通量PDE求解，并具备不确定性量化能力——同时捕捉认知不确定性与数据不确定性，且在未见边界条件和几何形状下具有泛化能力。

ABSTRACT

Solving partial differential equations (PDEs) is the canonical approach for understanding the behavior of physical systems. However, large scale solutions of PDEs using state of the art discretization techniques remains an expensive proposition. In this work, a new physics-constrained neural network (NN) approach is proposed to solve PDEs without labels, with a view to enabling high-throughput solutions in support of design and decision-making. Distinct from existing physics-informed NN approaches, where the strong form or weak form of PDEs are used to construct the loss function, we write the loss function of NNs based on the discretized residual of PDEs through an efficient, convolutional operator-based, and vectorized implementation. We explore an encoder-decoder NN structure for both deterministic and probabilistic models, with Bayesian NNs (BNNs) for the latter, which allow us to quantify both epistemic uncertainty from model parameters and aleatoric uncertainty from noise in the data. For BNNs, the discretized residual is used to construct the likelihood function. In our approach, both deterministic and probabilistic convolutional layers are used to learn the applied boundary conditions (BCs) and to detect the problem domain. As both Dirichlet and Neumann BCs are specified as inputs to NNs, a single NN can solve for similar physics, but with different BCs and on a number of problem domains. The trained surrogate PDE solvers can also make interpolating and extrapolating (to a certain extent) predictions for BCs that they were not exposed to during training. Such surrogate models are of particular importance for problems, where similar types of PDEs need to be repeatedly solved for many times with slight variations. We demonstrate the capability and performance of the proposed framework by applying it to steady-state diffusion, linear elasticity, and nonlinear elasticity.

研究动机与目标

• 开发一种高通量代理PDE求解器，与传统有限元法（FEM）相比显著降低重复模拟的计算成本。
• 通过将贝叶斯神经网络（BNNs）与物理信息学习相结合，实现在PDE求解中的不确定性量化。
• 基于离散弱形式残差构建损失函数，利用高效的卷积操作实现，避免强形式约束。
• 仅使用一个神经网络即可实现对新边界条件和问题域的泛化，无需重新训练。
• 通过提供快速、可靠且具备不确定性感知的PDE求解能力，支持工程与材料科学中的设计与决策。

提出的方法

• 该方法以PDE弱形式的离散残差为核心损失函数，其推导基于有限元组装原理。
• 采用基于卷积算子的向量化实现方式，利用二维卷积核从局部节点值中提取所有单元的残差。
• 在每个高斯积分点上，通过B算子矩阵和加权积分计算残差，再通过求和操作将其缩减为全局残差。
• 狄利克雷和诺伊曼边界条件通过输入张量编码，并通过掩码和重分配直接在解矩阵中强制施加。
• 采用编码器-解码器神经网络架构，其中确定性层用于生成确定性解，贝叶斯层用于不确定性量化。
• 对于BNN，似然函数基于离散残差构建，从而实现对网络权重的后验推断及不确定性传播。

实验结果

研究问题

• RQ1 在弱形式PDE上训练的物理约束神经网络，是否能在多种边界条件和几何形状下实现高精度与强泛化能力？
• RQ2 贝叶斯神经网络在无需标注数据的情况下，能否有效量化PDE求解中的认知不确定性与数据不确定性？
• RQ3 单个训练好的神经网络在未见的边界条件和问题域上，其插值或外推能力能达到何种程度？
• RQ4 与传统FEM或强形式PINNs相比，该方法的残差向量化卷积实现方式在效率与精度方面表现如何？
• RQ5 该框架能否推广至非线性PDE，如非线性弹性与扩散方程？

主要发现

• 所提方法仅使用弱形式残差作为损失函数，即可实现对稳态扩散、线性弹性与非线性弹性问题的精确求解。
• 模型在未见的边界条件和问题域上表现出良好泛化能力，展现出强大的外推性能。
• 贝叶斯神经网络变体成功量化了PDE求解中的认知不确定性（来自模型权重）与数据不确定性（来自数据噪声）。
• 向量化卷积实现方式实现了对所有单元残差的高效计算，无需显式循环，其精度与传统FEM组装方法相当。
• 通过训练单一代理模型，该框架支持高通量PDE求解，可快速应用于新配置。
• 该方法在复杂几何形状与非线性材料行为下仍保持高精度，已在弹性与扩散问题的基准测试中得到验证。