[论文解读] Correcting model misspecification in physics-informed neural networks (PINNs)

Zongren Zou, Xuhui Meng+1 more|arXiv (Cornell University)|Oct 16, 2023

Model Reduction and Neural Networks被引用 4

一句话总结

该论文提出一种方法，通过引入一个额外的深度神经网络（DNN）来学习有偏差的物理模型与噪声或稀疏观测数据之间的差异，从而校正物理信息神经网络（PINNs）中的模型误设问题。该方法在物理理解不完整复杂系统中提升了预测精度，并通过B-PINNs或集成PINNs实现不确定性量化，显著降低了反应-扩散问题与非牛顿流体问题中的误差。

ABSTRACT

Data-driven discovery of governing equations in computational science has emerged as a new paradigm for obtaining accurate physical models and as a possible alternative to theoretical derivations. The recently developed physics-informed neural networks (PINNs) have also been employed to learn governing equations given data across diverse scientific disciplines. Despite the effectiveness of PINNs for discovering governing equations, the physical models encoded in PINNs may be misspecified in complex systems as some of the physical processes may not be fully understood, leading to the poor accuracy of PINN predictions. In this work, we present a general approach to correct the misspecified physical models in PINNs for discovering governing equations, given some sparse and/or noisy data. Specifically, we first encode the assumed physical models, which may be misspecified, then employ other deep neural networks (DNNs) to model the discrepancy between the imperfect models and the observational data. Due to the expressivity of DNNs, the proposed method is capable of reducing the computational errors caused by the model misspecification and thus enables the applications of PINNs in complex systems where the physical processes are not exactly known. Furthermore, we utilize the Bayesian PINNs (B-PINNs) and/or ensemble PINNs to quantify uncertainties arising from noisy and/or gappy data in the discovered governing equations. A series of numerical examples including non-Newtonian channel and cavity flows demonstrate that the added DNNs are capable of correcting the model misspecification in PINNs and thus reduce the discrepancy between the physical models and the observational data. We envision that the proposed approach will extend the applications of PINNs for discovering governing equations in problems where the physico-chemical or biological processes are not well understood.

研究动机与目标

• 解决在物理过程理解不充分时PINNs中模型误设的挑战。
• 通过使用辅助DNN学习不完善物理模型与观测数据之间的差异，提升PINNs的预测精度。
• 量化由噪声和数据缺失引起的发现控制方程中的不确定性。
• 通过与符号回归结合，实现显式控制方程的发现。
• 将PINNs的应用范围扩展至物理化学或生物过程尚未完全明确的复杂系统。

提出的方法

• 使用自动微分将假设的、可能存在误设的物理模型编码进PINNs。
• 引入一个额外的DNN来建模不完善模型与观测数据之间的差异。
• 通过联合训练PINN与差异DNN，校正因模型误设导致的误差。
• 利用贝叶斯PINNs（B-PINNs）或集成PINNs，量化由于数据噪声或不完整导致的预测不确定性。
• 将训练后的模型与符号回归结合，提取显式的控制方程。
• 将该框架应用于已知存在模型误设的反应-扩散系统与非牛顿流体问题。

实验结果

研究问题

• RQ1 当底层物理模型不完整或不准确时，深度神经网络能否有效校正PINNs中的模型误设？
• RQ2 在已知存在模型误差的系统中，引入差异DNN在多大程度上提升了PINN预测的准确性？
• RQ3 B-PINNs或集成PINNs在噪声或数据缺失条件下，能否为发现的物理模型提供可靠的不确定性量化？
• RQ4 所提出的方法能否与符号回归无缝集成，以推导出可解释的控制方程？
• RQ5 在粘度被误设的复杂物理系统（如非牛顿流体）中，该方法是否保持鲁棒性与准确性？

主要发现

• 在非牛顿腔流问题中，速度预测的相对L2误差分别从7.79%降低至1.33%，从11.28%降低至2.29%。
• 力分量（fx和fy）的误差分别从3.9153×10⁻⁴和3.1466×10⁻⁴降低至7.8389×10⁻⁷和8.2283×10⁻⁷，显示出显著改进。
• 差异DNN有效捕捉了未建模的物理机制，使得即使初始物理模型存在误设，也能实现准确预测。
• B-PINNs与集成PINNs提供了合理的不确定性边界，反映了由数据噪声与缺失引起的模型不确定性。
• 该方法成功实现了在训练后与符号回归的结合，提取出显式的控制方程，增强了模型的可解释性。
• 该方法在多种系统中表现出稳健性能，包括反应-扩散系统与非牛顿流体，其中物理模型被故意误设。