[论文解读] Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets

Sifan Wang, Hanwen Wang+1 more|arXiv (Cornell University)|Mar 19, 2021

Model Reduction and Neural Networks被引用 61

一句话总结

该论文通过将物理信息正则化扩展到 DeepONet，以学习参数化偏微分方程（PDE）的解算子，从而实现数据高效的训练和符合 PDE 的预测，包括在仅有初始/边界条件的情况下实现零样本数据无学习，并以数量级提升 PDE 求解速度。

ABSTRACT

Deep operator networks (DeepONets) are receiving increased attention thanks to their demonstrated capability to approximate nonlinear operators between infinite-dimensional Banach spaces. However, despite their remarkable early promise, they typically require large training data-sets consisting of paired input-output observations which may be expensive to obtain, while their predictions may not be consistent with the underlying physical principles that generated the observed data. In this work, we propose a novel model class coined as physics-informed DeepONets, which introduces an effective regularization mechanism for biasing the outputs of DeepOnet models towards ensuring physical consistency. This is accomplished by leveraging automatic differentiation to impose the underlying physical laws via soft penalty constraints during model training. We demonstrate that this simple, yet remarkably effective extension can not only yield a significant improvement in the predictive accuracy of DeepOnets, but also greatly reduce the need for large training data-sets. To this end, a remarkable observation is that physics-informed DeepONets are capable of solving parametric partial differential equations (PDEs) without any paired input-output observations, except for a set of given initial or boundary conditions. We illustrate the effectiveness of the proposed framework through a series of comprehensive numerical studies across various types of PDEs. Strikingly, a trained physics informed DeepOnet model can predict the solution of $\mathcal{O}(10^3)$ time-dependent PDEs in a fraction of a second -- up to three orders of magnitude faster compared a conventional PDE solver. The data and code accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/Physics-informed-DeepONets}.

研究动机与目标

• motivation 学习参数化 PDE 的解算子并解决标准 DeepONet 在数据效率和物理一致性方面的不足。
• 引入一个物理信息正则化框架，在训练过程中通过自动微分强制满足控制 PDE。
• 展示在有限或无成对数据下，算子学习的数 据效率和零样本学习能力。
• 展示在扩散-反应、常微分方程（ODE）以及高频输入情形下的性能，包括分布外测试。

提出的方法

• 将解映射 G 表示为未堆叠的 DeepONet，分支网络用于输入，主干网络用于坐标，通过点积融合。
• 使用复合损失 L = L_operator + L_physics 进行训练，其中 L_operator 将 DeepONet 的输出与真实解（若可用）对齐。
• 将 L_physics 表述为偏微分方程残差惩罚，利用自动微分强制执行控制规律。
• 使用高斯随机场输入产生多样化的参数化 PDE 场景，评估数据效率和泛化能力。
• 在具有挑战性的情形下，采用傅里叶特征嵌入以捕捉输入和输出中的高频内容。
• 展示数据无关学习，即模型在不成对输出的情况下仍然遵循初始/边界条件和 PDE 约束。

实验结果

研究问题

• RQ1 物理信息正则化能否在没有大量成对数据的情况下引导 DeepONet 尊重潜在 PDE？
• RQ2 物理信息训练如何影响参数化 PDE 运营的数 据效率、泛化和外推能力？
• RQ3 哪些架构（如傅里叶特征）能够从不规则输入学习高频解成分？
• RQ4 与传统求解器相比，这类模型在多大程度上加速 PDE 求解？
• RQ5 对于具有现实初始/边界条件的复杂参数化 PDE，零样本或数据稀缺训练是否可行？

主要发现

• 物理信息 DeepONet 相较于传统 DeepONet 显著提高预测精度并降低对数据的需求。
• 该物理信息框架能够在没有成对输入-输出数据的情况下学习算子，依赖初始/边界条件与 PDE 残差。
• 在适当的架构（如傅里叶特征）下，该方法能够准确处理高频输入函数和不规则输入。
• 预测解和残差与真实值对齐良好，零样本或数据稀缺训练也能获得具竞争力的精度。
• 物理信息 DeepONet 可以比常规求解器快几个数量级地预测 PDE 解。